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Given the integral

$$\int_{-\infty}^{\infty} dx \;\psi(1/4+ix/2)\exp(-ax^2)$$

How can I evaluate that in the limit $ a\to 0$ and $ a\to \infty$?

Here $\psi(x)$ is the digamma function.


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Mathematica calculation suggests that it divesges to $+\infty$ as $a \downarrow 0$. Indeed, I expect that $$ \int_{-\infty}^{\infty} \psi \left( \frac{1}{4} + \frac{ix}{2} \right) e^{-a x^2} \; dx \ \sim \ - \frac{1}{2} \sqrt{\frac{\pi}{a}} \, (\gamma + \log (16a))$$ as $a \to 0$ and $$ \int_{-\infty}^{\infty} \psi \left( \frac{1}{4} + \frac{ix}{2} \right) e^{-a x^2} \; dx \ \sim \ \sqrt{\frac{\pi}{a}} \, \psi \left( \frac{1}{4} \right)$$ as $a \to \infty$. – Sangchul Lee Nov 27 '11 at 13:09

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