Articles of exponential sum

log of summation expression

I am curious about simplifying the following expression: $\log \left(\sum_\limits{i=0}^{n}x_i \right)$ Is there any rule to simplify a summation inside the log?

Expected time for the queue to become empty.

Question: Suppose, there is a queue A having i customers initially. The service time of the queue is Exponentially distributed with parameter $\mu$ (i.e., mean service time of one customer in Queue A is 1/$\mu$). Arrival to queue A has Poisson distribution with mean $\lambda$. What is the expected time before queue A becomes empty […]

Determining the Value of a Gauss Sum.

Can we evaluate the exact form of $$g\left(k,n\right)=\sum_{r=0}^{n-1}\exp\left(2\pi i\frac{r^{2}k}{n}\right) $$ for general $k$ and $n$? For $k=1$, on MathWorld we have that $$g\left(1,n\right)=\left\{ \begin{array}{cc} (1+i)\sqrt{n} & \ \text{when}\ n\equiv0\ \text{mod}\ 4\\ \sqrt{n} & \text{when}\ n\equiv1\ \text{mod}\ 4\\ 0 & \text{when}\ n\equiv2\ \text{mod}\ 4\\ i\sqrt{n} & \text{when}\ n\equiv3\ \text{mod}\ 4 \end{array}\right\} .$$ I know how to […]

Plotting exponential partial sums in the complex plane

I was plotting the following sequence of points $(a_n)_{n = 0}^\infty$ in the complex plane for various reals $\alpha > 1$: $$a_n = \sum_{k = 0}^n e^{i k^\alpha}$$ I found that for many values of $\alpha > 1$ the path traced by $(a_n)$ displayed interesting nonrandom and even sometimes chaotic behaviour. The following images show […]

Closed form for an infinite sum over Gamma functions?

I am having quite a bit of trouble trying to find a closed form (or a really fast way to compute) for the infinite sum $$\sum_{n=1}^{\infty} a^n \dfrac{\gamma(n+1,b)}{\Gamma(n+1)\Gamma(n)}$$ where $\gamma(n,b) = \int_0^bz^{n-1}e^{-z} dz$ is the lower incomplete Gamma function and $a,b > 0$. Any ideas how to solve this?

An alternating series identity with a hidden hyperbolic tangent

How does one use the inverse Mellin transform to prove that the following identity holds? $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n(e^{n\pi} + 1)} = \frac{1}{8}(\pi – 5\log(2))$$ The identity follows from MO189199 and the penultimate identity on this page (for $x=\frac{1}{2}$). Scratch-work: I computed the Mellin transform of $$f(x) = \frac{(-1)^x}{x(e^{x\pi} + 1)}$$ and re-wrote the function in terms of […]

Prove the following series $\sum\limits_{s=0}^\infty \frac{1}{(sn)!}$

This question already has an answer here: Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$ 2 answers

Baker-Hausdorff Lemma from Sakurai's book

I’d like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] + \frac{\left(i\lambda\right)^2}{2!}[G,[G,A]]+\ldots+\frac{(i\lambda)^n}{n!}\underbrace{[G,[G,[G,\ldots[G}_{n\ times},A]]]\ldots]+\ldots $$ where $\lambda$ denotes a real parameter and $[\ \!,]$ indicates the commutator. This is a proof left to the reader by Sakurai in his […]

Solve equation with unknown in exponents

This is in continuation of this but not related to it completely. I am interested in finding a solution to the equation: $m’ = m – \sum \limits_{j=1}^{m} (1 – d_{O_j}/n)^k$. where $m,m’,n$ and $d_{O_j}$ for $j \in {1,2,…m}$ are known and are positive. The only unknown is $k$. How can I approach this problem? […]