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I need a 2D harmonic function which is always real, never diverges to $\infty$ anywhere, goes to 0 as one goes infinitely far away from the origin, and isn't f=0. $$\begin{matrix}0=\nabla^2f&0=\lim_{r\rightarrow\infty}{f\left(r,\varphi\right)}\\\nexists r,\varphi\ni\left|f\left(r,\varphi\right)\right|\geq\infty&f\left(r,\varphi\right)\in\mathbb{R}\\\end{matrix}$$ Does such a function exist? If so what is it? Polar coordinates are prefered.

Laff70
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  • $142$ rep, you should know that this isnt how you ask a question by now. – Some Guy Mar 17 '21 at 19:34
  • @Some Guy You know, originally the question was going to be way more complex involving the wave equation and the derivatives of various functions but I managed to whittle that larger, more difficult problem down into this. I've done a lot of work to get here, showing it all is irrelevant to this simple problem though. As such, I only asked what I needed to. Sorry for not putting my entire problem in the question when I only needed this small remainder solved. – Laff70 Mar 17 '21 at 19:52
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    I vaguely remember that a harmonic function in a circle reaches its extreme values on the border. – Ivan Neretin Mar 17 '21 at 20:09
  • @IvanNeretin $f(r,φ)=r^n (c_1 sin(nφ)+c_2 cos(nφ))$ should be a harmonic function. Negative values of n are valid so the divergence happens at the origin as opposed to infinity. That formula isn't all encompassing though as it neglects the harmonic function ln(r). – Laff70 Mar 17 '21 at 20:32
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    The harmonic function $\ln r$ isn't harmonic at $0$. Anyway, your condition "never diverges to $\infty$" takes care of that. – Ivan Neretin Mar 17 '21 at 20:38
  • I'm pretty sure this duplicates a previously asked Question. Short answer: There is no such function (not identically zero). You said you "need" this function. It would greatly improve your asking if an explanation for the need were given. – hardmath Mar 18 '21 at 17:49

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