Let 
denote the Euler’s Gamma function and 
denote the digamma function. Evaluate
![\displaystyle \bigintsss_{0}^{2}\frac{\Gamma^2\left ( \frac{1}{t} \right )}{2t^3 \Gamma \left ( \frac{2}{t} \right )} \bigg[ t + 2\psi\left ( \frac{1}{t} \right) - 2 \psi \left ( \frac{2}{t} \right ) \bigg] \, {\rm d}t](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-0c6a9bd51c742215428e89c5db8639ce_l3.png "Rendered by QuickLaTeX.com")
Solution
We have successively
![\begin{aligned} \int_{0}^{2}\frac{\Gamma^2\left ( \frac{1}{t} \right )}{2t^3 \Gamma \left ( \frac{2}{t} \right )} \bigg[ t + 2\psi\left ( \frac{1}{t} \right) - 2 \psi \left ( \frac{2}{t} \right ) \bigg]\, {\rm d}t & =\int_{0}^{2} \frac{\Gamma^2\left ( \frac{1}{t} \right )}{t^3 \Gamma\left ( \frac{2}{t} \right )} \bigg[ \frac{t}{2}+ \psi\left ( \frac{1}{t} \right ) - \psi\left ( \frac{2}{t} \right ) \bigg] \, {\rm d}t \\ &\!\!\overset{u=2/t}{=\! =\! =\!} \frac{1}{4}\int_{1}^{\infty} \frac{\Gamma \left ( \frac{u}{2} \right ) \Gamma \left ( \frac{u}{2} \right )}{\Gamma \left ( \frac{u}{2}+ \frac{u}{2} \right )} \bigg[ 1+ u \psi \left ( \frac{u}{2} \right ) - u \psi (u) \bigg] \, {\rm d}u\\ &= \frac{1}{4}\int_{1}^{\infty} {\rm B}\left ( \frac{u}{2}, \frac{u}{2} \right ) \left ( 1+ u \psi \left ( \frac{u}{2} \right )- u \psi (u) \right ) \, {\rm d}u\\ &=\frac{1}{4}\int_{1}^{\infty} {\rm B}\left ( \frac{u}{2}, \frac{u}{2} \right ) \bigg( 1+ u \left ( \psi \left ( \frac{u}{2} \right ) - \psi\left ( \frac{u}{2}+ \frac{u}{2} \right ) \right ) \bigg )\, {\rm d}u \\ &=\frac{1}{4}\int_{1}^{\infty } \bigg[{\rm B} \left ( \frac{u}{2}, \frac{u}{2} \right ) + u {\rm B}\left ( \frac{u}{2}, \frac{u}{2} \right ) \left ( \psi\left ( \frac{u}{2} \right ) - \psi \left ( \frac{u}{2}+ \frac{u}{2} \right ) \right )\bigg]\, {\rm d}u \\ &=-\frac{1}{4}\int_{1}^{\infty} \left ( u {\rm B}\left ( \frac{u}{2}, \frac{u}{2} \right ) \right )'\, {\rm d}u \\ &= \frac{\pi}{4} \end{aligned}](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-0c4db007e40a1e75ba35b9f479be8628_l3.png "Rendered by QuickLaTeX.com")
The exercise can also be found at Aops.com .
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Let denote the Euler’s Gamma function and denote the digamma function. Evaluate
Solution
The exercise can also be found at Aops.com .