Beskonačna membrana s rupom u obliku kruga

Sada moramo izbaciti rješenja $r^n$ i $\ln r,$ pa sličnim rezoniranjem dolazimo do rješenja

$$u(r,\varphi) = C_{10} + \sum_{n=1}^{\infty} \frac{1}{r^n}\,\left(C_{1n}\,\cos n\varphi + C_{2n}\,\sin n\varphi\right),$$

gdje je, koristeći rubni uvjet,

$$C_{10} = \frac{1}{2\,\pi}\,\int_{-\pi}^{\pi}\,\alpha(\varphi)\,d\... ... \frac{R^n}{\pi}\,\int_{-\pi}^{\pi}\,\alpha(\varphi)\,\cos n\varphi\,d\varphi,$$

$$C_{2n} = \frac{R^n}{\pi}\,\int_{-\pi}^{\pi}\,\alpha(\varphi)\,\sin n\varphi\,d\varphi.$$