0 GENERAL CONSIDERATIONS ....................................... 1
What functions are. Organization of the Atlas. Notational
conventions. Rules of the calculus.
1 THE CONSTANT FUNCTION с ..................................... 13
J. Mathematical constants. Complex numbers. Pulse
functions. Series of powers of natural numbers.
2 THE FACTORIAL FUNCTION n! ................................... 21
Double and triple factorial functions. Combinatorics.
Stirling numbers of the second kind.
3 THE ZETA NUMBERS AND RELATED FUNCTIONS ...................... 29
Special values. Apéry's constant. The Debye functions
of classical physics.
4 THE BERNOULLI NUMBERS Bn .................................... 39
Dual definitions. Relationship to zeta numbers. The Euler-
Maclaurin sum formulas.
5 THE EULER NUMBERS En ........................................ 45
Relationship to beta numbers and Bernoulli numbers.
6 THE BINOMIAL COEFFICIENTS (νm) ............................... 49
Binomial expansion. Pascal's triangle. The Laplace-de
Moivre formula. Multinomial coefficients.
7 THE LINEAR FUNCTION bx+c AND ITS RECIPROCAL ................. 57
How to fit data to a "best straight line". Errors
attaching to the fitted parameters.
8 MODIFYING FUNCTIONS ......................................... 67
Selecting features of numbers. Rounding. Base conversion.
9 THE HEAVISIDE u(x-a) AND DIRAC δ(x-a) FUNCTIONS ............. 75
Window and other discontinuous functions. The comb
function. Green's functions.
10 THE INTEGER POWERS xn AND (bx+с)n ............................ 81
Summing power series. Euler transformation.
Transformations through lozenge diagrams.
11 THE SQUARE-ROOT FUNCTION √bx + c AND ITS RECIPROCAL ......... 95
Behavior of the semiparabolic function in the complex
plane. The parabola and its geometry.
12 THE NONINTEGER POWER xν .................................... 103
Behavior in four quadrants. Mellin transforms. De
Moivre's theorem. The fractional calculus.
13 THE SEMIELLIPTIC FUNCTION (b/d)√α2-x2 AND ITS
RECIPROCAL ................................................. 113
Ellipticity. Geometric properties of the ellipse,
ellipsoid, and semicircle. Superellipses.
14 THE (b/d)√x2±α2 FUNCTIONS AND THEIR RECIPROCALS ............ 121
Vertical, horizontal and diagonal varieties of
hyperbolas. Operations that interrelate functions
graphically.
15 THE QUADRATIC FUNCTION ax2+bx+c AND ITS RECIPROCAL ......... 131
Zeros, real and complex. The root-quadratic function.
Conic sections. Trajectory of a projectile.
16 THE CUBIC FUNCTION x3+ax2+bx+c ............................. 139
Zeros of cubics and quarries. "Joining the dots" with
sliding cubics and cubic splines.
17 POLYNOMIAL FUNCTIONS ....................................... 147
Finding zeros. Rational functions. Partial fractions.
Polynomial optimization and regression.
18 THE POCHHAMMER POLYNOMIALS (x)n ............................ 159
Stirling numbers of the first kind. Hypergeometric
functions.
19 THE BERNOULLI POLYNOMIALS Bn(x) ............................ 175
Sums of monotonic power series.
20 THE EULER POLYNOMIALS En(x) ................................ 181
Sums of alternating power series.
21 THE LEGENDRE POLYNOMIALS Pn(x) ............................. 187
Orthogonality. The Legendre differential equation and its
other solution, the Qn(x) function.
22 THE CHEBYSHEV POLYNOMIALS Tn(x) AND Un(x) .................. 197
Gegenbauer and Jacobi polynomials. Fitting data sets with
discrete Chebyshev polynomials.
23 THE LAGUERRE POLYNOMIALS Ln(x) ............................. 209
Associated Laguerre polynomials. Fibonacci numbers and
the golden section.
24 THE HERMITE POLYNOMIALS Hn(x) .............................. 217
Gauss integration. The systematic solution of second-
order differential equations.
25 THE LOGARITHMIC FUNCTION ln(x) ............................. 229
Logarithms to various bases. The dilogarithm and
polylogarithms. Logarithmic integral.
26 THE EXPONENTIAL FUNCTION exp(±x) ........................... 241
Exponential growth/decay. Self-exponential function.
Exponential polynomial. Laplace transforms.
27 EXPONENTIAL OF POWERS exp(±xν) ............................. 255
Exponential theta functions. Various distributions; their
probability and cumulative versions.
28 THE HYPERBOLIC COSINE cosh(x) AND SINE sinh(x) FUNCTIONS ... 269
О Algebraic and geometric interpretations. The catenary.
29 THE HYPERBOLIC SECANT AND COSECANT FUNCTIONS ............... 281
Is Interrelations between hyperbolic functions via
similar triangles. Interesting inverse Laplace
transformations.
30 THE HYPERBOLIC TANGENT AND COTANGENT FUNCTIONS ............. 289
The Langevin function, important in the theory of
electrical or magnetic dipoles.
31 THE INVERSE HYPERBOLIC FUNCTIONS ........................... 297
Synthesis from reciprocal linear functions. Logarithmic
equivalence. Expansion as hypergeometric functions.
32 THE COSINE cos(x) AND SINE sin(x) FUNCTIONS ................ 309
Sinusoids. Periodicity, frequency, phase and amplitude.
Fourier transforms. Clausen's integral.
33 THE SECANT sec(x) AND COSECANT csc(x) FUNCTIONS ............ 329
Interrelationships between circular functions via similar
triangles. The Gudermannian function and its inverse.
34 THE TANGENT tan(x) AND COTANGENT cot(x) FUNCTIONS .......... 339
Tangent and cotangent roots. Utility of half-argument
formulas. Rules of trigonometry.
35 THE INVERSE CIRCULAR FUNCTIONS ............................. 351
Synthesis from reciprocal linear functions. Two-
dimensional coordinate systems and scale factors.
36 PERIODIC FUNCTIONS ......................................... 367
Expansions in sines and cosines. Euler's formula and
Parseval's relationship. Waveforms.
37 THE EXPONENTIAL INTEGRALS Ei(x) AND Ein(x) ................. 375
Cauchy limits. Functions defined as indefinite integrals.
Table of popular integrals.
38 SINE AND COSINE INTEGRALS .................................. 385
Entire cosine versions. Auxiliary sine and cosine
integrals.
39 THE FRESNEL INTEGRALS C(x) AND S(x) ........................ 395
Bohmer integrals. Auxiliary Fresnel integrals. Curvatures
and lengths of plane curves. Cornu's spiral.
40 THE ERROR FUNCTION erf(x) AND ITS COMPLEMENT erfc(x) ....... 405
Inverse error function. Repeated integrals. Normal
probability. Random numbers. Monte Carlo.
41 THE exp(x)erfc(√x) AND RELATED FUNCTIONS .................. 417
Properties in the complex plane and the Voigt function.
42 DAWSON'S INTEGRAL daw(x) ................................... 427
The closely related erfi function. Intermediacy to
exponentials. Gaussian integrals of complex argument.
43 THE GAMMA FUNCTION Г(ν) .................................... 435
Gauss-Legendre formula. Complete beta function. Function
synthesis and basis hypergeometric functions.
44 THE DIGAMMA FUNCTION Ψ(ν) .................................. 449
Polygamma functions. Bateman's G function and its
derivatives. Sums of reciprocal linear functions.
45 THE INCOMPLETE GAMMA FUNCTIONS ............................. 461
The Mittag-Leffler, or generalized exponential, function.
46 THE PARABOLIC CYLINDER FUNCTION Dν(x) ...................... 471
Three-dimensional coordinate systems. The Laplacian,
separability, and an exemplary application.
47 THE KUMMER FUNCTION M(α,c,x) ............................... 485
The confluent hypergeometric differential equation.
Kummer's transformation. Zeros.
48 THE TRICOMI FUNCTION U(α,c,x) .............................. 497
О Numbers of zeros and extrema. The two Whittaker
functions. Bateman's confluent function.
49 THE MODIFIED BESSEL FUNCTIONS In(x) OF INTEGER ORDER ....... 507
Bessel's modified differential equation. Cylinder
functions generally and their classification.
51 THE MODIFIED BESSEL FUNCTION Iν(x) OF ARBITRARY ORDER ...... 519
Simplifications when the order is a multiple of 1/2, 1/3,
or 1/4 Auxiliary cylinder functions.
51 THE MACDONALD FUNCTION Kν(x) ............................... 527
Alternative solutions to Bessel's modified differential
equation. Spherical Macdonald functions.
52 THE BESSEL FUNCTIONS Jn(x) OF INTEGER ORDER ................ 537
Am Zeros, extrema, and their associated values. Miller's
method. The Newton-Raphson root-finding method.
53 THE BESSEL FUNCTION Jν(x) OF ARBITRARY ORDER ............... 553
The Bessel-Clifford equation. Hankel transforms. Neumann
series. Discontinuous Bessel integrals.
54 THE NEUMANN FUNCTION Yν(x) ................................. 567
Behavior close to zero argument. Hankel functions.
Asymptotic expansions of cylinder functions.
55 THE KELVIN FUNCTIONS ....................................... 577
Complex-plane relationships to Bessel functions.
56 THE AIRY FUNCTIONS Ai(x) AND Bi(x) ......................... 585
Hyperbolic/circular chimera. Airy's differential
equation. Auxiliary Airy functions. Airy derivatives.
57 THE STRUVE FUNCTION hν(x) .................................. 593
Kinship with Neumann functions. The modified Struve
function.
58 THE INCOMPLETE BETA FUNCTION B(ν,μ,x) ...................... 603
Role as a hypergeometric function. Integrals of circular
and hyperbolic functions raised to an arbitrary power.
59 THE LEGENDRE FUNCTIONS Pν(x) AND Qν(x) ...................... 611
The associated Legendre functions. Solving the Laplace
equation in spherical coordinates.
60 THE GAUSS HYPERGEOMETRIC FUNCTION F(α,b,c,x) ............... 627
Plethora of special cases. Contiguity relationships.
Linear transformations.
61 THE COMPLETE ELLIPTIC INTEGRALS K(k) AND E(k) .............. 637
The third kind of complete elliptic integral. Means.
The elliptic nome. Theta functions of various kinds.
62 THE INCOMPLETE ELLIPTIC INTEGRALS F(k,φ) AND Е(k,φ) ........ 653
The Landen transformations. II(ν,k,φ). Integrals of
reciprocal cubic functions. Romberg integration.
63 THE JACOBIAN ELLIPTIC FUNCTIONS ............................ 671
Trigonometric interpretation. Circular/hyperbolic
intermediacy. Double periodicity in the complex plane.
64 THE HURWITZ FUNCTION ζ(ν,u) ................................ 685
Bivariate eta functiota. The Lerch function. Weyl
differintegration and its application to periodic
functions.
APPENDIX A: USEFUL DATA ....................................... 697
SI units and prefixes. Universal constants. Terrestrial
constants and standards. The Greek alphabet.
APPENDIX В: BIBLIOGRAPHY ...................................... 703
Cited sources and supporting publications. Books and web
sources but not original research articles.
APPENDIX C: EQUATOR, THE ATLAS FUNCTION CALCULATOR ............ 705
Disk installation. Basic operations and additional
features. Input/output formats. Accuracy. Keywords.
SYMBOL INDEX .................................................. 723
Notation used here and elsewhere.
SUBJECT INDEX ................................................. 735
A comprehensive directory to the topics in this Atlas.
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