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Cauchy-Riemann Equations in Polar Form

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8. Let a function f (z) = u + i v be differentiable at a nonzero point z0 = r0 e(i&#952;0). Use the expressions for ux and vx found in Exercise 7, together with the polar form (6) of Cauchy-Riemann equations, to rewrite the expression
f &#900;(z0) = ux + i vx

as

f &#900;(z0) = e-i&#952; (ur + i vr),
where ur and vr are to be evaluated at (r0, &#952;0).

Exercise 7. Solve equations (2) for ux and uy to show that

ux = ur cos &#952; - (u&#952; sin &#952;) / r , uy = ur sin &#952; + (u&#952; cos &#952;) / r.

Then use these equations and similar ones for vx and vy to show that equations (4) are satisfied at a point z0 if equations (6) are satisfied there. Thus complete the verification that equations (6) are the Cauchy-Riemann equations in polar form.

Eqn (2): ur = ux cos &#952; + uy sin &#952;, u&#952; = - ux r sin &#952; + uy r cos &#952;

Eqn (4): ux = vy, uy = - vx

Eqn (6): r ur = v&#952;, u&#952; = - r vr