Index

absolute convergence, 50

absolute value, 3

analytic, 17

analytic continuation, 45

arc length, 67

argument, 6

argument principle, 137, 138

Bernoulli numbers, 104

bijection, 28

branch

function, 44

of arg z, 40

of log z, 40

of a holomorphic function, 42

of an inverse, 44

of the logarithm, 42

Burckel, R. B., 85

Carath´ eodory, C., 143

Casorati-Weierstrass theorem, 106

Cauchy integral, 83, 99

Cauchy product, 61, 84

Cauchy’s formula, 130

for a circle, 79

Cauchy’s formula for an annulus, 101

Cauchy’s theorem, 75, 109, 118, 121, 122

for a convex region, 78

for a triangle, 75

for simply connected domains, 126

for two concentric circles, 100

Cauchy-Hadamard theorem, 55

Cauchy-Riemann equations, 16

in polar coordinates, 16

chain rule, 14, 153

clircle, 31

complex conjugate, 4

complex integral, 67

complex number system, 1

complex partial diﬀerential operators, 18

complex plane, 3

complex projective space, 27

conformal automorphism, 142

conformal equivalence, 141

conformal map, 22

contour, 113

simple, 114

convergence

local uniform, 51

of an inﬁnite series, 49

uniform, 51

cross ratio, 30, 32

curve, 20

diﬀerentiable, 20

regular, 20

De Moivre’s formula, 7

diﬀerentation of power series, 58

diﬀerentiable

in the complex sense, 13

in the real sense, 15, 151

diﬀerentiation under the integral sign, 157

division of power series, 63

domain, 125

elementary function, 35

entire function, 86

Euler, L., 35

exponential function

(ez

), 35

extended complex plane, 11

Fej´ er, L., 143

161