Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.10 Cauchy’s integral formula y x C 1 γ C 2 C Figure 24.11 The contour used to prove the result (24.43).
Consider two closed ...

COMPLEX VARIABLES contourCandz 0 is a point withinCthen f(z 0 )= 1 2 πi ∮ C f(z) z−z 0 dz. (24.46) This formula is saying that t ...

24.11 Taylor and Laurent series Further, it may be proved by induction that thenth derivative off(z) is also given by a Cauchy i ...

COMPLEX VARIABLES whereanis given byf(n)(z 0 )/n!. The Taylor expansion is valid inside the region of analyticity and, for any p ...

24.11 TAYLOR AND LAURENT SERIES of orderpatz=z 0 but is analytic at every other point inside and onC.Then the functiong(z)=(z−z ...

COMPLEX VARIABLES x y R z 0 C 1 C 2 Figure 24.12 The region of convergenceRfor a Laurent series off(z) about a pointz=z 0 wheref ...

24.11 TAYLOR AND LAURENT SERIES denominator take the form (1−αz), whereαis some constant, and thus obtain f(z)=− 1 8 z(1−z/2)^3 ...

COMPLEX VARIABLES Differentiating both sidesm−1times,weobtain dm−^1 dzm−^1 [(z−z 0 )mf(z)] = (m−1)!a− 1 + ∑∞ n=1 bn(z−z 0 )n, fo ...

24.12 RESIDUE THEOREM Suppose the functionf(z) has a pole of ordermat the pointz=z 0 ,andso can be written as a Laurent series a ...

COMPLEX VARIABLES C C C′ (a) (b) Figure 24.13 The contours used to prove the residue theorem: (a) the original contour; (b) the ...

24.13 Definite integrals using contour integration We note that result (24.60) is a special case of (24.63) in whichθ 2 is equal ...

COMPLEX VARIABLES formula (24.56) withm= 2. Choosing the latter method and denoting the integrand by f(z), we have d dz [z^2 f(z ...

24.13 DEFINITE INTEGRALS USING CONTOUR INTEGRATION y −R O R x Γ Figure 24.14 A semicircular contour in the upper half-plane.
Ev ...

COMPLEX VARIABLES y −R O R x Γ γ Figure 24.15 An indented contour used when the integrand has a simple pole on the real axis. Wh ...

24.13 DEFINITE INTEGRALS USING CONTOUR INTEGRATION The proof of the lemma is straightforward once it has been observed that, for ...

COMPLEX VARIABLES y x A B C D Γ γ Figure 24.16 A typical cut-plane contour for use with multivalued functions that have a single ...

24.14 Exercises We have seen that ∫ Γand ∫ γvanish, and if we denotezbyxalong the lineABthen it has the valuez=xexp 2πialong the ...

COMPLEX VARIABLES 24.7 Find the real and imaginary parts of the functions (i)z^2 , (ii)ez, and (iii) coshπz. By considering the ...

24.14 EXERCISES 24.14 Prove that, forα>0, the integral ∫∞ 0 tsinαt 1+t^2 dt has the value (π/2) exp(−α). 24.15 Prove that ∫∞ ...

COMPLEX VARIABLES 24.22 The equation of an ellipse in plane polar coordinatesr, θ, with one of its foci at the origin, is l r =1 ...