Chapter 1 Ultrametric fields ................................... 1
Chapter 2 Monotonous and circular filters ..................... 10
Chapter 3 Ultrametric absolute values for rational functions .. 20
Chapter 4 Hensel Lemma ........................................ 29
Chapter 5 Extensions of ultrametric fields: the field  p ...... 34
Chapter 6 Normal extensions of Qp inside p ................... 41
Chapter 7 Spherically complete extensions ..................... 48
Chapter 8 Transcendence order over  p in p ................... 52
Chapter 9 Transcendence type over in p ..................... 59
Chapter 10 Algebras  ( ) ....................................... 66
Chapter 11 Analytic elements ................................... 72
Chapter 12 Composition of analytic elements .................... 79
Chapter 13 Mult(H(), UD) ...................................... 84
Chapter 14 Power and Laurent series ............................ 88
Chapter 15 Mittag-Leffier Theorem and dual of a space H(D) ..... 97
Chapter 16 Factorization of analytic elements ................. 108
Chapter 17 Algebras H() ...................................... 112
Chapter 18 Derivative of analytic elements .................... 119
Chapter 19 Properties of the function Ф for analytic
elements ........................................... 131
Chapter 20 Vanishing along a monotonous filter ................ 136
Chapter 21 Quasi-minorated elements ........................... 142
Chapter 22 Zeros of power series .............................. 148
Chapter 23 Maximum principle .................................. 159
Chapter 24 Image of a disk .................................... 163
Chapter 25 Logarithm and exponential in a p-adic field ........ 174
Chapter 26 Problems on p-adic exponentials .................... 178
Chapter 27 Quasi-invertible analytic elements ................. 183
Chapter 28 Divisors of analytic functions ..................... 191
Chapter 29 Michel Lazard's problem ............................ 201
Chapter 30 Sets of range uniqueness in p-adic fields .......... 208
Chapter 31 Motzkin factorization, roots of analytic
functions .......................................... 220
Chapter 32 Meromorphic functions .............................. 237
Chapter 33 Residues of meromorphic functions .................. 245
Chapter 34 Identity sequences for p-adic functions ............ 253
Chapter 35 The set Mult( b(d(0, -)),ǁ • ǁd(0, -)) .............. 259
Chapter 36 The Corona problem on b(d(0,1-))................... 266
Chapter 37 Shilov boundary for algebras H() ................. 276
Chapter 38 Mappings from Φ() to the tree Φ( ) ............... 288
Chapter 39 Injective analytic elements ........................ 296
Chapter 40 Nevanlinna Theory .................................. 308
Chapter 41 Immediate applications of Nevanlinna Theory ........ 319
Chapter 42 Applications to curves ............................. 324
Chapter 43 Small functions .................................... 332
Chapter 44 Exceptional values of functions and derivatives .... 341
Chapter 45 The p-adic Hayman conjecture ....................... 351
Chapter 46 Optimal functions .................................. 362
Chapter 47 Order of growth for entire functions ............... 367
Chapter 48 Type of growth for entire functions ................ 375
Chapter 49 Growth of the derivative of an entire function ..... 386
Chapter 50 Branched values .................................... 391
Chapter 51 Affinely rigid sets ................................ 404
Chapter 52 Composition of meromorphic functions ............... 410
Chapter 53 Functions of uniqueness ............................ 417
Chapter 54 Urscm and ursim .................................... 429
Chapter 55 Other urscm, ursim and non-urscm ................... 442
Chapter 56 Functions ƒ'P'(ƒ) .................................. 451
Chapter 57 Functions sharing values ........................... 462
Chapter 58 Analytic functions sharing a small function ........ 469
Chapter 59 Meromorphic functions sharing a small function ..... 473
Chapter 60 Nevanlinna Theory in characteristic p .............. 488
Chapter 61 Strong uniqueness and URSCM in characteristic p .... 495
Chapter 62 The Functional Equation P(ƒ) = Q(g) ................ 500
Chapter 63 Rational decomposition for entire functions ........ 508
Chapter 64 Yosida's equation .................................. 511
Chapter 65 Yoshida's equation inside a disk ................... 515
Bibliography .................................................. 523
Definitions ................................................... 533
Notations ..................................................... 539
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