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The function [tex] u(r,\theta) [/tex]

satisfies Laplace's equation in the wedge [tex] 0 \leq r \leq a, 0 \leq \theta \leq \beta [/tex]

with boundary conditions [tex] u(r,0) = u(r,\beta) =0, u_r(a,\theta)=h(\theta) [/tex]. Show that

[tex] u(r,\theta) = \sum_{n=0}^\infty A_nr^{n\pi/\beta}sin(\frac{n\pi\theta}{\beta}) [/tex]

[tex]A_n=a^{1-\frac{n\pi}{\beta}\frac{2}{n\pi}\int_{0}^{\beta}h(\theta)sin\frac{n\pi\theta}{\beta}d\theta [/tex]

satisfies Laplace's equation in the wedge [tex] 0 \leq r \leq a, 0 \leq \theta \leq \beta [/tex]

with boundary conditions [tex] u(r,0) = u(r,\beta) =0, u_r(a,\theta)=h(\theta) [/tex]. Show that

[tex] u(r,\theta) = \sum_{n=0}^\infty A_nr^{n\pi/\beta}sin(\frac{n\pi\theta}{\beta}) [/tex]

[tex]A_n=a^{1-\frac{n\pi}{\beta}\frac{2}{n\pi}\int_{0}^{\beta}h(\theta)sin\frac{n\pi\theta}{\beta}d\theta [/tex]

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