302 Index

bounded function, 89

canonical product, 260

order, 264

Cauchy’s estimates, 87

Cauchy’s Integral Formula, 65

classical form, 123

for convex sets, 65

general form, 119

Cauchy’s Integral Theorem, 54

classical form, 123

for a triangle, 57

for convex sets, 63

general form, 120

Cauchy-Riemann equations, 38

Cauchy-Schwarz inequality, 6

chain, 110

chain rule, 36

characteristic polynomial, 92

circles in

S2,

199

circles meeting at right angles, 237

closed set, 28

closure, 30

compact set, 54

comparison test, 11

complex derivative, 34

properties, 36

complex numbers, 2

field properties, 3

graphing, 5

why needed, 1

complex powers, 23

Complex Projective Space, 194

component, 148

conformal automorphism, 197

of

S2,

198

of the unit disc, 201

conformal equivalence, 185, 187

conformal map, 185, 186

examples, 188

has complex derivative, 186

conjugate pairs, 5

conjugation in C, 5

connected component, 67

connected set, 67

continuous function, 32

alternate characterization, 32

contour, 41

converge

absolutely, 160

pointwise, 75

uniformly, 75

convergence, 8

of improper integrals, 160

of infinite products, 246

uniform, 247

of sequences, 9

of series, 10

convex set, 61

cos function, 17

cotan function

properties, 179

counting zeroes and poles, 134

curve, 41

closed, 49

differentiable, 41

linear, 143

parameterized, 41

piecewise linear, 143

piecewise smooth, 41

simple closed, 121

smooth, 41

curvilinear triangle, 238

cycle, 113

de la Vall´ ee-Poussin, 288

deleted neighborhood, 31

derivative, 34

logarithmic, 248

of a curve, 41

of a power series, 82

differentiable, 34

differential equations, 92

exponential solution, 92

Dirichlet problem, 208

solution on D, 211

discrete subset, 97

distance between curves, 143

division in C, 7

eigenvalues of matrices, 92

electrostatics, 214

entire function

of finite order, 258

with no zeroes, 259

equivalent, 111

E-equivalent, 111

essential singularity, 98

Euclidean distance, 5

Euclidean norm, 5

Euler’s identity, 16

exponent of convergence, 260

exponential function, 14

law of exponents, 15