Author Archives: Shobhit Bhatnagar
Introduction to Theta Functions I
The Jacobi theta functions are defined for all complex variables of $z$ and $q$ such that $|q| < 1$, as follows: $$ \begin{aligned} \vartheta_1 (z,q) &= -i \sum_{n=-\infty}^{\infty} (-1)^n q^{(n+1/2)^2} e^{i(2n+1)z} \\ \vartheta_2 (z,q) &= \sum_{n=-\infty}^{\infty} q^{(n+1/2)^2} e^{i(2n+1)z} \\ \vartheta_3 … Continue reading
When is the group of units in the integers modulo n cyclic?
It is easy to see with the help of Bezout’s identity that the integers co-prime to $n$ from the set $\{0,1,\cdots, n-1\}$ form a group under multiplication modulo $n$. This group is denoted by $(\mathbb{Z}/n\mathbb{Z})^*$ and it’s order is given … 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