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19.21. (a) Show that S f(z) dz can be written as [ A · dr + i B. dr, where A (u, –v, 0), B = (v, u, 0), and dr (dx, dy, 0). (b) Show that both A and B have vanishing curls when f is analytic. (c) Now use the Stokes' theorem to prove the Cauchy-Goursat theorem.

19.21. (a) Show that S f(z) dz can be written as [ A · dr + i B. dr, where A (u, –v, 0), B = (v, u, 0), and dr (dx, dy, 0). (b) Show that both A and B have vanishing curls when f is analytic. (c) Now use the Stokes' theorem to prove the Cauchy-Goursat theorem.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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19.21. (a) Show that f(z) dz can be written as A · dr + i B. dr, where A = (u, -v,0), B = (v, u, 0), and dr : (b) Show that both A and B have vanishing curls when f is analytic. (c) Now use the Stokes' theorem to prove the Cauchy-Goursat theorem. (dx, dy, 0).
Transcribed Image Text:19.21. (a) Show that f(z) dz can be written as A · dr + i B. dr, where A = (u, -v,0), B = (v, u, 0), and dr : (b) Show that both A and B have vanishing curls when f is analytic. (c) Now use the Stokes' theorem to prove the Cauchy-Goursat theorem. (dx, dy, 0).
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