Derivation of the Poisson Kernel

From Geometric Series to Harmonic Functions

The Poisson kernel is frequently used and studied in the context of harmonic analysis and partial differential equations. Below is its derivation and why it shows up naturally.

The Foundation: Geometric Series

The absolute convergence of the infinite series below is well known:

\[ \sum_{k = 0}^{\infty}p^k = \frac{1}{1-p} \quad \text{ where } |p| < 1 \]

By letting \(p = re^{i\theta}\), where \(|r| < 1\), our expression becomes:

\[ \sum_{k = 0}^{\infty}r^ke^{ik\theta} = \frac{1}{1-re^{i\theta}} \]

Extracting the Real Part

To analyze the real part, we multiply by the complex conjugate over itself:

\[ \frac{1}{1-re^{i\theta}}\cdot \frac{1-re^{-i\theta}}{1-re^{-i\theta}} = \frac{1-re^{-i\theta}}{1 - re^{-i\theta}-re^{i\theta} + r^2e^{-i\theta}e^{i\theta}} \]

Using Euler's Formula (\(e^{i\theta} = \cos\theta + i\sin\theta\)) and the symmetry of trigonometric functions (\(\cos\) is even, \(\sin\) is odd), the denominator simplifies:

\[ 1 - r(e^{-i\theta}+e^{i\theta}) + r^2 = 1 - 2r\cos(\theta) + r^2 \]

Plugging this back into our fraction and expanding the numerator:

\[ \frac{1 - r\cos(\theta) + i r\sin(\theta)}{r^2 - 2r\cos(\theta) + 1} \]

The real part \(P_r(\theta)\) is therefore:

\[ P_r(\theta) = \frac{1 - r\cos(\theta)}{r^2 - 2r\cos(\theta) + 1} \]

Symmetrizing the Sum

Since we took the real part of the series, we know:

\[ \frac{1-r\cos(\theta)}{r^2-2r\cos(\theta) + 1} = \sum_{k = 0}^{\infty}r^k\cos(k\theta) = 1 + \sum_{k=1}^{\infty}r^k\cos(k\theta) \]

To reach the standard form of the kernel, we recognize that \(\cos(k\theta)\) is even, allowing us to relate this to a sum over all integers \(\mathbb{Z}\). After algebraic manipulation to account for the \(k=0\) term and combining terms:

\[ \sum_{k \in \mathbb{Z}}r^{|k|}e^{ik\theta} = 2\left(\frac{1-r\cos(\theta)}{1+r^2-2r\cos(\theta)}\right) - 1 \]

Continuing the simplification:

\[ \frac{2-2r\cos(\theta) - (1+r^2-2r\cos(\theta))}{1+r^2-2r\cos(\theta)} = \frac{1-r^2}{1+r^2-2r\cos(\theta)} \]

The Poisson Kernel

\[ P_r(\theta) = \frac{1-r^2}{1+r^2-2r\cos(\theta)} = \sum_{k \in \mathbb{Z}}r^{|k|}e^{ik\theta} \]

This kernel is the fundamental tool for solving the Dirichlet problem on the disk, essentially providing a way to find a harmonic function given its values on the boundary.

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