Tag Archives: difficult

Creating Difficult integrals by the residue theorem

Theorem Let $$f$$ be analytic function in the unit circle $$|z|\leq 1$$  such that $$f\neq 0$$ . Then $$\int^{2\pi}_0f(e^{it})\,dt =2\pi \, f(0)$$ $$\textit{proof}$$ Since the function $$f$$ is analytic in and on the … Continue reading

Prove the following $$I= \log \left\{\frac{\Gamma(b+c+1) \Gamma(c+a+1)\Gamma(a+b+1)}{\Gamma(a+1) \Gamma(b+1) \Gamma(c+1) \Gamma(a+b+c+1)} \right\}$$   where   $$I = \int_0^1 \frac{(1-x^a)(1-x^b)(1-x^c)}{(1-x)(-\log x)}dx$$ $$\textit{proof}$$ First note that since there is a log in the denominator that gives as an idea to use differentiation under … Continue reading