Polyanin A.D. Handbook of integral equations / A.D.Polyanin, A.V. Manzhirov. - 2nd ed. - Boca Raton: Chapman & Hall/CRC, 2008. - xxxiii, 1108 p.: ill. - (Handbooks of mathematical equations). - Bibliogr.: p.1071-1108. - ISBN 978-1-58488-507-8  Место хранения:   02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents

Authors ...................................................... xxix Preface ...................................................... xxxi Some Remarks and Notation .................................. xххііі Part I. Exact Solutions of Integral Equations 1 Linear Equations of the First Kind with Variable Limit of Integration ....................................... 3 1.1 Equations Whose Kernels Contain Power-Law Functions ........ 4 1.1-1 Kernels Linear in the Arguments x and t ............. 4 1.1-2 Kernels Quadratic in the Arguments x and t .......... 4 1.1-3 Kernels Cubic in the Arguments x and t .............. 5 1.1-4 Kernels Containing Higher-Order Polynomials in x and t ............................................... 6 1.1-5 Kernels Containing Rational Functions ............... 7 1.1-6 Kernels Containing Square Roots ..................... 9 1.1-7 Kernels Containing Arbitrary Powers ................ 12 1.1-8 Two-Dimensional Equation of the Abel Type .......... 15 1.2 Equations Whose Kernels Contain Exponential Functions ..... 15 1.2-1 Kernels Containing Exponential Functions ........... 15 1.2-2 Kernels Containing Power-Law and Exponential Functions .......................................... 19 1.3 Equations Whose Kernels Contain Hyperbolic Functions ...... 22 1.3-1 Kernels Containing Hyperbolic Cosine ............... 22 1.3-2 Kernels Containing Hyperbolic Sine ................. 28 1.3-3 Kernels Containing Hyperbolic Tangent .............. 36 1.3-4 Kernels Containing Hyperbolic Cotangent ............ 38 1.3-5 Kernels Containing Combinations of Hyperbolic Functions .......................................... 39 1.4 Equations Whose Kernels Contain Logarithmic Functions ..... 42 1.4-1 Kernels Containing Logarithmic Functions ........... 42 1.4-2 Kernels Containing Power-Law and Logarithmic Functions .......................................... 45 1.5 Equations Whose Kernels Contain Trigonometric Functions ... 46 1.5-1 Kernels Containing Cosine .......................... 46 1.5-2 Kernels Containing Sine ............................ 52 1.5-3 Kernels Containing Tangent ......................... 60 1.5-4 Kernels Containing Cotangent ....................... 62 1.5-5 Kernels Containing Combinations of Trigonometric Functions .......................................... 63 1.6 Equations Whose Kernels Contain Inverse Trigonometric Functions ................................................. 66 1.6-1 Kernels Containing Arccosine ....................... 66 1.6-2 Kernels Containing Arcsine ......................... 68 1.6-3 Kernels Containing Arctangent ...................... 70 1.6-4 Kernels Containing Arccotangent .................... 71 1.7 Equations Whose Kernels Contain Combinations of Elementary Functions ...................................... 73 1.7-1 Kernels Containing Exponential and Hyperbolic Functions .......................................... 73 1.7-2 Kernels Containing Exponential and Logarithmic Functions .......................................... 77 1.7-3 Kernels Containing Exponential and Trigonometric Functions .......................................... 78 1.7-4 Kernels Containing Hyperbolic and Logarithmic Functions .......................................... 83 1.7-5 Kernels Containing Hyperbolic and Trigonometric Functions .......................................... 84 1.7-6 Kernels Containing Logarithmic and Trigonometric Functions .......................................... 85 1.8 Equations Whose Kernels Contain Special Functions ......... 86 1.8-1 Kernels Containing Error Function or Exponential Integral ........................................... 86 1.8-2 Kernels Containing Sine and Cosine Integrals ....... 87 1.8-3 Kernels Containing Fresnel Integrals ............... 87 1.8-4 Kernels Containing Incomplete Gamma Functions ...... 88 1.8-5 Kernels Containing Bessel Functions ................ 88 1.8-6 Kernels Containing Modified Bessel Functions ....... 97 1.8-7 Kernels Containing Legendre Polynomials ........... 105 1.8-8 Kernels Containing Associated Legendre Functions .. 107 1.8-9 Kernels Containing Confluent Hypergeometric Functions ......................................... 107 1.8-10 Kernels Containing Hermite Polynomials ............ 108 1.8-11 Kernels Containing Chebyshev Polynomials .......... 109 1.8-12 Kernels Containing Laguerre Polynomials ........... 110 1.8-13 Kernels Containing Jacobi Theta Functions ......... 110 1.8-14 Kernels Containing Other Special Functions ........ 111 1.9 Equations Whose Kernels Contain Arbitrary Functions ...... 111 1.9-1 Equations with Degenerate Kernel: K(x,t) = g1(x)h1(t) + g2(x)h2(t) ............................ 111 1.9-2 Equations with Difference Kernel: K(x,t) = K(x-t) ............................................ 114 1.9-3 Other Equations ................................... 122 1.10 Some Formulas and Transformations ........................ 124 2 Linear Equations of the Second Kind with Variable Limit of Integration ........................................... 127 2.1 Equations Whose Kernels Contain Power-Law Functions ...... 127 2.1-1 Kernels Linear in the Arguments x and t ........... 127 2.1-2 Kernels Quadratic in the Arguments x and t ........ 129 2.1-3 Kernels Cubic in the Arguments x and t ............ 132 2.1-4 Kernels Containing Higher-Order Polynomials in x and t ............................................. 133 2.1-5 Kernels Containing Rational Functions ............. 136 2.1-6 Kernels Containing Square Roots and Fractional Powers ............................................ 138 2.1-7 Kernels Containing Arbitrary Powers ............... 139 2.2 Equations Whose Kernels Contain Exponential Functions .... 144 2.2-1 Kernels Containing Exponential Functions .......... 144 2.2-2 Kernels Containing Power-Law and Exponential Functions ......................................... 151 2.3 Equations Whose Kernels Contain Hyperbolic Functions ..... 154 2.3-1 Kernels Containing Hyperbolic Cosine .............. 154 2.3-2 Kernels Containing Hyperbolic Sine ................ 156 2.3-3 Kernels Containing Hyperbolic Tangent ............. 161 2.3-4 Kernels Containing Hyperbolic Cotangent ........... 162 2.3-5 Kernels Containing Combinations of Hyperbolic Functions ......................................... 164 2.4 Equations Whose Kernels Contain Logarithmic Functions .... 164 2.4-1 Kernels Containing Logarithmic Functions .......... 164 2.4-2 Kernels Containing Power-Law and Logarithmic Functions ......................................... 165 2.5 Equations Whose Kernels Contain Trigonometric Functions ................................................ 166 2.5-1 Kernels Containing Cosine ......................... 166 2.5-2 Kernels Containing Sine ........................... 169 2.5-3 Kernels Containing Tangent ........................ 174 2.5-4 Kernels Containing Cotangent ...................... 175 2.5-5 Kernels Containing Combinations of Trigonometric Functions ......................................... 176 2.6 Equations Whose Kernels Contain Inverse Trigonometric Functions ................................................ 176 2.6-1 Kernels Containing Arccosine ...................... 176 2.6-2 Kernels Containing Arcsine ........................ 177 2.6-3 Kernels Containing Arctangent ..................... 178 2.6-4 Kernels Containing Arccotangent ................... 178 2.7 Equations Whose Kernels Contain Combinations of Elementary Functions ..................................... 179 2.7-1 Kernels Containing Exponential and Hyperbolic Functions ......................................... 179 2.7-2 Kernels Containing Exponential and Logarithmic Functions ......................................... 180 2.7-3 Kernels Containing Exponential and Trigonometric Functions ......................................... 181 2.7-4 Kernels Containing Hyperbolic and Logarithmic Functions ......................................... 185 2.7-5 Kernels Containing Hyperbolic and Trigonometric Functions ......................................... 186 2.7-6 Kernels Containing Logarithmic and Trigonometric Functions ......................................... 187 2.8 Equations Whose Kernels Contain Special Functions ........ 187 2.8-1 Kernels Containing Bessel Functions ............... 187 2.8-2 Kernels Containing Modified Bessel Functions ...... 189 2.9 Equations Whose Kernels Contain Arbitrary Functions ...... 191 2.9-1 Equations with Degenerate Kernel: K(x,t) = g1(x)h1(t) + ... + gn(x)hn(t) ................... 191 2.9-2 Equations with Difference Kernel: K(x,t) = K(x-t) ............................................ 203 2.9-3 Other Equations ................................... 212 2.10 Some Formulas and Transformations ........................ 215 3 Linear Equations of the First Kind with Constant Limits of Integration ........................................... 217 3.1 Equations Whose Kernels Contain Power-Law Functions ...... 217 3.1-1 Kernels Linear in the Arguments x and t ........... 217 3.1-2 Kernels Quadratic in the Arguments x and t ........ 219 3.1-3 Kernels Containing Integer Powers of x and t or Rational Functions ................................ 220 3.1-4 Kernels Containing Square Roots ................... 222 3.1-5 Kernels Containing Arbitrary Powers ............... 223 3.1-6 Equations Containing the Unknown Function of a Complicated Argument ............................ 227 3.1-7 Singular Equations ................................ 228 3.2 Equations Whose Kernels Contain Exponential Functions .... 231 3.2-1 Kernels Containing Exponential Functions of the Form ℮λӀx-tӀ ........................................ 231 3.2-2 Kernels Containing Exponential Functions of the Forms ℮λx and ℮μt ................................ 234 3.2-3 Kernels Containing Exponential Functions of the Form ℮λxt ......................................... 234 3.2-4 Kernels Containing Power-Law and Exponential Functions ......................................... 236 3.2-5 Kernels Containing Exponential Functions of the Form ℮λ(x±t)2 ...................................... 236 3.2-6 Other Kernels ..................................... 237 3.3 Equations Whose Kernels Contain Hyperbolic Functions ..... 238 3.3-1 Kernels Containing Hyperbolic Cosine .............. 238 3.3-2 Kernels Containing Hyperbolic Sine ................ 238 3.3-3 Kernels Containing Hyperbolic Tangent ............. 241 3.3-4 Kernels Containing Hyperbolic Cotangent ........... 242 3.4 Equations Whose Kernels Contain Logarithmic Functions .... 242 3.4-1 Kernels Containing Logarithmic Functions .......... 242 3.4-2 Kernels Containing Power-Law and Logarithmic Functions ......................................... 244 3.4-3 Equation Containing the Unknown Function of a Complicated Argument .............................. 246 3.5 Equations Whose Kernels Contain Trigonometric Functions .. 246 3.5-1 Kernels Containing Cosine ......................... 246 3.5-2 Kernels Containing Sine ........................... 247 3.5-3 Kernels Containing Tangent ........................ 251 3.5-4 Kernels Containing Cotangent ...................... 252 3.5-5 Kernels Containing a Combination of Trigonometric Functions ........................... 252 3.5-6 Equations Containing the Unknown Function of a Complicated Argument ............................ 254 3.5-7 Singular Equations ................................ 255 3.6 Equations Whose Kernels Contain Combinations of Elementary Functions ..................................... 255 3.6-1 Kernels Containing Hyperbolic and Logarithmic Functions ......................................... 255 3.6-2 Kernels Containing Logarithmic and Trigonometric Functions ......................................... 256 3.6-3 Kernels Containing Combinations of Exponential and Other Elementary Functions .................... 257 3.7 Equations Whose Kernels Contain Special Functions ........ 258 3.7-1 Kernels Containing Error Function, Exponential Integral or Logarithmic Integral .................. 258 3.7-2 Kernels Containing Sine Integrals, Cosine Integrals, or Fresnel Integrals ................... 258 3.7-3 Kernels Containing Gamma Functions ................ 260 3.7-4 Kernels Containing Incomplete Gamma Functions ..... 260 3.7-5 Kernels Containing Bessel Functions of the First Kind .............................................. 261 3.7-6 Kernels Containing Bessel Functions of the Second Kind ....................................... 264 3.7-7 Kernels Containing Combinations of the Bessel Functions ......................................... 265 3.7-8 Kernels Containing Modified Bessel Functions of the First Kind .................................... 266 3.7-9 Kernels Containing Modified Bessel Functions of the Second Kind ................................... 266 3.7-10 Kernels Containing a Combination of Bessel and Modified Bessel Functions ......................... 269 3.7-11 Kernels Containing Legendre Functions ............. 270 3.7-12 Kernels Containing Associated Legendre Functions .. 271 3.7-13 Kernels Containing Kummer Confluent Hypergeometric Functions .......................... 272 3.7-14 Kernels Containing Tricomi Confluent Hypergeometric Functions .......................... 274 3.7-15 Kernels Containing Whittaker Confluent Hypergeometric Functions .......................... 274 3.7-16 Kernels Containing Gauss Hypergeometric Functions ......................................... 276 3.7-17 Kernels Containing Parabolic Cylinder Functions ... 276 3.7-18 Kernels Containing Other Special Functions ........ 277 3.8 Equations Whose Kernels Contain Arbitrary Functions ...... 278 3.8-1 Equations with Degenerate Kernel .................. 278 3.8-2 Equations Containing Modulus ...................... 279 3.8-3 Equations with Difference Kernel: K(x,t) = K(x-t) ............................................ 284 3.8-4 Other Equations of the Form ∫αβ K(x,t)y(t)dt = F(x) .............................................. 285 3.8-5 Equations of the Form ∫αβK(x,t)y(...)dt = F(x) .............................................. 289 3.9 Dual Integral Equations of the First Kind ................ 295 3.9-1 Kernels Containing Trigonometric Functions ........ 295 3.9-2 Kernels Containing Bessel Functions of the First Kind .............................................. 297 3.9-3 Kernels Containing Bessel Functions of the Second Kind ....................................... 299 3.9-4 Kernels Containing Legendre Spherical Functions of the First Kind, i2 = -1 ........................ 299 4 Linear Equations of the Second Kind with Constant Limits of Integration .................................... 301 4.1 Equations Whose Kernels Contain Power-Law Functions ...... 301 4.1-1 Kernels Linear in the Arguments x and t ........... 301 4.1-2 Kernels Quadratic in the Arguments x and t ........ 304 4.1-3 Kernels Cubic in the Arguments x and t ............ 307 4.1-4 Kernels Containing Higher-Order Polynomials in x and t ............................................. 311 4.1-5 Kernels Containing Rational Functions ............. 314 4.1-6 Kernels Containing Arbitrary Powers ............... 317 4.1-7 Singular Equations ................................ 319 4.2 Equations Whose Kernels Contain Exponential Functions .... 320 4.2-1 Kernels Containing Exponential Functions .......... 320 4.2-2 Kernels Containing Power-Law and Exponential Functions ......................................... 326 4.3 Equations Whose Kernels Contain Hyperbolic Functions ..... 327 4.3-1 Kernels Containing Hyperbolic Cosine .............. 327 4.3-2 Kernels Containing Hyperbolic Sine ................ 329 4.3-3 Kernels Containing Hyperbolic Tangent ............. 332 4.3-4 Kernels Containing Hyperbolic Cotangent ........... 333 4.3-5 Kernels Containing Combination of Hyperbolic Functions ......................................... 334 4.4 Equations Whose Kernels Contain Logarithmic Functions .... 334 4.4-1 Kernels Containing Logarithmic Functions .......... 334 4.4-2 Kernels Containing Power-Law and Logarithmic Functions ......................................... 335 4.5 Equations Whose Kernels Contain Trigonometric Functions .. 335 4.5-1 Kernels Containing Cosine ......................... 335 4.5-2 Kernels Containing Sine ........................... 337 4.5-3 Kernels Containing Tangent ........................ 342 4.5-4 Kernels Containing Cotangent ...................... 343 4.5-5 Kernels Containing Combinations of Trigonometric Functions ......................................... 344 4.5-6 Singular Equation ................................. 344 4.6 Equations Whose Kernels Contain Inverse Trigonometric Functions ................................................ 344 4.6-1 Kernels Containing Arccosine ...................... 344 4.6-2 Kernels Containing Arcsine ........................ 345 4.6-3 Kernels Containing Arctangent ..................... 346 4.6-4 Kernels Containing Arccotangent ................... 347 4.7 Equations Whose Kernels Contain Combinations of Elementary Functions ..................................... 348 4.7-1 Kernels Containing Exponential and Hyperbolic Functions ......................................... 348 4.7-2 Kernels Containing Exponential and Logarithmic Functions ......................................... 349 4.7-3 Kernels Containing Exponential and Trigonometric Functions ......................................... 349 4.7-4 Kernels Containing Hyperbolic and Logarithmic Functions ......................................... 351 4.7-5 Kernels Containing Hyperbolic and Trigonometric Functions ......................................... 352 4.7-6 Kernels Containing Logarithmic and Trigonometric Functions ......................................... 353 4.8 Equations Whose Kernels Contain Special Functions ........ 353 4.8-1 Kernels Containing Bessel Functions ............... 353 4.8-2 Kernels Containing Modified Bessel Functions ...... 355 4.9 Equations Whose Kernels Contain Arbitrary Functions ...... 357 4.9-1 Equations with Degenerate Kernel: K(x,t) = g1(x)h1(t) + ... + gn(x)hn(t) ...................... 357 4.9-2 Equations with Difference Kernel: K(x,t) = K(x-t) ............................................ 372 4.9-3 Other Equations of the Form y(x) + ∫αbK(x,t)y(t) dt = F(x) ......................................... 374 4.9-4 Equations of the Form y(x) +∫αbK(x,t)y(...)dt = F(x) .............................................. 381 4.10 Some Formulas and Transformations ........................ 390 5 Nonlinear Equations of the First Kind with Variable Limit of Integration ..................................... 393 5.1 Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters ..................................... 393 5.1-1 Equations of the Form ∫0x y(t)y(x-t)dt = ƒ(x) ...... 393 5.1-2 Equations of the Form ∫0xK(x,t)y(t)y(x-t)dt = ƒ(x) .............................................. 395 5.1-3 Equations of the Form ∫0xy(t)y(...)dt = ƒ(x) ....... 396 5.2 Equations with Quadratic Nonlinearity That Contain Arbitrary Functions ...................................... 397 5.2-1 Equations of the Form ∫αxK(x,t)[Ay(t)+By2(t)]dt = ƒ(x) ............................................ 397 5.2-2 Equations of the Form ∫αxK(x,t)y(t)y(αx+bt)dt = ƒ(x) .............................................. 398 5.3 Equations with Nonlinearity of General Form .............. 399 5.3-1 Equations of the Form ∫αxK(x,t)ƒ(t,y(t))dt = g(x) .............................................. 399 5.3-2 Other Equations ................................... 401 6 Nonlinear Equations of the Second Kind with Variable Limit of Integration ..................................... 403 6.1 Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters ..................................... 403 6.1-1 Equations of the Form y(x) + ∫αxK(x,t)y2(t)dt = Fix) .............................................. 403 6.1-2 Equations of the Form y(x) + ∫αx K(x,t)y(t)y(x-t)dt = F(x) ......................... 406 6.2 Equations with Quadratic Nonlinearity That Contain Arbitrary Functions ...................................... 406 6.2-1 Equations of the Form y(x) + ∫αxK(x,t)y2(t)dt = F(x) .............................................. 406 6.2-2 Other Equations ................................... 407 6.3 Equations with Power-Law Nonlinearity .................... 408 6.3-1 Equations Containing Arbitrary Parameters ......... 408 6.3-2 Equations Containing Arbitrary Functions .......... 410 6.4 Equations with Exponential Nonlinearity .................. 411 6.4-1 Equations Containing Arbitrary Parameters ......... 411 6.4-2 Equations Containing Arbitrary Functions .......... 413 6.5 Equations with Hyperbolic Nonlinearity ................... 414 6.5-1 Integrands with Nonlinearity of the Form cosh[βy(t)] ....................................... 414 6.5-2 Integrands with Nonlinearity of the Form sinh[βy(t)] ....................................... 415 6.5-3 Integrands with Nonlinearity of the Form tanh[βy(t)] ....................................... 416 6.5-4 Integrands with Nonlinearity of the Form coth[βy(t)] ....................................... 418 6.6 Equations with Logarithmic Nonlinearity .................. 419 6.6-1 Integrands Containing Power-Law Functions of x and t ............................................. 419 6.6-2 Integrands Containing Exponential Functions of x and t ............................................. 419 6.6-3 Other Integrands .................................. 420 6.7 Equations with Trigonometric Nonlinearity ................ 420 6.7-1 Integrands with Nonlinearity of the Form cos[(βy(t)] ....................................... 420 6.7-2 Integrands with Nonlinearity of the Form sin[βy(t)] ........................................ 422 6.7-3 Integrands with Nonlinearity of the Form tan[βy(t)] ........................................ 423 6.7-4 Integrands with Nonlinearity of the Form cot[βy(t)] ........................................ 424 6.8 Equations with Nonlinearity of General Form .............. 425 6.8-1 Equations of the Form y(x) + ∫αxK(x,t)G(y(t))dt = F(x) .............................................. 425 6.8-2 Equations of the Form y(x) + ∫αxK(x-t)G(t,y(t))dt = F(x) .............................................. 428 6.8-3 Other Equations ................................... 431 7 Nonlinear Equations of the First Kind with Constant Limits of Integration .................................... 433 7.1 Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters ..................................... 433 7.1-1 Equations of the Form ∫αbK(t)y(x)y(t)dt = F(x) .... 433 7.1-2 Equations of the Form ∫αbK(t)y(t)y(xt)dt = F(x) .............................................. 435 7.1-3 Other Equations ................................... 436 7.2 Equations with Quadratic Nonlinearity That Contain Arbitrary Functions ...................................... 437 7.2-1 Equations of the Form ∫αb K(t)y(t)y(...)dt = F(x) .............................................. 437 7.2-2 Equations of the Form ∫αbK(х,t)y(t) + M(x,t)y2(t)]dt = F(x) ............................. 443 7.3 Equations with Power-Law Nonlinearity That Contain Arbitrary Functions ...................................... 444 7.3-1 Equations of the Form ∫αbK(t)yμ(x)yγ(t) dt = F(x) .............................................. 444 7.3-2 Equations of the Form ∫αbK(t)yγ(t)y(xt)dt = F(x) .............................................. 444 7.3-3 Equations of the Form ∫αbK(t)yγ(t)y(x+βt)dt = F(x) .............................................. 445 7.3-4 Equations of the Form ∫αb[K(x,t)y(t) + M(x,t)yγ(t)]dt = ƒ(x) ............................ 446 7.3-5 Other Equations ................................... 446 7.4 Equations with Nonlinearity of General Form .............. 447 7.4-1 Equations of the Form ∫αbφ(y(x))K(t,y(t))dt = F(x) .............................................. 447 7.4-2 Equations of the Form ∫αb y(xt)K (t,y(t))dt = F(x) .............................................. 447 7.4-3 Equations of the Form ∫αby(x + βt)K(t,y(t))dt = F(x) .............................................. 449 7.4-4 EquationsofmeForm ∫αbK(x,t)y(t) + φ(x)ψ(t,y(t))]dt = F(x) .......................... 450 7.4-5 Other Equations ................................... 451 8 Nonlinear Equations of the Second Kind with Constant Limits of Integration .................................... 453 8.1 Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters ..................................... 453 8.1-1 Equations of the Form y(x) + ∫αbK(x,t)y2(t)dt = F(x) .............................................. 453 8.1-2 Equations of the Form y(x) + ∫αbK(x,t)y(x)y(t) dt = F(x) ......................................... 454 8.1-3 Equations of the Form y(x) + ∫αbK(t)y(t)y(...) dt = F(x) ......................................... 455 8.2 Equations with Quadratic Nonlinearity That Contain Arbitrary Functions ...................................... 456 8.2-1 Equations of the Form y(x) +∫αbK(x,t)y2(t)dt = F(x) .............................................. 456 8.2-2 Equations of the Form y(x) + ∫αb∑Knm(x,t)yn(x)ym(t)dt = F(x), n+m≤2 ............ 457 8.2-3 Equations of the Form y(x) + ∫αbK(t)y(t)y(...) dt = F(x) ......................................... 460 8.3 Equations with Power-Law Nonlinearity .................... 464 8.3-1 Equations of the Form y(x) + ∫αbK(x,t)yβ(t) dt = F(x) ......................................... 464 8.3-2 Other Equations ................................... 465 8.4 Equations with Exponential Nonlinearity .................. 467 8.4-1 Integrands with Nonlinearity of the Form exp[βy(t)] ....................................... 467 8.4-2 Other Integrands .................................. 468 8.5 Equations with Hyperbolic Nonlinearity ................... 468 8.5-1 Integrands with Nonlinearity of the Form cosh[βy(t)] ...................................... 468 8.5-2 Integrands with Nonlinearity of the Form sinh[βy(t)] ....................................... 469 8.5-3 Integrands with Nonlinearity of the Form tanh[βy(t)] ....................................... 469 8.5-4 Integrands with Nonlinearity of the Form coth[βy(t)] ....................................... 470 8.5-5 Other Integrands .................................. 471 8.6 Equations with Logarithmic Nonlinearity .................. 472 8.6-1 Integrands with Nonlinearity of the Form In[βy(t)] ........................................ 472 8.6-2 Other Integrands .................................. 473 8.7 Equations with Trigonometric Nonlinearity ................ 473 8.7-1 Integrands with Nonlinearity of the Form cos[βy(t)] ........................................ 473 8.7-2 Integrands with Nonlinearity of the Form sin[βy(t)] ........................................ 474 8.7-3 Integrands with Nonlinearity of the Form tan[βy(t)] ........................................ 475 8.7-4 Integrands with Nonlinearity of the Form cot[βy(t)] ........................................ 475 8.7-5 Other Integrands .................................. 476 8.8 Equations with Nonlinearity of General Form .............. 477 8.8-1 Equations of the Form y(x) +∫αbK(Ӏx-tӀ)G(y(t)) dt = F(x) ......................................... 477 8.8-2 Equations of the Form y(x) + ∫αbK(x,t)G(t,y(t)) dt = F(x) ......................................... 479 8.8-3 Equations of the Form y(x) + ∫αbG(x,t,y(t))dt = F(x) .............................................. 483 8.8-4 Equations of the Form y(x) + ∫αby(xt)G(t,y(t)) dt = F(x) ......................................... 485 8.8-5 Equations of the Form y(x) + ∫αbу(х+βt)G(t,y(t)) dt = F(x) ......................................... 487 8.8-6 Other Equations ................................... 494 Part II Methods for Solving Integral Equations 9 Main Definitions and Formulas Integral Transforms ....... 501 9.1 Some Definitions, Remarks, and Formulas .................. 501 9.1-1 Some Definitions .................................. 501 9.1-2 Structure of Solutions to Linear Integral Equations ......................................... 502 9.1-3 Integral Transforms ............................... 503 9.1-4 Residues Calculation Formulas Cauchy's Residue Theorem ........................................... 504 9.1-5 Jordan Lemma ...................................... 505 9.2 Laplace Transform ........................................ 505 9.2-1 Definition Inversion Formula ..................... 505 9.2-2 Inverse Transforms of Rational Functions .......... 506 9.2-3 Inversion of Functions with Finitely Many Singular Points ................................... 507 9.2-4 Convolution Theorem Main Properties of the Laplace Transform ................................. 507 9.2-5 Limit Theorems .................................... 507 9.2-6 Representation of Inverse Transforms as Convergent Series ................................. 509 9.2-7 Representation of Inverse Transforms as Asymptotic Expansions as x → ∞ .................... 509 9.2-8 Post-Widder Formula ............................... 510 9.3 Mellin Transform ......................................... 510 9.3-1 Definition Inversion Formula ..................... 510 9.3-2 Main Properties of the Mellin Transform ........... 511 9.3-3 Relation Among the Mellin, Laplace, and Fourier Transforms ........................................ 511 9.4 Fourier Transform ........................................ 512 9.4-1 Definition Inversion Formula ..................... 512 9.4-2 Asymmetric Form of the Transform .................. 512 9.4-3 Alternative Fourier Transform ..................... 512 9.4-4 Convolution Theorem Main Properties of the Fourier Transforms ................................ 513 9.5 Fourier Cosine and Sine Transforms ....................... 514 9.5-1 Fourier Cosine Transform .......................... 514 9.5-2 Fourier Sine Transform ............................ 514 9.6 Other Integral Transforms ................................ 515 9.6-1 Hankel Transform .................................. 515 9.6-2 Meijer Transform .................................. 516 9.6-3 Kontorovich-Lebedev Transform ..................... 516 9.6-4 y-transform ....................................... 516 9.6-5 Summary Table of Integral Transforms .............. 517 10 Methods for Solving Linear Equations of the Form ∫αx K(x,t)y(t)dt = ƒ(x) ...................................... 519 10.1 Volterra Equations of the First Kind ..................... 519 10.1-1 Equations of the First Kind Function and Kernel Classes ........................................... 519 10.1-2 Existence and Uniqueness of a Solution ............ 520 10.1-3 Some Problems Leading to Volterra Integral Equations of the First Kind ....................... 520 10.2 Equations with Degenerate Kernel: K(x,t) = g1(x)h1(t) + ... + gn(х)hn(t) ......................................... 522 10.2-1 Equations with Kernel of the Form K(x,t) = g1(x)h1(t) + g2(х)h2(t) ............................ 522 10.2-2 Equations with General Degenerate Kernel .......... 523 10.3 Reduction of Volterra Equations of the First Kind to Volterra Equations of the Second Kind .................... 524 10.3-1 First Method ...................................... 524 10.3-2 Second Method ..................................... 524 10.4 Equations with Difference Kernel: K(x,t) = K(x-t) ........ 524 10.4-1 Solution Method Based on the Laplace Transform .... 524 10.4-2 Case in Which the Transform of the Solution is a Rational Function ............................... 525 10.4-3 Convolution Representation of a Solution .......... 526 10.4-4 Application of an Auxiliary Equation .............. 527 10.4-5 Reduction to Ordinary Differential Equations ...... 527 10.4-6 Reduction of a Volterra Equation to a Wiener-Hopf Equation .......................................... 528 10.5 Method of Fractional Differentiation ..................... 529 10.5-1 Definition of Fractional Integrals ................ 529 10.5-2 Definition of Fractional Derivatives .............. 529 10.5-3 Main Properties ................................... 530 10.5-4 Solution of the Generalized Abel Equation ......... 531 10.5-5 Erdelyi-Kober Operators ........................... 532 10.6 Equations with Weakly Singular Kernel .................... 532 10.6-1 Method of Transformation of the Kernel ............ 532 10.6-2 Kernel with Logarithmic Singularity ............... 533 10.7 Method of Quadratures ............................... 534 10.7-1 Quadrature Formulas ............................... 534 10.7-2 General Scheme of the Method ...................... 535 10.7-3 Algorithm Based on the Trapezoidal Rule ........... 536 10.7-4 Algorithm for an Equation with Degenerate Kernel .. 536 10.8 Equations with Infinite Integration Limit ................ 537 10.8-1 Equation of the First Kind with Variable Lower Limit of Integration .............................. 537 10.8-2 Reduction to a Wiener-Hopf Equation of the First Kind .............................................. 538 11 Methods for Solving Linear Equations of the Form y(x) - ∫αbK(x,t)y(t)dt = ƒ(x) ................................... 539 11.1 Volterra Integral Equations of the Second Kind ...... 539 11.1-1 Preliminary Remarks Equations for the Resolvent ... 539 11.1-2 Relationship Between Solutions of Some Integral Equations ......................................... 540 11.2 Equations with Degenerate Kernel: K(x,t) = g1(x)h1(t) + ... + gn(x)hn(t) ......................................... 540 11.2-1 Equations with Kernel of the Form K(x,t) = φ(x) + ψ(х)(х-t) .................................. 540 11.2-2 Equations with Kernel of the Form K(x,t) = φ(t) + ψ(t)(t-x) .................................. 541 11.2-3 Equations with Kernel of the Form K(x,t) = ∑nm=1 φm(x)(x-t)m-1 ................................. 542 11.2-4 Equations with Kernel of the Form K(x,t) = ∑nm=1 φm(t)(t-x)m-1 ................................. 543 11.2-5 Equations with Degenerate Kernel of the General Form .............................................. 543 11.3 Equations with Difference Kernel: K(x,t) = K(x-1) ........ 544 11.3-1 Solution Method Based on the Laplace Transform .... 544 11.3-2 Method Based on the Solution of an Auxiliary Equation .......................................... 546 11.3-3 Reduction to Ordinary Differential Equations ...... 547 11.3-4 Reduction to a Wiener-Hopf Equation of the Second Kind .............................................. 547 11.3-5 Method of Fractional Integration for the Generalized Abel Equation ......................... 548 11.3-6 Systems of Volterra Integral Equations ............ 549 11.4 Operator Methods for Solving Linear Integral Equations ... 549 11.4-1 Application of a Solution of a "Truncated" Equation of the First Kind ........................ 549 11.4-2 Application of the Auxiliary Equation of the Second Kind ....................................... 551 11.4-3 Method for Solving "Quadratic" Operator Equations . 552 11.4-4 Solution of Operator Equations of Polynomial Form .............................................. 553 11.4-5 Some Generalizations .............................. 554 11.5 Construction of Solutions of Integral Equations with Special Right-Hand Side .................................. 555 11.5-1 General Scheme .................................... 555 11.5-2 Generating Function of Exponential Form ........... 555 11.5-3 Power-Law Generating Function ..................... 557 11.5-4 Generating Function Containing Sines and Cosines .. 558 11.6 Method of Model Solutions ................................ 559 11.6-1 Preliminary Remarks ............................... 559 11.6-2 Description of the Method ......................... 560 11.6-3 Model Solution in the Case of an Exponential Right-Hand Side ................................... 561 11.6-4 Model Solution in the Case of a Power-Law Right-Hand Side ................................... 562 11.6-5 Model Solution in the Case of a Sine-Shaped Right-Hand Side ................................... 562 11.6-6 Model Solution in the Case of a Cosine-Shaped Right-Hand Side ................................... 563 11.6-7 Some Generalizations .............................. 563 11.7 Method of Differentiation for Integral Equations ......... 564 11.7-1 Equations with Kernel Containing a Sum of Exponential Functions ............................. 564 11.7-2 Equations with Kernel Containing a Sum of Hyperbolic Functions .............................. 564 11.7-3 Equations with Kernel Containing a Sum of Trigonometric Functions ........................... 564 11.7-4 Equations Whose Kernels Contain Combinations of Various Functions ................................. 565 11.8 Reduction of Volterra Equations of the Second Kind to Volterra Equations of the First Kind ..................... 565 11.8-1 First Method ...................................... 565 11.8-2 Second Method ..................................... 566 11.9 Successive Approximation Method .......................... 566 11.9-1 General Scheme .................................... 566 11.9-2 Formula for the Resolvent ......................... 567 11.10 Method of Quadratures ................................... 568 11.10-1 General Scheme of the Method ..................... 568 11.10-2 Application of the Trapezoidal Rule .............. 568 11.10-3 Case of a Degenerate Kernel ...................... 569 11.11 Equations with Infinite Integration Limit ............... 569 11.11-1 Equation of the Second Kind with Variable Lower Integration Limit ................................ 570 11.11-2 Reduction to a Wiener-Hopf Equation of the Second Kind ...................................... 571 12 Methods for Solving Linear Equations of the Form ∫αbK(x,t)y(t)dt = ƒ(x) ................................... 573 12.1 Some Definition and Remarks .............................. 573 12.1-1 Fredholm Integral Equations of the First Kind ..... 573 12.1-2 Integral Equations of the First Kind with Weak Singularity ....................................... 574 12.1-3 Integral Equations of Convolution Type ............ 574 12.1-4 Dual Integral Equations of the First Kind ......... 575 12.1-5 Some Problems Leading to Integral Equations of the First Kind .................................... 575 12.2 Integral Equations of the First Kind with Symmetric Kernel ................................................... 577 12.2-1 Solution of an Integral Equation in Terms of Series in Eigenfunctions of Its Kernel ............ 577 12.2-2 Method of Successive Approximations ............... 579 12.3 Integral Equations of the First Kind with Nonsymmetric Kernel ................................................... 580 12.3-1 Representation of a Solution in the Form of Series General Description ........................ 580 12.3-2 Special Case of a Kernel That is a Generating Function .......................................... 580 12.3-3 Special Case of the Right-Hand Side Represented in Terms of Orthogonal Functions .................. 582 12.3-4 General Case Galerkin's Method .................... 582 12.3-5 Utilization of the Schmidt Kernels for the Construction of Solutions of Equations ............ 582 12.4 Method of Differentiation for Integral Equations ......... 583 12.4-1 Equations with Modulus ............................ 583 12.4-2 Other Equations Some Generalizations .............. 585 12.5 Method of Integral Transforms ............................ 586 12.5-1 Equation with Difference Kernel on the Entire Axis .............................................. 586 12.5-2 Equations with Kernel K(x,t) = K(x/t) on the Semiaxis .......................................... 587 12.5-3 Equation with Kernel K(x,t) = K(xt) and Some Generalizations ................................... 587 12.6 Krein's Method and Some Other Exact Methods for Integral Equations of Special Types ............................... 588 12.6-1 Krein's Method for an Equation with Difference Kernel with a Weak Singularity .................... 588 12.6-2 Kernel is the Sum of a Nondegenerate Kernel and an Arbitrary Degenerate Kernel ....................... 589 12.6-3 Reduction of Integral Equations of the First Kind to Equations of the Second Kind .............. 591 12.7 Riemann Problem for the Real Axis ........................ 592 12.7-1 Relationships Between the Fourier Integral and the Cauchy Type Integral .......................... 592 12.7-2 One-Sided Fourier Integrals ....................... 593 12.7-3 Analytic Continuation Theorem and the Generalized Liouville Theorem ................................. 595 12.7-4 Riemann Boundary Value Problem .................... 595 12.7-5 Problems with Rational Coefficients ............... 601 12.7-6 Exceptional Cases The Homogeneous Problem ......... 602 12.7-7 Exceptional Cases The Nonhomogeneous Problem ...... 604 12.8 Carleman Method for Equations of the Convolution Type of the First Kind ........................................... 606 12.8-1 Wiener-HopfEquationof the First Kind .............. 606 12.8-2 Integral Equations of the First Kind with Two Kernels ........................................... 607 12.9 Dual Integral Equations of the First Kind ................ 610 12.9-1 Carleman Method for Equations with Difference Kernels ........................................... 610 12.9-2 General Scheme of Finding Solutions of Dual Integral Equations ................................ 611 12.9-3 Exact Solutions of Some Dual Equations of the First Kind ........................................ 613 12.9-4 Reduction of Dual Equations to a Fredholm Equation .......................................... 615 12.10 Asymptotic Methods for Solving Equations with Logarithmic Singularity .................................. 618 12.10-1 Preliminary Remarks .............................. 618 12.10-2 Solution for Large λ ............................. 619 12.10-3 Solution for Small λ ............................. 620 12.10-4 Integral Equation of Elasticity .................. 621 12.11 Regularization Methods .................................. 621 12.11-1 Lavrentiev Regularization Method ................. 621 12.11-2 Tikhonov Regularization Method ................... 622 12.12 Fredholm Integral Equation of the First Kind as an Ill-Posed Problem ........................................ 623 12.12-1 General Notions of Well-Posed and Ill-Posed Problems ......................................... 623 12.12-2 Integral Equation of the First Kind is an Ill-Posed Problem ................................ 624 13 Methods for Solving Linear Equations of the Form y(x) - ∫αbK(x,t)y(t)dt = ƒ(x) ................................... 625 13.1 Some Definition and Remarks .............................. 625 13.1-1 Fredholm Equations and Equations with Weak Singularity of the Second Kind .................... 625 13.1-2 Structure of the Solution ......................... 626 13.1-3 Integral Equations of Convolution Type of the Second Kind ....................................... 626 13.1-4 Dual Integral Equations of the Second Kind ........ 627 13.2 Fredholm Equations of the Second Kind with Degenerate Kernel Some Generalizations .............................. 627 13.2-1 Simplest Degenerate Kernel ........................ 627 13.2-2 Degenerate Kernel in the General Case ............. 628 13.2-3 Kernel is the Sum of a Nondegenerate Kernel and an Arbitrary Degenerate Kernel .................... 631 13.3 Solution as a Power Series in the Parameter Method of Successive Approximations ................................ 632 13.3-1 Iterated Kernels .................................. 632 13.3-2 Method of Successive Approximations ............... 633 13.3-3 Construction of the Resolvent ..................... 633 13.3-4 Orthogonal Kernels ................................ 634 13.4 Method of Fredholm Determinants .......................... 635 13.4-1 Formula for the Resolvent ......................... 635 13.4-2 Recurrent Relations ............................... 636 13.5 Fredholm Theorems and the Fredholm Alternative ........... 637 13.5-1 Fredholm Theorems ................................. 637 13.5-2 Fredholm Alternative .............................. 638 13.6 Fredholm Integral Equations of the Second Kind with Symmetric Kernel ......................................... 639 13.6-1 Characteristic Values and Eigenfunctions .......... 639 13.6-2 Bilinear Series ................................... 640 13.6-3 Hilbert-Schmidt Theorem ........................... 641 13.6-4 Bilinear Series of Iterated Kernels ............... 642 13.6-5 Solution of the Nonhomogeneous Equation ........... 642 13.6-6 Fredholm Alternative for Symmetric Equations ...... 643 13.6-7 Resolvent of a Symmetric Kernel ................... 644 13.6-8 Extremal Properties of Characteristic Values and Eigenfunctions .................................... 644 13.6-9 Kellog's Method for Finding Characteristic Values in the Case of Symmetric Kernel ................... 645 13.6-10 Trace Method for the Approximation of Characteristic Values ............................ 646 13.6-11 Integral Equations Reducible to Symmetric Equations ........................................ 647 13.6-12 Skew-Symmetric Integral Equations ................ 647 13.6-13 Remark on Nonsymmetric Kernels ................... 647 13.7 Integral Equations with Nonnegative Kernels .............. 648 13.7-1 Positive Principal Eigenvalues Generalized Jentzch Theorem ................................... 648 13.7-2 Positive Solutions of a Nonhomogeneous Integral Equation .......................................... 649 13.7-3 Estimates for the Spectral Radius ................. 649 13.7-4 Basic Definition and Theorems for Oscillating Kernels ........................................... 651 13.7-5 Stochastic Kernels ................................ 654 13.8 Operator Method for Solving Integral Equations of the Second Kind .............................................. 655 13.8-1 Simplest Scheme ................................... 655 13.8-2 Solution of Equations of the Second Kind on the Semiaxis .......................................... 655 13.9 Methods of Integral Transforms and Model Solutions ....... 656 13.9-1 Equation with Difference Kernel on the Entire Axis .............................................. 656 13.9-2 Equation with the Kernel K(x,t) = t-1Q(x/t) on the Semiaxis ...................................... 657 13.9-3 Equation with the Kernel K(x,t) = tβQ(xt) on the Semiaxis .......................................... 658 13.9-4 Method of Model Solutions for Equations on the Entire Axis ....................................... 659 13.10 Carleman Method for Integral Equations of Convolution Type of the Second Kind .................................. 660 13.10-1 Wiener-Hopf Equation of the Second Kind .......... 660 13.10-2 Integral Equation of the Second Kind with Two Kernels .......................................... 664 13.10-3 Equations of Convolution Type with Variable Integration Limit ................................ 668 13.10-4 Dual Equation of Convolution Type of the Second Kind ............................................. 670 13.11 Wiener-Hopf Method ...................................... 671 13.11-1 Some Remarks ..................................... 671 13.11-2 Homogeneous Wiener-Hopf Equation of the Second Kind ............................................. 673 13.11-3 General Scheme of the Method The Factorization Problem .......................................... 676 13.11-4 Nonhomogeneous Wiener-Hopf Equation of the Second Kind ...................................... 677 13.11-5 Exceptional Case of a Wiener-Hopf Equation of the Second Kind .................................. 678 13.12 Krein's Method for Wiener-Hopf Equations ................ 679 13.12-1 Some Remarks The Factorization Problem ........... 679 13.12-2 Solution of the Wiener-Hopf Equations of the Second Kind ...................................... 681 13.12-3 Hopf-Fock Formula ................................ 683 13.13 Methods for Solving Equations with Difference Kernels on a Finite Interval .................................... 683 13.13-1 Krein's Method ................................... 683 13.13-2 Kernels with Rational Fourier Transforms ......... 685 13.13-3 Reduction to Ordinary Differential Equations ..... 686 13.14 Method of Approximating a Kernel by a Degenerate One .... 687 13.14-1 Approximation of the Kernel ...................... 687 13.14-2 Approximate Solution ............................. 688 13.15 Bateman Method .......................................... 689 13.15-1 General Scheme of the Method ..................... 689 13.15-2 Some Special Cases ............................... 690 13.16 Collocation Method ...................................... 692 13.16-1 General Remarks .................................. 692 13.16-2 Approximate Solution ............................. 693 13.16-3 Eigenfunctions of the Equation ................... 694 13.17 Method of Least Squares ................................. 695 13.17-1 Description of the Method ........................ 695 13.17-2 Construction of Eigenfunctions ................... 696 13.18 Bubnov-Galerkin Method .................................. 697 13.18-1 Description of the Method ........................ 697 13.18-2 Characteristic Values ............................ 697 13.19 Quadrature Method ....................................... 698 13.19-1 General Scheme for Fredholm Equations of the Second Kind ...................................... 698 13.19-2 Construction of the Eigenfunctions ............... 699 13.19-3 Specific Features of the Application of Quadrature Formulas .............................. 700 13.20 Systems of Fredholm Integral Equations of the Second Kind .................................................... 701 13.20-1 Some Remarks ..................................... 701 13.20-2 Method of Reducing a System of Equations to a Single Equation ................................ 701 13.21 Regularization Method for Equations with Infinite Limits of Integration ................................... 702 13.21-1 Basic Equation and Fredholm Theorems ............. 702 13.21-2 Regularizing Operators ........................... 703 13.21-3 Regularization Method ............................ 704 14 Methods for Solving Singular Integral Equations of the First Kind ............................................... 707 14.1 Some Definitions and Remarks ............................. 707 14.1-1 Integral Equations of the First Kind with Cauchy Kernel ............................................ 707 14.1-2 Integral Equations of the First Kind with Hilbert Kernel ............................................ 707 14.2 Cauchy Type Integral ..................................... 708 14.2-1 Definition of the Cauchy Type Integral ............ 708 14.2-2 Holder Condition .................................. 709 14.2-3 Principal Value of a Singular Integral ............ 709 14.2-4 Multivalued Functions ............................. 711 14.2-5 Principal Value of a Singular Curvilinear Integral .......................................... 712 14.2-6 Poincare-Bertrand Formula ......................... 714 14.3 Riemann Boundary Value Problem ........................... 714 14.3-1 Principle of Argument The Generalized Liouville Theorem ........................................... 714 14.3-2 Hermite Interpolation Polynomial .................. 716 14.3-3 Notion of the Index ............................... 716 14.3-4 Statement of the Riemann Problem .................. 718 14.3-5 Solution of the Homogeneous Problem ............... 720 14.3-6 Solution of the Nonhomogeneous Problem ............ 721 14.3-7 Riemann Problem with Rational Coefficients ........ 723 14.3-8 Riemann Problem for a Half-Plane .................. 725 14.3-9 Exceptional Cases of the Riemann Problem .......... 727 14.3-10 Riemann Problem for a Multiply Connected Domain .. 731 14.3-11 Riemann Problem for Open Curves .................. 734 14.3-12 Riemann Problem with a Discontinuous Coefficient . 739 14.3-13 Riemann Problem in the General Case .............. 741 14.3-14 Hilbert Boundary Value Problem ................... 742 14.4 Singular Integral Equations of the First Kind ............ 743 14.4-1 Simplest Equation with Cauchy Kernel .............. 743 14.4-2 Equation with Cauchy Kernel on the Real Axis ...... 743 14.4-3 Equation of the First Kind on a Finite Interval ... 744 14.4-4 General Equation of the First Kind with Cauchy Kernel ............................................ 745 14.4-5 Equations of the First Kind with Hilbert Kernel ... 746 14.5 Multhopp-Kalandiya Method ................................ 747 14.5-1 Solution That is Unbounded at the Endpoints of the Interval ...................................... 747 14.5-2 Solution Bounded at One Endpoint of the Interval .. 749 14.5-3 Solution Bounded at Both Endpoints of the Interval .......................................... 750 14.6 Hypersingular Integral Equations ......................... 751 14.6-1 Hypersingular Integral Equations with Cauchy- and Hilbert-Type Kernels .............................. 751 14.6-2 Definition of Hypersingular Integrals ............. 751 14.6-3 Exact Solution of the Simplest Hypersingular Equation with Cauchy-Type Kernel .................. 753 14.6-4 Exact Solution of the Simplest Hypersingular Equation with Hilbert-Type Kernel ................. 754 14.6-5 Numerical Methods for Hypersingular Equations ..... 754 15 Methods for Solving Complete Singular Integral Equations ................................................ 757 15.1 Some Definitions and Remarks ............................. 757 15.1-1 Integral Equations with Cauchy Kernel ............. 757 15.1-2 Integral Equations with Hilbert Kernel ............ 759 15.1-3 Fredholm Equations of the Second Kind on a Contour ......................................... 759 15.2 Carleman Method for Characteristic Equations ............. 761 15.2-1 Characteristic Equation with Cauchy Kernel ........ 761 15.2-2 Transposed Equation of a Characteristic Equation .. 764 15.2-3 Characteristic Equation on the Real Axis .......... 765 15.2-4 Exceptional Case of a Characteristic Equation ..... 767 15.2-5 Characteristic Equation with Hilbert Kernel ....... 769 15.2-6 Tricomi Equation .................................. 769 15.3 Complete Singular Integral Equations Solvable in a Closed Form ............................................ 770 15.3-1 Closed-Form Solutions in the Case of Constant Coefficients ...................................... 770 15.3-2 Closed-Form Solutions in the General Case ......... 771 15.4 Regularization Method for Complete Singular Integral Equations ................................................ 772 15.4-1 Certain Properties of Singular Operators .......... 772 15.4-2 Regularizer ....................................... 774 15.4-3 Methods of Left and Right Regularization .......... 775 15.4-4 Problem of Equivalent Regularization .............. 776 15.4-5 Fredholm Theorems ................................. 777 15.4-6 Carleman-Vekua Approach to the Regularization ..... 778 15.4-7 Regularization in Exceptional Cases ............... 779 15.4-8 Complete Equation with Hilbert Kernel ............. 780 15.5 Analysis of Solutions Singularities for Complete Integral Equations with Generalized Cauchy Kernels ....... 783 15.5-1 Statement of the Problem and Preliminary Remarks .. 783 15.5-2 Auxiliary Results ................................. 784 15.5-3 Equations for the Exponents of Singularity of a Solution ........................................ 787 15.5-4 Analysis of Equations for Singularity Exponents ... 789 15.5-5 Application to an Equation Arising in Fracture Mechanics ......................................... 791 15.6 Direct Numerical Solution of Singular Integral Equations with Generalized Kernels ................................. 792 15.6-1 Preliminary Remarks ............................... 792 15.6-2 Quadrature Formulas for Integrals with the Jacobi Weight Function ................................... 793 15.6-3 Approximation of Solutions in Terms of a System of Orthogonal Polynomials ......................... 795 15.6-4 Some Special Functions and Their Calculations ..... 797 15.6-5 Numerical Solution of Singular Integral Equations . 799 15.6-6 Numerical Solutions of Singular Integral Equations of Bueckner Type ........................ 801 16 Methods for Solving Nonlinear Integral Equations ......... 805 16.1 Some Definitions and Remarks ............................. 805 16.1-1 Nonlinear Equations with Variable Limit of Integration (Volterra Equations) .................. 805 16.1-2 Nonlinear Equations with Constant Integration Limits (Urysohn Equations) ........................ 806 16.1-3 Some Special Features of Nonlinear Integral Equations ......................................... 807 16.2 Exact Methods for Nonlinear Equations with Variable Limit of Integration ..................................... 809 16.2-1 Method of Integral Transforms ..................... 809 16.2-2 Method of Differentiation for Nonlinear Equations with Degenerate Kernel ............................ 810 16.3 Approximate and Numerical Methods for Nonlinear Equations with Variable Limit of Integration ............. 811 16.3-1 Successive Approximation Method ................... 811 16.3-2 Newton-Kantorovich Method ......................... 813 16.3-3 Collocation Method ................................ 815 16.3-4 Quadrature Method ................................. 816 16.4 Exact Methods for Nonlinear Equations with Constant Integration Limits ....................................... 817 16.4-1 Nonlinear Equations with Degenerate Kernels ....... 817 16.4-2 Method of Integral Transforms ..................... 819 16.4-3 Method of Differentiating for Integral Equations .. 820 16.4-4 Method for Special Urysohn Equations of the First Kind .............................................. 821 16.4-5 Method for Special Urysohn Equations of the Second Kind ....................................... 822 16.4-6 Some Generalizations .............................. 824 16.5 Approximate and Numerical Methods for Nonlinear Equations with Constant Integration Limits ............... 826 16.5-1 Successive Approximation Method ................... 826 16.5-2 Newton-Kantorovich Method ......................... 827 16.5-3 Quadrature Method ................................. 829 16.5-4 Tikhonov Regularization Method .................... 829 16.6 Existence and Uniqueness Theorems for Nonlinear Equations ................................................ 830 16.6-1 Hammerstein Equations ............................. 830 16.6-2 Urysohn Equations ................................. 832 16.7 Nonlinear Equations with a Parameter: Eigenfunctions, Eigenvalues, Bifurcation Points .......................... 834 16.7-1 Eigenfunctions and Eigenvalues of Nonlinear Integral Equations ................................ 834 16.7-2 Local Solutions of a Nonlinear Integral Equation with a Parameter .................................. 835 16.7-3 Bifurcation Points of Nonlinear Integral Equations ......................................... 835 17 Methods for Solving Multidimensional Mixed Integral Equations ................................................ 839 17.1 Some Definition and Remarks .............................. 839 17.1-1 Basic Classes of Functions ........................ 839 17.1-2 Mixed Equations on a Finite Interval .............. 840 17.1-3 Mixed Equation on a Ring-Shaped (Circular) Domain . 841 17.1-4 Mixed Equations on a Closed Bounded Set ........... 842 17.2 Methods of Solution of Mixed Integral Equations on a Finite Interval ........................................ 843 17.2-1 Equation with a Hilbert-Schmidt Kernel and a Given Right-Hand Side ........................... 843 17.2-2 Equation with Hilbert-Schmidt Kernel and Auxiliary Conditions .............................. 845 17.2-3 Equation with a Schmidt Kernel and a Given Right-Hand Side on an Interval .................... 848 17.2-4 Equation with a Schmidt Kernel and Auxiliary Conditions ........................................ 851 17.3 Methods of Solving Mixed Integral Equations on a Ring-Shaped Domain ..................................... 855 17.3-1 Equation with a Hilbert-Schmidt Kernel and a Given Right-Hand Side ........................... 855 17.3-2 Equation with a Hilbert-Schmidt Kernel and Auxiliary Conditions .............................. 856 17.3-3 Equation with a Schmidt Kernel and a Given Right-Hand Side ................................... 859 17.3-4 Equation with a Schmidt Kernel and Auxiliary Conditions on Ring-Shaped Domain .................. 862 17.4 Projection Method for Solving Mixed Equations on a Bounded Set ............................................ 866 17.4-1 Mixed Operator Equation with a Given Right-Hand Side .............................................. 866 17.4-2 Mixed Operator Equations with Auxiliary Conditions ........................................ 869 17.4-3 General Projection Problem for Operator Equation .. 873 18 Application of Integral Equations for the Investigation of Differential Equations ................................ 875 18.1 Reduction of the Cauchy Problem for ODEs to Integral Equations ................................................ 875 18.1-1 Cauchy Problem for First-Order ODEs Uniqueness and Existence Theorems ............................ 875 18.1-2 Cauchy Problem for First-Order ODEs Method of Successive Approximations ......................... 876 18.1-3 Cauchy Problem for Second-Order ODEs Method of Successive Approximations ......................... 876 18.1-4 Cauchy Problem for a Special n-Order Linear ODE ... 876 18.2 Reduction of Boundary Value Problems for ODEs to Volterra Integral Equations. Calculation of Eigenvalues .. 877 18.2-1 Reduction of Differential Equations to Volterra Integral Equations ................................ 877 18.2-2 Application of Volterra Equations to the Calculation of Eigenvalues ........................ 879 18.3 Reduction of Boundary Value Problems for ODEs to Fredholm Integral Equations with the Help of the Green's Function ................................................. 881 18.3-1 Linear Ordinary Differential Equations Fundamental Solutions ............................. 881 18.3-2 Boundary Value Problems for nth Order Differential Equations Green's Function ........... 882 18.3-3 Boundary Value Problems for Second-Order Differential Equations Green's Function ........... 883 18.3-4 Nonlinear Problem of Nonisothermal Flow in Plane Channel ........................................... 884 18.4 Reduction of PDEs with Boundary Conditions of the Third Kind to Integral Equations ............................... 887 18.4-1 Usage of Particular Solutions of PDEs for the Construction of Other Solutions ................... 887 18.4-2 Mass Transfer to a Particle in Fluid Flow Complicated by a Surface Reaction ................. 888 18.4-3 Integral Equations for Surface Concentration and Diffusion Flux .................................... 890 18.4-4 Method of Numerical Integration of the Equation for Surface Concentration ......................... 891 18.5 Representation of Linear Boundary Value Problems in Terms of Potentials ...................................... 892 18.5-1 Basic Types of Potentials for the Laplace Equation and Their Properties ..................... 892 18.5-2 Integral Identities Green's Formula ............... 895 18.5-3 Reduction of Interior Dirichlet and Neumann Problems to Integral Equations .................... 895 18.5-4 Reduction of Exterior Dirichlet and Neumann Problems to Integral Equations .................... 896 18.6 Representation of Solutions of Nonlinear PDEs in Terms of Solutions of Linear Integral Equations (Inverse Scattering) .............................................. 898 18.6-1 Description of the Zakharov-Shabat Method ......... 898 18.6-2 Korteweg-de Vries Equation and Other Nonlinear Equations ......................................... 899 Supplements Supplement 1 Elementary Functions and Their Properties ........ 905 1.1 Power, Exponential, and Logarithmic Functions ............ 905 1.1-1 Properties of the Power Function .................. 905 1.1-2 Properties of the Exponential Function ............ 905 1.1-3 Properties of the Logarithmic Function ............ 906 1.2 Trigonometric Functions .................................. 907 1.2-1 Simplest Relations ................................ 907 1.2-2 Reduction Formulas ................................ 907 1.2-3 Relations Between Trigonometric Functions of Single Argument ................................... 908 1.2-4 Addition and Subtraction of Trigonometric Functions ......................................... 908 1.2-5 Products of Trigonometric Functions ............... 908 1.2-6 Powers of Trigonometric Functions ................. 908 1.2-7 Addition Formulas ................................. 909 1.2-8 Trigonometric Functions of Multiple Arguments ..... 909 1.2-9 Trigonometric Functions of Half Argument .......... 909 1.2-10 Differentiation Formulas .......................... 910 1.2-11 Integration Formulas .............................. 910 1.2-12 Expansion in Power Series ......................... 910 1.2-13 Representation in the Form of Infinite Products ... 910 1.2-14 Euler and de Moivre Formulas Relationship with Hyperbolic Functions .............................. 911 1.3 Inverse Trigonometric Functions .......................... 911 1.3-1 Definitions of Inverse Trigonometric Functions .... 911 1.3-2 Simplest Formulas ................................. 912 1.3-3 Some Properties ................................... 912 1.3-4 Relations Between Inverse Trigonometric Functions . 912 1.3-5 Addition and Subtraction of Inverse Trigonometric Functions ......................................... 912 1.3-6 Differentiation Formulas .......................... 913 1.3-7 Integration Formulas .............................. 913 1.3-8 Expansion in Power Series ......................... 913 1.4 Hyperbolic Functions ..................................... 913 1.4-1 Definitions of Hyperbolic Functions ............... 913 1.4-2 Simplest Relations ................................ 913 1.4-3 Relations Between Hyperbolic Functions of Single Argument (x≥0) .................................... 914 1.4-4 Addition and Subtraction of Hyperbolic Functions .. 914 1.4-5 Products of Hyperbolic Functions .................. 914 1.4-6 Powers of Hyperbolic Functions .................... 914 1.4-7 Addition Formulas ................................. 915 1.4-8 Hyperbolic Functions of Multiple Argument ......... 915 1.4-9 Hyperbolic Functions of Half Argument ............. 915 1.4-10 Differentiation Formulas .......................... 916 1.4-11 Integration Formulas .............................. 916 1.4-12 Expansion in Power Series ......................... 916 1.4-13 Representation in the Form of Infinite Products ... 916 1.4-14 Relationship with Trigonometric Functions ......... 916 1.5 Inverse Hyperbolic Functions ............................. 917 1.5-1 Definitions of Inverse Hyperbolic Functions ....... 917 1.5-2 Simplest Relations ................................ 917 1.5-3 Relations Between Inverse Hyperbolic Functions .... 917 1.5-4 Addition and Subtraction of Inverse Hyperbolic Functions ......................................... 917 1.5-5 Differentiation Formulas .......................... 917 1.5-6 Integration Formulas .............................. 918 1.5-7 Expansion in Power Series ......................... 918 Supplement 2 Finite Sums and Infinite Series .................. 919 2.1 Finite Numerical Sums .................................... 919 2.1-1 Progressions ...................................... 919 2.1-2 Sums of Powers of Natural Numbers Having the Form ∑km ............................................... 919 2.1-3 Alternating Sums of Powers of Natural Numbers, ∑(-l)kkm .......................................... 920 2.1-4 Other Sums Containing Integers .................... 920 2.1-5 Sums Containing Binomial Coefficients ............. 920 2.1-6 Other Numerical Sums .............................. 921 2.2 Finite Functional Sums ................................... 922 2.2-1 Sums Involving Hyperbolic Functions ............... 922 2.2-2 Sums Involving Trigonometric Functions ............ 922 2.3 Infinite Numerical Series ................................ 924 2.3-1 Progressions ...................................... 924 2.3-2 Other Numerical Series ............................ 924 2.4 Infinite Functional Series ............................... 925 2.4-1 Power Series ...................................... 925 2.4-2 Trigonometric Series in One Variable Involving Sine .............................................. 927 2.4-3 Trigonometric Series in One Variable Involving Cosine ............................................ 928 2.4-4 Trigonometric Series in Two Variables ............. 930 Supplement 3 Tables of Indefinite Integrals ................... 933 3.1 Integrals Involving Rational Functions ................... 933 3.1-1 Integrals Involving a + bx ........................ 933 3.1-2 Integrals Involving α+x and b+x ................... 933 3.1-3 Integrals Involving α2+x2 ......................... 934 3.1-4 Integrals Involving α2-x2 ......................... 935 3.1-5 Integrals Involving α3+X3 ......................... 936 3.1-6 Integrals Involving α3-x3 ......................... 936 3.1-7 Integrals Involving α4±x4 ......................... 937 3.2 Integrals Involving Irrational Functions ................. 937 3.2-1 Integrals Involving x1/2 .......................... 937 3.2-2 Integrals Involving (α+bx)p/2 ..................... 938 3.2-3 Integrals Involving (x2+α2)1/2 ..................... 938 3.2-4 Integrals Involving (x2-α2)1/2 ..................... 938 3.2-5 Integrals Involving (α2-x2)1/2 ..................... 939 3.2-6 Integrals Involving Arbitrary Powers Reduction Formulas .......................................... 939 3.3 Integrals Involving Exponential Functions ................ 940 3.4 Integrals Involving Hyperbolic Functions ................. 940 3.4-1 Integrals Involving cosh x ........................ 940 3.4-2 Integrals Involving sinh x ........................ 941 3.4-3 Integrals Involving tanh x or coth x .............. 942 3.5 Integrals Involving Logarithmic Functions ................ 943 3.6 Integrals Involving Trigonometric Functions .............. 944 3.6-1 Integrals Involving cos x (n = 1,2,...) ........... 944 3.6-2 Integrals Involving sin x (n = 1,2,...) ........... 945 3.6-3 Integrals Involving sin x and cos x ............... 947 3.6-4 Reduction Formulas ................................ 947 3.6-5 Integrals Involving tan x and cos x ............... 947 3.7 Integrals Involving Inverse Trigonometric Functions ...... 948 Supplement 4 Tables of Definite Integrals ..................... 951 4.1 Integrals Involving Power-Law Functions .................. 951 4.1-1 Integrals Over a Finite Interval .................. 951 4.1-2 Integrals Over an Infinite Interval ............... 952 4.2 Integrals Involving Exponential Functions ................ 954 4.3 Integrals Involving Hyperbolic Functions ................. 955 4.4 Integrals Involving Logarithmic Functions ................ 955 4.5 Integrals Involving Trigonometric Functions .............. 956 4.5-1 Integrals Over a Finite Interval .................. 956 4.5-2 Integrals Over an Infinite Interval ............... 957 4.6 Integrals Involving Bessel Functions ..................... 958 4.6-1 Integrals Over an Infinite Interval ............... 958 4.6-2 Other Integrals ................................... 959 Supplement 5 Tables of Laplace Transforms .................... 961 5.1 General Formulas ......................................... 961 5.2 Expressions with Power-Law Functions ..................... 963 5.3 Expressions with Exponential Functions ................... 963 5.4 Expressions with Hyperbolic Functions .................... 964 5.5 Expressions with Logarithmic Functions ................... 965 5.6 Expressions with Trigonometric Functions ................. 966 5.7 Expressions with Special Functions ....................... 967 Supplement 6 Tables of Inverse Laplace Transforms ............. 969 6.1 General Formulas ......................................... 969 6.2 Expressions with Rational Functions ...................... 971 6.3 Expressions with Square Roots ............................ 975 6.4 Expressions with Arbitrary Powers ........................ 977 6.5 Expressions with Exponential Functions ................... 978 6.6 Expressions with Hyperbolic Functions .................... 979 6.7 Expressions with Logarithmic Functions ................... 980 6.8 Expressions with Trigonometric Functions ................. 981 6.9 Expressions with Special Functions ....................... 981 Supplement 7 Tables of Fourier Cosine Transforms .............. 983 7.1 General Formulas ......................................... 983 7.2 Expressions with Power-Law Functions ..................... 983 7.3 Expressions with Exponential Functions ................... 984 7.4 Expressions with Hyperbolic Functions .................... 985 7.5 Expressions with Logarithmic Functions ................... 985 7.6 Expressions with Trigonometric Functions ................. 986 7.7 Expressions with Special Functions ....................... 987 Supplement 8 Tables of Fourier Sine Transforms ................ 989 8.1 General Formulas ......................................... 989 8.2 Expressions with Power-Law Functions ..................... 989 8.3 Expressions with Exponential Functions ................... 990 8.4 Expressions with Hyperbolic Functions .................... 991 8.5 Expressions with Logarithmic Functions ................... 992 8.6 Expressions with Trigonometric Functions ................. 992 8.7 Expressions with Special Functions ....................... 993 Supplement 9 Tables of Mellin Transforms ...................... 997 9.1 General Formulas ......................................... 997 9.2 Expressions with Power-Law Functions ..................... 998 9.3 Expressions with Exponential Functions ................... 998 9.4 Expressions with Logarithmic Functions ................... 999 9.5 Expressions with Trigonometric Functions ................. 999 9.6 Expressions with Special Functions ...................... 1000 Supplement 10 Tables of Inverse Mellin Transforms ............ 1001 10.1 Expressions with Power-Law Functions .................... 1001 10.2 Expressions with Exponential and Logarithmic Functions .. 1002 10.3 Expressions with Trigonometric Functions ................ 1003 10.4 Expressions with Special Functions ...................... 1004 Supplement 11 Special Functions and Their Properties ......... 1007 11.1 Some Coefficients, Symbols, and Numbers ................. 1007 11.1-1 Binomial Coefficients ............................ 1007 11.1-2 Pochhammer Symbol ................................ 1007 11.1-3 Bernoulli Numbers ................................ 1008 11.1-4 Euler Numbers .................................... 1008 11.2 Error Functions Exponential and Logarithmic Integrals ... 1009 11.2-1 Error Function and Complementary Error Function .. 1009 11.2-2 Exponential Integral ............................. 1010 11.2-3 Logarithmic Integral ............................. 1010 11.3 Sine Integral and Cosine Integral Fresnel Integrals ..... 1011 11.3-1 Sine Integral .................................... 1011 11.3-2 Cosine Integral .................................. 1011 11.3-3 Fresnel Integrals and Generalized Fresnel Integrals ........................................ 1012 11.4 Gamma Function, Psi Function, and Beta Function ......... 1012 11.4-1 Gamma Function ................................... 1012 11.4-2 Psi Function (Digamma Function) .................. 1013 11.4-3 Beta Function .................................... 1014 11.5 Incomplete Gamma and Beta Functions ..................... 1014 11.5-1 Incomplete Gamma Function ........................ 1014 11.5-2 Incomplete Beta Function ......................... 1015 11.6 Bessel Functions (Cylindrical Functions) ................ 1016 11.6-1 Definitions and Basic Formulas ................... 1016 11.6-2 Integral Representations and Asymptotic Expansions ....................................... 1017 11.6-3 Zeros of Bessel Functions ........................ 1019 11.6-4 Orthogonality Properties of Bessel Functions ..... 1019 11.6-5 Hankel Functions (Bessel Functions of the Third Kind) ............................................ 1020 11.7 Modified Bessel Functions ............................... 1021 11.7-1 Definitions Basic Formulas ....................... 1021 11.7-2 Integral Representations and Asymptotic Expansions ....................................... 1022 11.8 Airy Functions .......................................... 1023 11.8-1 Definition and Basic Formulas .................... 1023 11.8-2 Power Series and Asymptotic Expansions ........... 1023 11.9 Confluent Hypergeometric Functions ...................... 1024 11.9-1 Kummer and Tricomi Confluent Hypergeometric Functions ........................................ 1024 11.9-2 Integral Representations and Asymptotic Expansions ....................................... 1027 11.9-3 Whittaker Confluent Hypergeometric Functions ..... 1027 11.10 Gauss Hypergeometric Functions ......................... 1028 11.10-1 Various Representations of the Gauss Hypergeometric Function ......................... 1028 11.10-2 Basic Properties ................................ 1028 11.11 Legendre Polynomials, Legendre Functions, and Associated Legendre Functions .......................... 1030 11.11-1 Legendre Polynomials and Legendre Functions ..... 1030 11.11-2 Associated Legendre Functions with Integer Indices and Real Argument ....................... 1031 11.11-3 Associated Legendre Functions General Case ...... 1032 11.12 Parabolic Cylinder Functions ........................... 1034 11.12-1 Definitions Basic Formulas ...................... 1034 11.12-2 Integral Representations, Asymptotic Expansions, and Linear Relations ............................ 1035 11.13 Elliptic Integrals ..................................... 1035 11.13-1 Complete Elliptic Integrals ..................... 1035 11.13-2 Incomplete Elliptic Integrals (Elliptic Integrals) ...................................... 1037 11.14 Elliptic Functions ..................................... 1038 11.14-1 Jacobi Elliptic Functions ....................... 1039 11.14-2 Weierstrass Elliptic Function ................... 1042 11.15 Jacobi Theta Functions ................................. 1043 11.15-1 Series Representation of the Jacobi Theta Functions Simplest Properties ................... 1043 11.15-2 Various Relations and Formulas Connection with Jacobi Elliptic Functions ....................... 1044 11.16 Mathieu Functions and Modified Mathieu Functions ....... 1045 11.16-1 Mathieu Functions ............................... 1045 11.16-2 Modified Mathieu Functions ...................... 1046 11.17 Orthogonal Polynomials ................................. 1047 11.17-1 Laguerre Polynomials and Generalized Laguerre Polynomials ..................................... 1047 11.17-2 Chebyshev Polynomials and Functions ............. 1048 11.17-3 Hermite Polynomials and Functions ............... 1050 11.17-4 Jacobi Polynomials .............................. 1051 11.17-5 Gegenbauer Polynomials .......................... 1051 11.18 Nonorthogonal Polynomials .............................. 1052 11.18-1 Bernoulli Polynomials ........................... 1052 11.18-2 Euler Polynomials ............................... 1053 Supplement 12 Some Notions of Functional Analysis ............ 1055 12.1 Functions of Bounded Variation .......................... 1055 12.1-1 Definition of a Function of Bounded Variation .... 1055 12.1-2 Classes of Functions of Bounded Variation ........ 1056 12.1-3 Properties of Functions of Bounded Variation ..... 1056 12.1-4 Criteria for Functions to Have Bounded Variation . 1057 12.1-5 Properties of Continuous Functions of Bounded Variation ........................................ 1057 12.2 Stieltjes Integral ...................................... 1057 12.2-1 Basic Definitions ................................ 1057 12.2-2 Properties of the Stieltjes Integral ............. 1058 12.2-3 Existence Theorems for the Stieltjes Integral .... 1058 12.3 Lebesgue Integral ....................................... 1059 12.3-1 Riemann Integral and the Lebesgue Integral ....... 1059 12.3-2 Sets of Zero Measure Notion of "Almost Every where" ........................................... 1060 12.3-3 Step Functions and Measurable Functions .......... 1060 12.3-4 Definition and Properties of the Lebesgue Integral ......................................... 1061 12.3-5 Measurable Sets .................................. 1062 12.3-6 Integration Over Measurable Sets ................. 1063 12.3-7 Case of an Infinite Interval ..................... 1063 12.3-8 Case of Several Variables ........................ 1064 12.3-9 Spaces Lp ........................................ 1064 12.4 Linear Normed Spaces .................................... 1065 12.4-1 Linear Spaces .................................... 1065 12.4-2 Linear Normed Spaces ............................. 1065 12.4-3 Space of Continuous Functions C(α,b) ............. 1066 12.4-4 Lebesgue Space Lp(α,b) ........................... 1066 12.4-5 Holder Space Cα(0,l) ............................. 1066 12.4-6 Space of Functions of Bounded Variation V(0,1) ... 1066 12.5 Euclidean and Hilbert Spaces Linear Operators in Hilbert Spaces .................................................. 1067 12.5-1 Preliminary Remarks .............................. 1067 12.5-2 Euclidean and Hilbert Spaces ..................... 1067 12.5-3 Linear Operators in Hilbert Spaces ............... 1068 References ................................................... 1071 Index ........................................................ 1081