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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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
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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)= 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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