Index 163

ratio test, 34, 41, 72, 77

rational function, 60, 70, 77, 79, 122,

147

rational numbers, 1, 7, 14, 20, 79

rational point, 147

real part, 25, 90, 93, 100

rearrangement, 34

reciprocal, 3, 61

recurrence relation, 74

reflexivity, 19

removable singularity, 119

residue, 116, 119, 121, 123, 124, 133

Riemann mapping theorem, 151

Riemann sphere, 63, 65

Riemann surface, 63

Riemann’s remark, 34

root test, 34, 71

rotation, 44, 51, 61, 152

Rouche’s theorem, 145

row operations, 61

simple curve, 103

simple pole, 120

simply connected, 23, 95, 145

sine, 41, 42

smooth curve, 101

smooth function, 36, 84, 85

stereographic projection, 64

Stirling’s formula, 135

Stokes’s theorem, 109

subharmonic function, 94

subsequence, 12, 77, 137, 138

supremum, 9, 13

symmetry, 19

tangent vector, 96

topological space, 51

topology, 51

transitive group, 151

transitivity, 19, 21

translation, 61

triangle inequality, 11, 22, 139

trichotomy, 7

uniform convergence, 35, 71, 114, 117

unit circle, 40

unitary, 158

variance, 130

vector ﬁeld, 101, 108

volume form, 97

wedge product, 97

whispering gallery, 53

winding number, 45, 141, 143

work, 101

zero-set, 49

zeroes, 141