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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(2)-1 (c) h(z) = c²

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(2)-1 (c) h(z) = c²

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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1. Verify directly that the real and imaginary parts of the following analytic functions satisfy Laplace's equation: (a) f(z)= 22 +22+1 (b) g(z) = ¹ (c) h(z) = c² 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) u = e sin y (c) u = xy-x+y (d) u = sin x cosh y (e) u = Ime=² 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 then f itself is a 6. Let f(z) be analytic. Show that if the real (imaginary) part of f is constant, constant.
Transcribed Image Text:1. Verify directly that the real and imaginary parts of the following analytic functions satisfy Laplace's equation: (a) f(z)= 22 +22+1 (b) g(z) = ¹ (c) h(z) = c² 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) u = e sin y (c) u = xy-x+y (d) u = sin x cosh y (e) u = Ime=² 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 then f itself is a 6. Let f(z) be analytic. Show that if the real (imaginary) part of f is constant, constant.
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