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
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