A question from the monograph: “ Number Theory, Physics and Geometry, Volume 1”. [closed]

In page 9 (first edition), we have next chain of equalities which the last one doesn't make sense to me, it's an equation for the trace of the eigenvalue for a rectangular with periodical boundary conditions Laplacian eigenvalue equation: $$\Delta \psi_n +E_n \psi_n =0$$

The trace is defined as: $$d(E)=\sum_{n_1 ,n_2=-\infty}^{\infty} \delta(E-E_{n_1 n_2})$$

where $E_{n_1 n_2}=(\frac{2\pi}{a} n_1)^2+(\frac{2\pi}{b} n_2)^2$, and $a$,$b$ are the sides of the rectangle.

Now my problem arises in the next second equality: $$d(E)=\frac{ab}{(2\pi)^2} \sum_{m_1,m_2=-\infty}^{\infty} \int \int e^{i(m_1 a cos(\phi)+m_2 b sin(\phi))r}\delta (k^2-r^2) rdrd\phi=$$ $$= \frac{ab}{2(2\pi)^2} \int_{0}^{2\pi} e^{ik\sqrt{(m_1 a)^2+(m_2 b)^2} cos(\phi)} d\phi$$

I don't understand how did they arrive at the last expression.

Besides the missing sum, $m_1a\cos(\phi)+m_2b\sin(\phi)=\sqrt{(m_1a)^2+(m_2b)^2}\cos(\phi-\phi_0)$ where $\phi_0=\tan^{-1}(\frac{m_2b}{m_1a})$. Voting to close as too localized. – Gjergji Zaimi Sep 13 2011 at 6:46