13. The polar form of the Cauchy-Riemann equations. (a) Use the coordinate transformation x = r cos 0 and y = r sin 0 and the chain rules U₁ = Ux Əx - ar + U. ду Ər and əx 20 Əy 20 etc.
13. The polar form of the Cauchy-Riemann equations. (a) Use the coordinate transformation x = r cos 0 and y = r sin 0 and the chain rules U₁ = Ux Əx - ar + U. ду Ər and əx 20 Əy 20 etc.
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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Problem 13 from Chapter 3: Analytic and Harmonic Functions, Section 3.2: The Couchy Riemann Equations from John H Matthew's Complex Analysis for Mathematics and Engineering Textbook, Third Edition
Transcribed Image Text:13. The polar form of the Cauchy-Riemann equations.
(a) Use the coordinate transformation
x = r cos 0 and y = r sin 0
and the chain rules
84
Ur
Əx
ar
+ Uy
ду
ar
and
Up = Ux
əx
20
+ Ur
Əy
to prove that
и₁ = 4₂cos + usin 0
and
v₁ = v₂cos 0 + v,sin 0 and
(b) Use the results of part (a) to prove that
ru, Ve and rv = -8.
-
20
etc.
Chapter 3 Analytic and Harmonic Functions
40 =
-ursin 0 + urcos 8
V=-vrsin 0 + v,rcos 0.
and
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