12. Let u be a real-valued function defined on the unit disc D. Suppose that u is twice continuously differentiable and harmonic, that is, Au(x, y) = 0 for all (x, y) = D. (a) Prove that there exists a holomorphic function f on the unit disc such that Re(f) = u. Also show that the imaginary part of f is uniquely defined up to an additive (real) constant. [Hint: From the previous chapter we would have ƒ'(z) = 20u/az. Therefore, let g(z) = 20u/az and prove that g is holomorphic. Why can one find F with F' = g? Prove that Re(F) differs from u by a real constant.]
12. Let u be a real-valued function defined on the unit disc D. Suppose that u is twice continuously differentiable and harmonic, that is, Au(x, y) = 0 for all (x, y) = D. (a) Prove that there exists a holomorphic function f on the unit disc such that Re(f) = u. Also show that the imaginary part of f is uniquely defined up to an additive (real) constant. [Hint: From the previous chapter we would have ƒ'(z) = 20u/az. Therefore, let g(z) = 20u/az and prove that g is holomorphic. Why can one find F with F' = g? Prove that Re(F) differs from u by a real constant.]
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 22E
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