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Let $u(x,y)$ be a solution of Laplace equation on $x^{2}+y^{2}\leq1.$ If $$u(cos(\theta),sin(\theta))=\begin{cases} sin\theta & 0\leq\theta\leq\pi \\ 0 & \pi\leq\theta\leq 2\pi \end{cases}$$ Then $u(0,0)$ equals

$(a.)\frac{1}{\pi}$

$(b.)\frac{2}{\pi}$

$(c.)\frac{1}{2\pi}$

$(d.)\frac{\pi}{2}$

I don't know how to solve this problem. General solution of Laplace equation is very lengthy. I am trying to find the required value directly but $cos(\theta)$ and $sin(\theta )$ are not simultaneously zero. How to solve it? Thanks in advance.

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Use the mean value property of harmonic functions, which simplifies for a function defined on the unit disk to the statement

$$u(0,0) = \frac{1}{2\pi}\int_0^{2\pi} u(\cos\theta, \sin\theta)\;d\theta.$$

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  • $\begingroup$ ok sir thanks...i am reading first time this property...thanks a lot... $\endgroup$ – neelkanth May 17 '16 at 5:32

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