(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Green's function [itex]G(x_0,y_0,x,y) =v(x_0,y_0,x,y) + h(x_0,y_0,x,y)[/itex] in a region [itex]\Omega \hbox { with boundary } \Gamma[/itex]. Also [itex]v(x_0,y_0,x,y) = -h(x_0,y_0,x,y)[/itex] on boundary [itex]\Gamma[/itex] and both [itex]v(x_0,y_0,x,y) \hbox { and }h(x_0,y_0,x,y)[/itex] are harmonic function in [itex]\Omega[/itex]

[tex]v=\frac{1}{2}ln[(x-x_0)^2 + (y-y_0)^2] [/tex]

Let u be continuous and h is harmonic on an open disk around [itex](x_0,y_0)[/itex] in [itex]\Omega [/itex]. Show that

[tex]_r\stackrel{lim}{\rightarrow}_0 \int_{C_r} u\frac{\partial h}{\partial n} ds = 0[/tex]

Hint from the book: Both |u| and [itex] |\frac{\partial h}{\partial n}|[/itex] are bounded near [itex](x_0,y_0)[/itex], say by M. If [itex] I_r[/itex] denotes the integral in question, then [itex] |I_r| \leq 2\pi M r \rightarrow 0 \hbox { as } r\rightarrow 0[/itex]

2. Relevant equations

Green's 1st identity:

[tex] \int _{\Omega} ( u\nabla^2 h + \nabla u \cdot \nabla h ) dx dy = \int_{\Gamma} u \frac{\partial h}{\partial n} ds[/tex]

3. The attempt at a solution

[tex] h=-v \hbox { on } \Gamma \Rightarrow\; \int _{\Omega} ( u\nabla^2 h + \nabla u \cdot \nabla h ) dx dy = -\int_{\Gamma} u \frac{\partial v}{\partial n} ds = -\int_{\Gamma} u \frac{1}{r} ds = -\int^{2\pi}_{0} u d\theta[/tex]

h is harmonic in [itex]\Omega\;\Rightarrow \nabla^2 h = 0[/itex]

[tex]\Rightarrow\; \int _{\Omega} \nabla u \cdot \nabla h \; dx dy = -\int^{2\pi}_{0} u d\theta[/tex]

I really don't know how to continue, please help.

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# Homework Help: Prove equation in Green's function.

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