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4. (a) Let u be a harmonic function in D. Show that if v is a harmonic conjugate of u in a domain D, then both u² - v² and u³ - 3uv² are harmonic in D. (b)Suppose that functions u and v are harmonic in D. Are the following functions harmonic? (1) u² - v²; (2) uv; (3) 9u + 13v; (4) Vry + Au

4. (a) Let u be a harmonic function in D. Show that if v is a harmonic conjugate of u in a domain D, then both u² - v² and u³ - 3uv² are harmonic in D. (b)Suppose that functions u and v are harmonic in D. Are the following functions harmonic? (1) u² - v²; (2) uv; (3) 9u + 13v; (4) Vry + Au

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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4. (a) Let u be a harmonic function in D. Show that if v is a harmonic conjugate of u in a domain D, then both u²v² and u³ - 3uv² are harmonic in D. (b)Suppose that functions u and v are harmonic in D. Are the following functions harmonic? (1) u² − v²; (2) uv; (3) 9u + 13v; (4) Vxy + Au (Assume that harmonic functions are smooth functions with all derivatives.)
Transcribed Image Text:4. (a) Let u be a harmonic function in D. Show that if v is a harmonic conjugate of u in a domain D, then both u²v² and u³ - 3uv² are harmonic in D. (b)Suppose that functions u and v are harmonic in D. Are the following functions harmonic? (1) u² − v²; (2) uv; (3) 9u + 13v; (4) Vxy + Au (Assume that harmonic functions are smooth functions with all derivatives.)
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