a. Explanation Given information: For the functions u = u ( x , y ) and v = v ( x , y ) the Cauchy-Riemann equation are given by ∂ u ∂ x = ∂ v ∂ y and ∂ u ∂ y = − ∂ v ∂ x . Formula used: The derivative with respect to a variable for the function of two or more than two variables is partially differentiating with respect to that variable and keeping all other variables as constants. For the function, u = u ( x , y ) , partial derivative with respect to x is , u x ( x , y ) = ∂ u ∂ x . The derivative of the function e a x is, d d x e a x = a e a x . Trigonometric derivatives are as follows: d d x sin a x = a cos a x and d d x cos a x = − a sin a x Calculation: Consider the functions u = e − x cos y , v = e − x sin y , we know find the partial derivatives to satisfy the Cauchy-Riemann equation. The derivative with respect to a variable for the function of two or more than two variables is partially differentiating with respect to that variable and keeping all other variables as constants. Partial derivative of u = e − x cos y with respect to x , ∂ u ∂ x = ∂ ∂ x ( e − x cos y ) = − e − x cos y Partial derivative of u = e − x cos y with respect to y , ∂ u ∂ y = ∂ ∂ y b. To determine To check: Whetherthe function u = x 2 + y 2 , v = 2 x y satisfies the Cauchy-Riemann equations. c. To determine To check: Whetherthe function u = ln ( x 2 + y 2 ) , v = 2 tan − 1 ( y x ) satisfies the Cauchy-Riemann equations.

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 11.3, Problem 51PS

To determine

To check: Whetherthe function u=e−xcosy,v=e−xsiny satisfies the Cauchy-Riemann equations.

To determine

To check: Whetherthe function u=x2+y2,v=2xy satisfies the Cauchy-Riemann equations.

To determine

To check: Whetherthe function u=ln(x2+y2),v=2tan−1(yx) satisfies the Cauchy-Riemann equations.

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Chapter 11.3, Problem 50PS

Chapter 11.3, Problem 52PS

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To check: Whetherthe function u = e − x cos y , v = e − x sin y satisfies the Cauchy-Riemann equations.

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To check: Whetherthe function u=e−xcosy,v=e−xsiny satisfies the Cauchy-Riemann equations.

To check: Whetherthe function u=x2+y2,v=2xy satisfies the Cauchy-Riemann equations.

To check: Whetherthe function u=ln(x2+y2),v=2tan−1(yx) satisfies the Cauchy-Riemann equations.

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