Let f(z) = u(x, y)+ i v(x, y) = U(r, 0) + iV (r, 0). By exploiting the relations x = r cos 0 , y = r sin 0 , we proved that if the Cauchy-Riemann equations hold true, that is, Ux = Vy Uy = -Vx (1) then, r U, = V9 r V, = -Ug . (2) One can actually show that the viceversa is also true, that is, if f satisfies (2) then f satisfies also (1). The equations in (2) are the polar form of the Cauchy-Riemann equations. Recalling that f'(z) = ux + i vx, check that f'(2) = e-iº (Ur + i V;).
Let f(z) = u(x, y)+ i v(x, y) = U(r, 0) + iV (r, 0). By exploiting the relations x = r cos 0 , y = r sin 0 , we proved that if the Cauchy-Riemann equations hold true, that is, Ux = Vy Uy = -Vx (1) then, r U, = V9 r V, = -Ug . (2) One can actually show that the viceversa is also true, that is, if f satisfies (2) then f satisfies also (1). The equations in (2) are the polar form of the Cauchy-Riemann equations. Recalling that f'(z) = ux + i vx, check that f'(2) = e-iº (Ur + i V;).
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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Transcribed Image Text:Let f(2) = u(x, y) + i v(x, y) = U (r, 0) + iV (r, 0). By exploiting the relations
x = r cos0 ,
y =r sin 0,
we proved that if the Cauchy-Riemann equations hold true, that is,
= Vy
Uy
Vx
(1)
then,
r Up = Ve
r Vr = -U .
(2)
One can actually show that the viceversa is also true, that is, if f satisfies (2) then f satisfies
also (1). The equations in (2) are the polar form of the Cauchy-Riemann equations.
Recalling that f' (2) :
= Ux +i vr, check that
f'(2) = e-i0 (U, + i V,).
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