Complex Analysis - Maths KU

IND-10 Word Index
evaluation by residues, 347,
352-353, 354-355, 357, 359
improper, 355, 359, 361-362
independent of the path, 265
line, 238-239
real, 236-240
Integral transform, 375
Integration along a branch cut, 361
Integration by parts, 270
Integration of power series, 314-315
Interior point of a set, 29
Invariance of Laplace’s equation
under a mapping, 162, 223
Invariant set under a mapping, 78
Inverse Fourier transform:
definition of, 381
evaluated by residue theory,
382-383
Inverse function, 88
Inverse hyperbolic functions, 219
derivatives of, 220
Inverse Laplace transform, 374
definition of, 377
evaluated by residue theory, 378
Inverse trigonometric functions,
215-216
derivatives of, 218
Inversion in the unit circle, 100
Irrotational flow, 167
Isolated singularity, 324
Isotherms, 167-168
Jordan’s inequality, 372
Joukowski airfoil, 442
Joukowski transformation, 442
Kernel of an integral transform, 375
Kinetic energy, 166
Lagrange’s identity, 23
Laplace transform:
analyticity of, 377
definition of, 374
existence of, 376
inverse of, 374, 377-378
Laplace’s equation, 159, 222
invariance of, 162, 223
in polar coordinates, 163
Laplacian, 159
Laurent series, 326-327
analytic part of, 325
principal part of, 325
Laurent’s theorem, 327
Level curves, 164
L’Hôpital’s rule, 147, 149
proofs of, 151, 324
Limit(s):
of a complex function, 112
imaginary part of, 116
infinite, 127
at infinity, 127
nonexistence of, 113
properties of, 117
of a real function, 111, 115
real part of, 116
of a sequence, 302-303
Linear approximation, 76
Linear fractional transformation:
circle preserving property of, 402
composition of, 404
cross ratio for, 406
definition of, 400
on extended complex plane, 401
inverse of, 404-405
as a matrix, 404
Linear function, 68, 71
as a composition of a rotation,
magnification, and translation,
72
as a magnification, 70
as a rotation, 69
as a translation, 68
Linear mappings, 68, 71-73
image of a point under, 72
Linear transformation, 384
Line equation of, 62
Line integrals in the plane, 238-241
Line(s):
of flow, 167, 287
of force, 167-168
of heat flux, 167-168
parametrization of, 62
segment, 63
Liouville’s theorem, 279
Location of zeros, 366-367
Logarithmic function:
algebraic properties of, 184
analyticity of, 187-188