Can we find $a$ complex such that:
$$\int_{0}^{\infty} x^{-c} e^{-a x^2} dx =0$$
($c$ is complex with $1>Re(c)>0$)
Same question with this other integral, can we find a complex $\alpha$ such that ($K_{\alpha}(x)$ is the K-Bessel function):
$$\int_{0}^{\infty} x^{-\frac{1}{2}} K_c (\alpha x) dx =0$$
($c$ is also fixed complex with $\frac{1}{2}>Re(c)>0$)
Any method or reference for ths type of problem ?
When it converges, your first integral is $$\frac{a^{(c-1)/2} \Gamma((1-c)/2)}{2}$$ But the Gamma function has no zeros.
EDIT: ... and I believe the second is $$\frac{\sqrt{2}\; \Gamma((1/4-c/2) \Gamma1/4+c/2)}{4\sqrt{\alpha}}$$ so again no zeros.