Author Archives: Shobhit Bhatnagar
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: $$ I_1 = \int_0^1 \frac{\log^2(1+x^2)}{1+x^2}dx … Continue reading
Evaluating very nasty logarithmic integrals: Part I
In this post, we’ll continue our exploration of logarithmic integrals and Euler sums. We’ll also discuss the contour integration method for evaluating Euler sums. It is recommended that the reader goes through the previous posts, (A) and (B), before reading … Continue reading
Euler Sums involving square of Harmonic numbers
In my previous post on Euler sums, we evaluated sums containing $H_n$ and $H_n^{(2)}$. In this post, we’ll derive some further results using the integral $\int_0^x \frac{\log^3(1-t)}{t}dt$. Our starting point is the following generating function identity: $$ \sum_{n=1}^\infty (H_n)^2 x^n … Continue reading