$\newcommand{\dx}{\mathrm dx\,}$I’m having lots of trouble figuring this out, so perhaps you guys can help me. For example, let’s take the integral
$$\int\limits_0^{\infty}\dx\frac {\sin x\log x}x=-\frac {\gamma\pi}2$$
Our integral can be computed using differentiation under the integral sign and taking the imaginary part of
$$\mathfrak{I}(s)=-\int\limits_0^{\infty}\dx x^{a-1}e^{-sx}=-s^{-a}\Gamma(a)$$
Set $s=i$ and take the imaginary part to get
$$\operatorname{Im}\mathfrak{I}(i)=\Gamma(a)\sin\frac {\pi a}2$$
But when I differentiate and set $a=0$, then the gamma function becomes undefined because $\Gamma’(0)$ doesn’t produce a determinate form.
I’m not exactly sure what went wrong. Perhaps you can help me?
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