5. A function u(x, y) is defined by u2³-3xy². (a) Show that u is harmonic, i.e. it satisfies Laplace's equation P²uP²u + -0. ər² dy² (b) Determine the conjugate harmonic function u of u and hence determine the function f(2)= u(x, y) +iv(x, y) as a function of z.

Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please solve the laplacian based question. If possible please solve all questions thank you.

5. A function u(x, y) is defined by u = x³ - 3xy².
(a) Show that u is harmonic, i.e. it satisfies Laplace's equation
J²u J²u
+ = 0.
əx² dy²
(b) Determine the conjugate harmonic function v of u and hence determine
the function f(z) = u(x, y) + iv(x, y) as a function of z.
Transcribed Image Text:5. A function u(x, y) is defined by u = x³ - 3xy². (a) Show that u is harmonic, i.e. it satisfies Laplace's equation J²u J²u + = 0. əx² dy² (b) Determine the conjugate harmonic function v of u and hence determine the function f(z) = u(x, y) + iv(x, y) as a function of z.
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