What is the “name” of this function?

There is a function I met in complex analysis. $$f(\lambda) = \int \limits_{-\infty}^{\infty}\frac{e^{i\lambda x}}{\sqrt{1 + x^{2n}}}dx$$

-
Maple does something complicated in terms of the Meijer G function. –  GEdgar Dec 16 '12 at 14:23
I read about McDonalds function, $$f(\lambda ) = \int \limits_{-\infty}^{\infty} \frac{e^{i \lambda x}dx}{\sqrt{1 + x^2}}.$$ It's one of the Bessel's function. Has the function from my question the name as the, maybe, Bessel's function? –  John Taylor Dec 16 '12 at 15:00

It's called the Fourier Transform of $\frac1{\sqrt{1+x^{2n}}}$.
I read about McDonald function, $$f( \lambda ) = \int \limits_{-\infty}^{\infty}\frac{e^{i \lambda x}dx}{\sqrt{1 + x^2}}.$$ It's one of Bessel's function. Has the function from my question the name as the Bessel's function? –  John Taylor Dec 16 '12 at 14:52