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We know that the operator $A=\Delta$ with domain $D(A)=\{u\in W^{2, 2}(\Omega): u=0 \ \ \text{on } \partial\Omega\}$ (say $\Omega$ is a bounded nice domain) has eigenvalues $\lambda_1>\lambda_2\ge \...

The eigenfunctions of the Laplace operator on the disc can be written in polar coordinates as $f(r,\theta)=R_{nk}(r)e^{ik\theta}$, where $k\in\mathbb Z$ and $n\in\mathbb N$ and the radial function is $...