Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.3 POWER SERIES IN A COMPLEX VARIABLE This series is absolutely convergent if ∑∞ n=0 |an|rn, (24.12) which is a series of posi ...

COMPLEX VARIABLES which is an alternating series whose terms decrease in magnitude and which therefore converges. The ratio test ...

24.4 SOME ELEMENTARY FUNCTIONS real-variable counterpart it is called theexponential function; also like its real counterpart it ...

COMPLEX VARIABLES derived from them (e.g. tan and tanh), the identities they satisfy and their derivative properties are also ju ...

24.5 Multivalued functions and branch cuts On the RHS let us writetas follows: t=rexp[i(θ+2kπ)], wherekis an integer. We then ob ...

COMPLEX VARIABLES y y y x x x C θ θ r r (a) (b) (c) C′ Figure 24.1 (a) A closed contour not enclosing the origin; (b) a closed c ...

24.6 Singularities and zeros of complex functions (a) (b) (c) −i −i −i i i i y y y x x x r 1 r 2 θ 1 θ 2 z Figure 24.2 (a) Coord ...

COMPLEX VARIABLES whereais a finite, non-zero complex number. We note that if the above limit is equal to zero, thenz=z 0 is a p ...

24.7 Conformal transformations Thus limz→ 0 f(z) = 1 independently of the way in whichz→0, and sof(z) has a removable singularit ...

COMPLEX VARIABLES y θ 2 θ 1 C 1 C 2 z 1 z 2 z 0 w=g(z) s r C 1 ′ C′ 2 φ 2 φ 1 w 0 w 1 w 2 x Figure 24.3 Two curvesC 1 andC 2 in ...

24.7 CONFORMAL TRANSFORMATIONS point of intersection. Since any finite-length tangent may be curved,wiis more strictly given byw ...

COMPLEX VARIABLES y Q R S T x i P w=g(z) R′ P′ S′ Q′ T′ s r Figure 24.4 Transforming the upper half of thez-plane into the inter ...

24.7 CONFORMAL TRANSFORMATIONS x 1 x 2 x 3 x 4 x 5 w 1 w 2 w 3 w 4 w 5 φ 1 φ 2 φ 3 φ 4 φ 5 y x s r w=g(z) Figure 24.5 Transformi ...

COMPLEX VARIABLES φ 1 φ 2 φ 3 x 1 x 2 − 11 y x w 1 w 2 ib w^3 −aa s r w=g(z) Figure 24.6 Transforming the upper half of thez-pla ...

24.8 Complex integrals x φ^1 φ^2 1 x 2 − 11 y x w 1 w 2 w 3 w 3 −aa s r w=g(z) Figure 24.7 Transforming the upper half of thez-p ...

COMPLEX VARIABLES A B C 1 C 2 C 3 x y Figure 24.8 Some alternative paths for the integral of a functionf(z) between AandB. The q ...

24.8 COMPLEX INTEGRALS (a) (b) (c) y y y x RRx x R R t t −R −R s=1 iR t=0 C 3 b C 3 a C 1 C 2 Figure 24.9 Different paths for an ...

COMPLEX VARIABLES must be made in terms of entirely real integrals. For example, the first is given by ∫ 1 0 −R+iR R(1−t)+itR dt ...

24.9 CAUCHY’S THEOREM namely Cauchy’s theorem, which is the cornerstone of the integral calculus of complex variables. Before di ...

COMPLEX VARIABLES A B R y x C 1 C 2 Figure 24.10 Two pathsC 1 andC 2 enclosing a regionR. analyticity off(z) within and onCbeing ...