I have a very basic question about prolate speroidal wave functions that can be defined as eigenfunctions to the following integral equation: $$ \lambda\cdot \psi(x) = \int_{-1}^{1}\frac{\sin 2c(x-y)}{\pi(x-y)}\psi(y)\, \mathrm{d}y, $$ where $c$ is a fixed parameter. The eigenfunction $\psi_0(x)$ ($||\psi_0||_{L_2}=1$) with the maximal eigenvalue $\lambda_0$ is quite remarkable as it has maximal energy ($L_2$-norm) concentrated on the interval $[-c,c]$ among the functions whose Fourier transform is supported on $[-1,1]$.
What is $\psi_0(0)$ in the regime when $c\to\infty$, i.e., when $\lambda_0\to 1$? An equivalent question would be to compute $\int_{-1}^{1}\psi_0(x)\, \mathrm{d}x$.
There is alternative definition via eigenfunctions to spheroidal differential equation http://dlmf.nist.gov/30 (defined in part 30.2, part 30.15 relates to the former definition). I tried to apply power series expansion 30.4.4, but failed to calculate the leading coefficient $g_0$ defined by 30.4.5.