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1. Verify directly that the real and imaginary parts of the following analytic functions satisfy Laplace's equation: (a) f(2)=z²+2z+1 (b) g(z) = ¹ (c) h(z) = e²

1. Verify directly that the real and imaginary parts of the following analytic functions satisfy Laplace's equation: (a) f(2)=z²+2z+1 (b) g(z) = ¹ (c) h(z) = e²

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Verify directly that the real and imaginary parts of the following analytic functions satisfy

Laplace’s equation:

(a) f(z) = z

2 + 2z + 1

(b) g(z) = 1

z

(c) h(z) = e

z

13:30 FxPro ← TUT-03-08-23.pdf Complex Analysis (MAT3241/3641) University of Venda Tutorial 3 • Attempt all the problems. Provide fully detailed responses. Vo)) Vo)) LTE LTE2 48% Ơ 1. Verify directly that the real and imaginary parts of the following analytic functions satisfy Laplace's equation: (a) f(2)= 2² + 2z+1 (b) g(z) = (c) h(z) = e² ||| 2. Verify that each given function u is harmonic (in the region where it is defined), then find a harmonic conjugate of u and the analytic function f(z) whose real part is u: (a) u = y (b) ue sin y (c) u = xy - x + y (d) u = sin x cosh y (e) u = Ime²² 3. Show that if v is a harmonic conjugate for u, then -u is a harmonic conjugate for v. 4. Show that if v is a harmonic conjugate of u in a domain D, then uv is harmonic in D. Y 5. Find an analytic function f(z) whose imaginary part is given by v(x, y) = y - x² + y² 6. Let f(z) be analytic. Show that if the real (imaginary) part of f is constant, then f itself is a constant. 2
Transcribed Image Text:13:30 FxPro ← TUT-03-08-23.pdf Complex Analysis (MAT3241/3641) University of Venda Tutorial 3 • Attempt all the problems. Provide fully detailed responses. Vo)) Vo)) LTE LTE2 48% Ơ 1. Verify directly that the real and imaginary parts of the following analytic functions satisfy Laplace's equation: (a) f(2)= 2² + 2z+1 (b) g(z) = (c) h(z) = e² ||| 2. Verify that each given function u is harmonic (in the region where it is defined), then find a harmonic conjugate of u and the analytic function f(z) whose real part is u: (a) u = y (b) ue sin y (c) u = xy - x + y (d) u = sin x cosh y (e) u = Ime²² 3. Show that if v is a harmonic conjugate for u, then -u is a harmonic conjugate for v. 4. Show that if v is a harmonic conjugate of u in a domain D, then uv is harmonic in D. Y 5. Find an analytic function f(z) whose imaginary part is given by v(x, y) = y - x² + y² 6. Let f(z) be analytic. Show that if the real (imaginary) part of f is constant, then f itself is a constant. 2
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