1. ## Parabolic cylinder functions

Hi everyone! I have a differential equation of the form

$\displaystyle \frac{d^2 f}{d x^2} + \lambda^2(1 + x^2)f(x) = 0.$

The solutions to this equation are the parabolic cylinder functions. Using the definition as stated in Wikipedia, the even and odd solutions are $\displaystyle y_1(i \lambda/2; \sqrt{2 i \lambda}x)$ and $\displaystyle y_2(i \lambda/2; \sqrt{2 i \lambda}x)$, respectively. For real, nonzero values of $\displaystyle \lambda$, these solutions are oscillating with decreasing amplitude and period. I am interested in calculating the value of

$\displaystyle \int_0^\infty dx\:y_1(i\lambda/2; \sqrt{2 i\lambda}x)$

as a function of $\displaystyle \lambda$. With the normalization as used in the Wikipedia article, this value is a positive and finite constant (when $\displaystyle \lambda \neq 0$). I found numerically for $\displaystyle \lambda = 1$ that this value is $\displaystyle \approx 0.57$. Can anyone help me with this problem?

Kind regards,
Simon

2. ## Re: Parabolic cylinder functions

I have made some progress, but the problem is not yet solved. Doing some further numerical studies, I found the value of the integral for a set of $\displaystyle \lambda$. For $\displaystyle \lambda \ll 1$ the value seems to be $\displaystyle \approx 0.85 \lambda^{-0.32}$ (dashed curve in the figure).

Edit: Never mind about the fit (dashed curve) =P. For even smaller $\displaystyle \lambda$ the integral curve has the same curvature in the log-log plot, so that the proportionality constant (0.85) and the exponent (-0.32) continuously increases.