In this paper by means of contour integration we will evaluate definite integrals of the form \begin{equation*} \int_{0}^{1}\left(\ln^k(ay)-\ln^k\left(\frac{a}{y}\right)\right)R(y)dy \end{equation*} in terms of a special function, where $R(y)$ is a general function and $k$ and $a$ are arbitrary complex numbers. These evaluations can be expressed in terms of famous mathematical constants such as the Euler's constant $\gamma$ and Catalan's constant $C$. Using derivatives, we will also derive new integral representations for some Polygamma functions such as the Digamma and Trigamma functions along with others.

中文翻译：

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Hurwitz zeta函数团队中对数积分的推导

在本文中，通过轮廓积分，我们将评估形式为\ begin {equation *} \ int_ {0} ^ {1} \ left（\ ln ^ k（ay）-\ ln ^ k \ left（\ frac {a} {y} \ right）\ right）R（y）dy \ end {equation *}的特殊功能，其中$ R（y）$是常规函数，$ k $和$ a $是任意复数。这些评估可以用著名的数学常数来表达，例如欧拉常数$ \ gamma $和加泰罗尼亚常数$ C $。使用导数，我们还将为某些Polygamma函数（例如Digamma和Trigamma函数以及其他函数）导出新的积分表示形式。