Category Archives: Logarithmic Integrals
Integral representations of the reciprocal beta function
In this post, we’ll prove a very interesting identity: $$ \int_0^\pi \left(\sin(\theta) \right)^{\alpha-1} e^{i \beta \theta} \; d\theta = \frac{\pi e^{\frac{i \pi}{2} \beta}}{\alpha 2^{\alpha-1} B \left(\frac{\alpha+\beta+1}{2}, \frac{\alpha-\beta+1}{2} \right)} $$ where $\beta + 1> \alpha > 0$ and $B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$ … Continue reading
Evaluating very nasty logarithmic integrals: Part III
This is part 3 of our series on very nasty logarithmic integrals. Please have a look at part 1 and part 2 before reading this post. Integral #5 The first integral that we will evaluate in this post is the … Continue reading
Evaluating very nasty logarithmic integrals: Part II
In this post, we’ll evaluate some more nasty logarithmic integrals. Please read part 1 of this series if you haven’t done so already. Integral #3 We’ll start by finding a closed form for the integral: $$ I_1 = \int_0^1 \frac{\log^2(1+x^2)}{1+x^2}dx … Continue reading