As in Example 1 in the text we have R(r)=c3 r^n+c4 r^-n. In...

As in Example 1 in the text we have $R(r)=c_{3} r^{n}+c_{4} r^{-n}$. In order that the solution be bounded as $r \rightarrow \infty$ we must define $c_{3}=0 .$ Hence $$u(r, \theta)=A_{0}+\sum_{n=1}^{\infty} r^{-n}\left(A_{n} \cos n \theta+B_{n} \sin n \theta\right)$$ where $$\begin{array}{l} A_{0}=\frac{1}{2 \pi} \int_{0}^{2 \pi} f(\theta) d \theta \\ A_{n}=\frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \cos n \theta d \theta \\ B_{n}=\frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \sin n \theta d \theta \end{array}$$.

As in Example 1 in the text we have $R(r)=c_{3} r^{n}+c_{4} r^{-n}$. In order that the solution be bounded as $r \rightarrow \infty$ we must define $c_{3}=0 .$ Hence $$u(r, \theta)=A_{0}+\sum_{n=1}^{\infty} r^{-n}\left(A_{n} \cos n \theta+B_{n} \sin n \theta\right)$$ where $$\begin{array}{l}
A_{0}=\frac{1}{2 \pi} \int_{0}^{2 \pi} f(\theta) d \theta \\
A_{n}=\frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \cos n \theta d \theta \\
B_{n}=\frac{c^{n}}{\pi} \int_{0}^{2 \pi} f(\theta) \sin n \theta d \theta
\end{array}$$.

Key Concepts

Separation of Variables

This is a common technique used to solve partial differential equations by assuming that the solution can be written as a product of functions, each depending on a single coordinate. In problems with polar coordinates, for example, the method allows the Laplace equation to be separated into radial and angular parts, leading to ordinary differential equations that are easier to solve.

Boundedness Conditions

When solving differential equations on unbounded domains, it is crucial to impose conditions that ensure the solution does not become unbounded. In the context of the radial component, terms that grow without limit as the radial variable increases must be eliminated to guarantee that the solution remains finite at infinity.

Fourier Series Representation

Fourier series decompose periodic functions into sums of sine and cosine terms. In boundary value problems, this representation is utilized to express the angular part of the solution, where the coefficients are determined by integrals involving the given boundary data. This approach helps match the solution to prescribed boundary conditions.

Harmonic Functions in Polar Coordinates

Harmonic functions are solutions of Laplace's equation and exhibit the mean value property. Expressing these functions in polar coordinates is particularly effective for problems with circular or cylindrical symmetry. The general solution is expressed as a series involving powers of the radial coordinate multiplied by sine and cosine functions of the angular coordinate.

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