Inside the Wolfram Documentation page for the secant function, an identity is given which involves the gamma function, polygamma function, and Catalan's constant.
Notes on documentation page:
Some special functions can be used to evaluate more complicated definite integrals. For example, polygamma and gamma functions and the Catalan constant are needed to express the following integral:
$$\int_0^{\pi/2} \sec^a(t)\,dt= \frac{\sqrt{\pi}}{2\Gamma\left(1-\frac{a}{2}\right)}\Gamma\left(\dfrac{1-a}{2}\right),\quad\text{$\operatorname{Re}(a)<1$} $$
I know that the Gamma function is defined as
$$\Gamma(z) = \int_0^\infty x^{z-1} e^{-x}\, dx, \quad\text{$\operatorname{Re}(z)>0$}$$
and Catalan's constant can be written as
$$G = \beta(2) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} – \frac{1}{3^2} + \frac{1}{5^2} – \frac{1}{7^2} + \frac{1}{9^2} – \cdots$$
but I don't see how this helps. I couldn't find a source in the Wolfram documentation page and couldn't find a duplicate question on Math SE. How did the author of the Wolfram page arrive at this identity?
Best Answer
To derive the identity, it is a straight up application of the Beta function in the form $$\operatorname{B}(m,n) = 2 \int_0^{\frac{\pi}{2}} \cos^{2m - 1} t \sin^{2n - 1} t \, dt.$$
For the secant integral, we have \begin{align} \int_0^{\frac{\pi}{2}} \sec^a t \, dt &= \int_0^{\frac{\pi}{2}} \cos^{-a} t \, dt\\ &= \frac{1}{2} \cdot 2 \int_0^{\frac{\pi}{2}} \cos^{2\left (\frac{1 - a}{2} \right ) - 1} t \sin^{2 \left (\frac{1}{2} \right ) - 1} t \, dt\\ &= \frac{1}{2} \operatorname{B} \left (\frac{1 - a}{2}, \frac{1}{2} \right )\\ &= \frac{1}{2} \frac{\Gamma \left (\frac{1 - a}{2} \right ) \Gamma \left (\frac{1}{2} \right )}{\Gamma \left (\frac{1 - a}{2} + \frac{1}{2} \right )}\\ &= \frac{\sqrt{\pi}}{2 \Gamma \left (1 - \frac{a}{2} \right )} \Gamma \left (\frac{1 - a}{2} \right ), \end{align} for $a < 1$, as required. Here the following well-known results of $$\operatorname{B} (a,b) = \frac{\Gamma (a) \Gamma (b)}{\Gamma (a + b)},$$ together with $\Gamma (\frac{1}{2}) = \sqrt{\pi}$, have been used.