Explanation Given: The given data is the function f ( z ) = u + i v is analytic and the equations are ∂ v ∂ y = ∂ u ∂ x ∂ v ∂ x = − ∂ u ∂ y Calculation: z = r e i θ Where z ( complex number ) is a function of length r and argument θ . f ( z ) = u + i v ⇒ f ( z ( r , θ ) ) = u ( r , θ ) + i v ( r , θ ) Use chain rule. ∂ f ∂ r = ∂ f ∂ z ∂ z ∂ r = ∂ f ∂ z ∂ ( r e i θ ) ∂ r = ∂ f ∂ z e i θ − − − ( i ) ∂ f ∂ θ = ∂ f ∂ z ∂ z ∂ θ = i ∂ f ∂ z ∂ ( r e i θ ) ∂ r = i ∂ f ∂ z r e i θ − − − ( i i ) From (i) and (ii). Function f has both partial derivatives. Function f is analytic. So, ∂ f ∂ z exist. ∂ f ∂ r = ∂ u ∂ r + i ∂ v ∂ r − − − ( i i i ) ∂ f ∂ θ = ∂ u ∂ θ + i ∂ v ∂ θ − − − ( i v ) Combine ( i ) and ( i i i )

Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Question

Chapter 14.2, Problem 46P

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Find the polar form of the Cauchy-Riemann equations.

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Chapter 14.2, Problem 45P

Chapter 14.2, Problem 47P

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Mathematical Methods in the Physical Sciences

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