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.]
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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![### Harmonic Functions and Holomorphic Functions on the Unit Disc
#### Problem Statement:
Let \( u \) be a real-valued function defined on the unit disc \( \mathbb{D} \). Suppose that \( u \) is twice continuously differentiable and harmonic, that is,
\[ \Delta u(x, y) = 0 \]
for all \( (x, y) \in \mathbb{D} \).
(a) Prove that there exists a holomorphic function \( f \) on the unit disc such that
\[ \operatorname{Re}(f) = u. \]
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 \( f'(z) = 2 \frac{\partial u}{\partial \bar{z}} \). Therefore, let \( g(z) = 2 \frac{\partial u}{\partial \bar{z}} \) and prove that \( g \) is holomorphic. Why can one find \( F \) with \( F' = g \)? Prove that \( \operatorname{Re}(F) \) differs from \( u \) by a real constant.
#### Explanation:
In this problem, we are asked to bridge the concepts of harmonic and holomorphic functions. Harmonic functions satisfy Laplace's equation, i.e., \( \Delta u = 0 \), which implies that the second partial derivatives of the function sum to zero. Holomorphic functions, or complex analytic functions, have derivatives that exist and are continuous in the complex plane.
1. **Existence of Holomorphic Function:**
To show that there exists a holomorphic function \( f \) on the unit disc \( \mathbb{D} \) such that \( \operatorname{Re}(f) = u \), we must construct such an \( f \).
2. **Relationship Between Harmonic and Holomorphic Functions:**
If \( u \) is harmonic, in terms of complex variables,
\[ f'(z) = 2 \frac{\partial u}{\partial \bar{z}} \].
This suggests considering \( g(z) = 2 \frac{\partial u}{\partial \bar{z}} \). Show that \( g \) is holomorphic by verifying the Cauchy-Riemann](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e1e2e4-888b-4182-8c02-fd46dda7f6b1%2F68493ba2-74dc-4f55-b568-190ba7f83e46%2Fr9gb8ch_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Harmonic Functions and Holomorphic Functions on the Unit Disc
#### Problem Statement:
Let \( u \) be a real-valued function defined on the unit disc \( \mathbb{D} \). Suppose that \( u \) is twice continuously differentiable and harmonic, that is,
\[ \Delta u(x, y) = 0 \]
for all \( (x, y) \in \mathbb{D} \).
(a) Prove that there exists a holomorphic function \( f \) on the unit disc such that
\[ \operatorname{Re}(f) = u. \]
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 \( f'(z) = 2 \frac{\partial u}{\partial \bar{z}} \). Therefore, let \( g(z) = 2 \frac{\partial u}{\partial \bar{z}} \) and prove that \( g \) is holomorphic. Why can one find \( F \) with \( F' = g \)? Prove that \( \operatorname{Re}(F) \) differs from \( u \) by a real constant.
#### Explanation:
In this problem, we are asked to bridge the concepts of harmonic and holomorphic functions. Harmonic functions satisfy Laplace's equation, i.e., \( \Delta u = 0 \), which implies that the second partial derivatives of the function sum to zero. Holomorphic functions, or complex analytic functions, have derivatives that exist and are continuous in the complex plane.
1. **Existence of Holomorphic Function:**
To show that there exists a holomorphic function \( f \) on the unit disc \( \mathbb{D} \) such that \( \operatorname{Re}(f) = u \), we must construct such an \( f \).
2. **Relationship Between Harmonic and Holomorphic Functions:**
If \( u \) is harmonic, in terms of complex variables,
\[ f'(z) = 2 \frac{\partial u}{\partial \bar{z}} \].
This suggests considering \( g(z) = 2 \frac{\partial u}{\partial \bar{z}} \). Show that \( g \) is holomorphic by verifying the Cauchy-Riemann
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