Articles of elliptic integrals

Elliptical Integrals

I was trying to figure out the length of the arc in a single cycle of a sinusoidal curve and I used the curve length formula to arrive at $$\int_0^{2\pi}\sqrt{1+\cos^2x}\ dx,$$ which I am fairly certain is correct. However, I have no idea how to evaluate this integral and when I looked it up, Mathematica […]

Approximation for elliptic integral of second kind

My (physics) book gives the following approximation: $\int_{-\pi/2}^{\pi/2} \sqrt{1-(1-a^2) \sin(k)^2} dk \approx 2 + (a_1 – b_1 \ln a^2) a^2 + O(a^2 \ln a^2)$ where a1 and b1 are “(unspecified) numerical constants.” I’ve been looking for either a derivation of this, or the same approximation listed elsewhere and have gotten nowhere. Can someone help me […]

An identity of an Elliptical Integral

Suppose $0<k<1$ and $\displaystyle K(k)=\int_0^1\frac{\mathrm{d}x}{\sqrt{(1-x^2)(1-k^2x^2)}}$. Let $\tilde{k}$ be $\tilde{k}^2=1-k^2$. Show that $$\displaystyle K(k)=\frac{2}{1+\tilde{k}}K\left(\frac{1-\tilde{k}}{1+\tilde{k}}\right)$$ There’s a hint in Stein’s Complex Analysis which is this change of variable : $x=\dfrac{2t}{1+\tilde{k}+(1-\tilde{k})t^2}$.

Closed form for the integral $\int_0^\infty \frac{dx}{\sqrt{(x+a)(x+b)(x+c)(x+d)}}$

Let’s consider the function defined by the integral: $$R(a,b,c,d)=\int_0^\infty \frac{dx}{\sqrt{(x+a)(x+b)(x+c)(x+d)}}$$ I’m interested in the case $a,b,c,d \in \mathbb{R}^+$. Obviously, the function is symmetric in all four parameters. This function has some really nice properties. $$R(ka,kb,kc,kd)=\frac{1}{k} R(a,b,c,d)$$ Thus: $$R(a,a,a,a)=\frac{1}{a}$$ Moreover: $$R(a,a,b,b)=\frac{\ln a-\ln b}{a-b}$$ This is the reciprocal of the logarithmic mean of the numbers $a$ and […]

Closed form for an almost-elliptic integral

Does $$\int_0^{2 \pi} \log\left(\frac{1}{2}[1+\sqrt{1-(a \sin\phi)^2}]\right) d\phi $$ have a closed form ? An approximation for small $a$ is $2E-\pi$, but it is the exact form that is needed for any $|a|<1$. The integration standing to Jack is $$ -\frac{\pi a^2}{4}\phantom{}_4 F_3\left(1,1,\frac{3}{2},\frac{3}{2};2,2,2;a^2\right) $$ The question now is: Can it be changed into a finite combination of […]

How to compute elliptic integrals in MATLAB

I need to calculate the complete elliptic integrals of the first and second kind , the incomplete elliptic integral of the first kind, and the incomplete elliptic integral of the second kind in MATLAB. MATLAB has built in functions to calculate these functions, as I have shown in the links above, however I am not […]

To find area of the curves that are extension of ellipse

I like to draw an ellipse via 2 fixed points and a rope between the fixed points (2 focuses). I wanted to extend the idea. Point A,B,C,D are fixed points and Point E can move freely. Point E,B,C have small pulleys without friction and also their perimeters are very small. (Take zero for theoretical calculation) […]

Evaluation of complete elliptic integrals $K(k) $ for $k=\tan(\pi/8),\sin(\pi/12)$

This is inspired from here. I will repeat some information from the linked question for the benefit of readers. Let $k\in(0,1)$ and the elliptic integrals $K, E$ are defined as follows: $$K(k)=\int_{0}^{\pi/2}\frac{dx}{\sqrt{1 – k^{2}\sin^{2}x}},\,E(k)=\int_{0}^{\pi/2}\sqrt{1-k^{2}\sin^{2}x}\,dx\tag{1}$$ The number $k$ is called the modulus and a complementary modulus $k’$ is defined by $k’=\sqrt{1-k^{2}}$ and if the value of […]

Closed-form of sums from Fourier series of $\sqrt{1-k^2 \sin^2 x}$

Consider the even $\pi$-periodic function $f(x,k)=\sqrt{1-k^2 \sin^2 x}$ with Fourier cosine series $$f(x,k)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos2nx,\quad a_n=\frac{2}{\pi}\int_0^{\pi} \sqrt{1-k^2 \sin^2 x}\cos 2nx \,dx.$$ This was considered in this earlier question, and in comments an observation was made: The Fourier coefficients all appear to be of the form $a_n= A_n(k) K(k)+B_n(k) E(k)$ where $K,E$ are the complete elliptic […]

Derivative of the elliptic integral of the first kind

The complete elliptic integral of the first kind is defined as $$K(k)=\int_0^{\pi/2} \frac{dx}{\sqrt{1-k^2\sin^2{x}}}$$ and the complete elliptic integral of the second kind is defined as $$E(k)=\int_0^{\pi/2}\sqrt{1-k^2\sin^2{x}}~dx$$ for $0\leq k<1$. I’m supposed to prove the following relation $$K'(k)=\frac{E(k)}{k(1-k^2)}-\frac{K(k)}{k}.$$ What I tried so far Without much thought about the exchange of integration and differentiation I tried to […]