Complex Analysis: Theory and Applications · Problem 3.2.10

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · polar-cauchy-riemann-equations

Problem

To prove Theorem 3.5, the polar form of the Cauchy-Riemann equations, (a) Let

f(z) = f(x,y) = f(re^{i\theta}) = u(re^{i\theta}) + iv(re^{i\theta}) = U(r,\theta) + iV(r,\theta)

. Use the transformation

x = r\cos\theta

and

y = r\sin\theta

(i.e.,

(x,y) = re^{i\theta}

) and the chain rules

U_r = u_x\frac{\partial x}{\partial r} + u_y\frac{\partial y}{\partial r} \quad \text{and} \quad U_\theta = u_x\frac{\partial x}{\partial \theta} + u_y\frac{\partial y}{\partial \theta} \quad \text{(similarly for } V\text{)}.

to prove that

U_r = u_x\cos\theta + u_y\sin\theta, \quad U_\theta = -u_x r\sin\theta + u_y r\cos\theta; \quad \text{and}

V_r = v_x\cos\theta + v_y\sin\theta, \quad V_\theta = -v_x r\sin\theta + v_y r\cos\theta.

(b) Use the original Cauchy-Riemann equations for

u

and

v

and the results of part (a) to prove that

rU_r = V_\theta

and

rV_r = -U_\theta

, thus verifying Equation (3.22). (c) Use part (a) and Equations (3.14) and (3.15) to show that the right sides of Equations (3.23) and (3.24) simplify to

f'(z_0)

Related theory

Cauchy Riemann Equations

Browse the problem library