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.
Thanks in advance.
