As follows, Consider the function = u + iv with the real and imaginary parts exp

Question

As follows,

Consider the function = u + iv with the real and imaginary parts expressed in terms of either X and y, or r and theta. The Cauchy-Riemann equations, (ux = vy, uy = - vx) are satisfied at a point z if and only if the polar form of the Cauchy-Riemann equations: ur = v theta/r, u theta/r = - vr is satisfied (assuming that the partial derivatives are continuous there). If f ( z ) is differentiable, show that the derivative may be evaluated using the expressions: f ' ( z ) = e - theta (ur + ivr) = - i/z { u theta + iv theta)

Explanation / Answer

if f is differentiable then we can use cauchy riemenn equation

u(r) = v(theta)/r--------------------------(a)

and u(theta)/r = -v(r)---------------------(b)

z= u+iv = e^-i(theta)

there fore f'(z) = df/dx = du/dx + dv/dy

= e^-(theta) ( u(r ) + i v(r))---------------(1)

(1) can be also written as

z= re^-i(theta)

hence 1 becomes = -i/z * (u(theta) + i v(theta)) (from a and b)

hence proved

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