I have 2 Humbert seriers special forms that arise in a quantum physics problem. $a$ and $x$ are real.
$$\phi_1(1+ia,ia,ia+3/2;1/2,ix)$$
$$\phi_2(ia,-ia,1/2;ix,-ix)$$
I want to find asymptotic expansions when $x \to \infty$ in order to compute precise approximations when x is sufficiently large.
N.B. $\phi_1$ can be written as an Euler type integral: https://en.wikipedia.org/wiki/Humbert_series
$$\phi_1(1+ia,ia,ia+3/2;1/2,ix)=\frac{\Gamma(3/2+ia)}{\Gamma(1+ia)\sqrt{\pi}}\int_{0}^{1} \frac{t^{ia}(1-t/2)^{-ia}}{\sqrt{1-t}} e^{ixt} dt$$
2026-04-30 02:02:52.1777514572
Asymptotics for 2 Humbert series special forms
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