Index 303

properties, 17

factoring polynomials, 91

finite order

of an entire function, 258

Fourier Inversion Formula, 172

Fourier transform, 167

of normal distribution, 170

using residues, 167

function φ, 290

function ψ, 288

functions of finite order, 258

zeroes, 263

Fundamental Theorem of Algebra, 90

gamma function, 270

has no zeroes, 273

infinite product formula, 274

meromorphic continuation, 271

geometric series, 11

greatest lower bound, 79

group homomorphism, 112

Hadamard, 288

Hadamard’s Theorem, 265

harmonic conjugate, 40

existence, 105

harmonic function, 39, 105

maximum principle, 106

mean value property, 106

Harnack’s Inequality, 213

heat equation, 214

heat flow, 214

Heine-Borel Theorem, 54

homeomorphic sets, 152

homeomorphism, 195

homologous cycles, 115

homologous to 0, 115

homotopic curves, 145

homotopy, 145

hydrodynamics, 214

Identity Theorem, 97

imaginary part, 5

improper integral, 160

converge absolutely, 160

principal value, 160

indented contours, 172

independence of path, 63

index, 64

for curves, 144

locally constant, 144

of a cycle, 114

properties, 67

inf, 79

infinite products

convergence, 246

uniform convergence, 247

integral

over a 1-chain, 111

over a path, 45

Riemann, 43

integrating using residues, 159

interior, 30

inverse Fourier transform, 172

inverse Laplace transform, 175

inversion in C, 7

inversion transformation, 198

isolated point, 31, 96

isolated singularity, 98

Jensen’s Formula, 262

join of two paths, 50

Jordan Curve Theorem, 121

kernel, 202

Laplace transform, 175

Laplace’s equation, 39

Laurent series expansion, 127

least upper bound, 78

length of a path, 52

lifting a map, 232

through a covering map, 234

liminf, 79

limit of a function, 31

limit of a sequence, 9

limsup, 79

linear fractional transformation, 197

Liouville’s Theorem, 89

log function, 21

branches, 22

is analytic, 39

principal branch, 22

properties, 22

logarithm of a function, 97

logarithmic derivative, 248

long division of power series, 155

Maximum Modulus Theorem, 103

Mellin transform, 176

using residues, 176

meromorphic function, 101

factorization, 255

inverse of, 101

with pole at ∞, 194