If ${c_n}$ is a bounded sequence, then $$ f(r, \theta)=\sum_{n=-\infty}^{\infty}c_nr^{|n|}e^{in\theta} $$ is harmonic in the disc.
Help me proving?
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- Show that each term is harmonic. 2) Show the Laplacian of the sum is the sum of the Laplacian.
– user10676 Mar 29 '13 at 18:40 -
What disk? Do you mean unit disk? – Thomas Andrews Mar 29 '13 at 18:45
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yes, a unit disc – Matema Tika Mar 29 '13 at 18:49
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$$f'_r=\sum_{-\infty}^\infty |n|c_nr^{|n|-1}e^{in\theta}\;,\;\;f''_{rr}=\sum_{-\infty}^\infty |n|\,(|n|-1)c_n\,r^{|n|-2}e^{in\theta}$$
$$f'_\theta=i\sum_{-\infty}^\infty n\,c_nr^{|n|}e^{in\theta}\;,\;\;\;f''_{\theta\theta}=-\sum_{-\infty}^\infty n^2\,c_nr^{|n|}e^{in\theta}$$
So Laplace's equation in polar coordinates:
$$r^2f''_{rr}+rf'_r+f''_{\theta\theta}=\ldots$$
And now you have to prove the above is zero...
Spoiler!:
$$\sum_{-\infty}^\infty \;c_n\left(|n|\,(|n|-1)+|n|-n^2\right)r^{|n|}e^{in\theta}=0$$
DonAntonio
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Well, it almost is, @MatemaTika, yet you should first try hard to reach it after reading some hints. That's the best way to learn... – DonAntonio Mar 29 '13 at 18:58
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Hint: In complex form your sum is $c_0 + \sum_{n = 1}^{\infty} c_nz^n + \sum_{n = 1}^{\infty} c_n\bar{z}^n$. Write this as a real part of an analytic function on the unit disc. (By the way, I'm assuming the $c_n$ are real, otherwise it's not necessarily a real-valued function).
Zarrax
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