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5. Let u be harmonic on the complex plane. Show that for any complex number a and r > 0, u(a) = 2²/7 ²* u(a + re¹0) do. 2π (This is the 'Mean value Theorem' for harmonic functions.) Conclude that |u(a)| ≤max_u(a + rei). 0≤0<2T

5. Let u be harmonic on the complex plane. Show that for any complex number a and r > 0, u(a) = 2²/7 ²* u(a + re¹0) do. 2π (This is the 'Mean value Theorem' for harmonic functions.) Conclude that |u(a)| ≤max_u(a + rei). 0≤0<2T

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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This is for a course in complex analysis. The main topics we have covered in this section are power series expansions and Cauchy's Formula. We have NOT covered residues. We must end with the part involving the inequality at the bottom.

 

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5. Let u be harmonic on the complex plane. Show that for any complex number a and r > 0, 1 2πT u(a) = 2/7 ²* u(a + re²⁰) do. 2π (This is the 'Mean value Theorem' for harmonic functions.) Conclude that |u(a)| ≤_max_ |u(a + rei). 0<0<2T
Transcribed Image Text:5. Let u be harmonic on the complex plane. Show that for any complex number a and r > 0, 1 2πT u(a) = 2/7 ²* u(a + re²⁰) do. 2π (This is the 'Mean value Theorem' for harmonic functions.) Conclude that |u(a)| ≤_max_ |u(a + rei). 0<0<2T
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