1
vote
0answers
52 views

Closed form of $\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1)$

Is there a known closed form of the series below? $$\sum_{n=1}^\infty(-1)^{n+1} \frac xn \ln\left(1+\frac xn\right), \quad x \in (0,1]$$
7
votes
0answers
129 views

The closed form of $\sum_{n=0}^{\infty} \arcsin\left(\frac{1}{e^n}\right)$

In my study on some type of integrals I met the series below that I don't how to approach it. Of course, one of the obvious questions is: does it have a closed form? Before answering that, I need to ...
10
votes
5answers
280 views
10
votes
1answer
164 views

How to prove $\int_1^\infty\frac{K(x)^2}x dx=\frac{i\,\pi^3}8$?

How can I prove the following identity? $$\int_1^\infty\frac{K(x)^2}x dx\stackrel{\color{#B0B0B0}?}=\frac{i\,\pi^3}8,\tag1$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind: ...
1
vote
1answer
94 views

The transformations on the nome and Landen's transformation

Could someone please explain how to transform the nome $q = e^{-\pi K'/K}$ from $q^2$ to $q$ and then to $-q$? In other words, how does changing $q^2$ to $q$ and then $q$ to $-q$ affect $k$ and $K$. ...
11
votes
2answers
258 views

Show that $K=K'$ if and only if $k=(\sqrt{2}-1)^2$ (Ahlfors)

In Ahlfors' Complex Analysis text, page 240, he defines the following two integrals: $$K=\int_{-1}^1 \frac{dt}{\sqrt{(1-t^2)(1-k^2 t^2)}}, $$ $$K'=\int_1^{1/k} \frac{dt}{\sqrt{(t^2-1)(1-k^2t^2)}}. $$ ...
3
votes
1answer
115 views

Difficulties with an elliptic integral (Ahlfors)

In Ahlfors' Complex Analysis text, page 239 he defines $$F(w)=\int_0^w \frac{dw}{\sqrt{(1-w^2)(1-k^2w^2)}} $$ where $0<k<1$ is a constant. He notes this time we agree that ...
2
votes
0answers
171 views

Separate incomplete elliptic integral into real and imaginary parts

I am working in a problem that involves Incomplete Elliptic Integrals of the First and Second kind of the form $F(\sin^{-1}x~|~m)$ and $E(\sin^{-1}x~|~m)$ where the parameters $m$, $x$ are real ...
1
vote
0answers
55 views

Looking for examples where $f(z)=\operatorname{inv} \int_{0}^{z} g(z)\, dz$ with $f(z)$ entire and $g(z)$ not meromorphic.

I'm looking for examples where $f(z)=\operatorname{inv}\int_{0}^{z} g(z) \, dz$ with $f(z)$ entire and $g(z)$ not meromorphic. For clarity, by $\operatorname{inv}$, I mean the functional inverse. ...
1
vote
0answers
149 views

Complete Elliptic Integral of the 3rd Kind - Residual Computation

Let us consider the following function $f(a,k)$ in the interval $a,k\in (0,1]$ : $$f(a,k)=\frac{2 \sqrt{1-a^2} \sqrt{a^2-k^2}}{\sqrt{a^2}}\Pi\left(a^2,k^2\right)$$ where $\Pi\left(a^2,k^2\right)$ is ...
5
votes
1answer
245 views

Topology of Branch Cuts and Elliptic Integrals

In reading these notes (elliptic curves starting from elliptic integrals) I came across a couple claims about the topology of some complex surfaces. On page 4, they discuss the integral $$\phi(x) = ...