Help needed with Elliptic Integrals

[TerryW]

Homework Statement

See Attempt at a Solution

Relevant Equations

See Attempt at a Solution

I want to find the solution to the integral

$\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}$

I can see that $\frac{d^2u}{d\theta^2} = A +Bu+Cu^2$ is a Weierstrass elliptic function, which can be generated from $\Large(\normalsize\frac{du}{d\theta}\Large)\normalsize^2 = c-u^2 +2u^3$ (A = 0, B=-1, C=3)

So does this make my integral an elliptic integral? I haven't been able to find a table of integrals anywhere which contains an integral of this form so I'm a bit stuck.

TerryW

 

[renormalize]

I want to find the solution to the integral

$\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}$

Mathematica can evaluate the indefinite integral on the right side:

Here, $\text{EllipticF}\left[\phi,m\right]$ is the elliptic integral of the 1rst kind $F\left(\phi\mid m\right)$ and the $r_k$ are the 3 roots of the cubic polynomial equation $c-r^2+2\,r^3=0$. According to the documentation, the roots are indexed as follows:
"The root indexing representation $\text{Root}\left[f,k\right]$ applies to polynomial functions $f$ only. The indexing of roots takes the real roots first, in increasing order. For polynomials with rational coefficients, the complex conjugate pairs of roots have consecutive indices."

 

[TerryW]

Many thanks for this.

I think I can console myself that I didn't miss an easy route to a solution! It's interesting that the elliptic integral of the first kind appears in the solution, but where the rest of it comes from may remain a mystery to me.

I'm still puzzled by the fact that on the one hand there are statements like:

"Modern mathematics defines an 'elliptic integral' as any function $f$ which can be expressed in the form $f(x) = \int_c^xR(t,\sqrt{P(t)})dt$" (from Wikipedia)

but then Wikipedia does not go beyond the elliptic integral of the third kind.

Do you know of any reference for elliptic integrals in general?

TerryW

 

[renormalize]

Do you know of any reference for elliptic integrals in general?

There are doubtless entire monographs devoted to elliptic integrals, but one concise reference is:
Handbook of Mathematical Functions by Abramowitz and Stegun, chap. 17
I happen to have a print copy (from 1968!) but you can find PDF versions floating around on the web.

 

[TerryW]

I've found a pdf of Abromowitz and Stegun. I'll see if it clarifies things for me.

Looking at the result from Mathematica, I wonder if it is actually correct? For example, the term $(r2-r3)^2$ on the denominator of the second square root cancels out the term $(r2-r3)$ at the beginning of the expression. Also, if r1, r2, and r3, are the roots of my cubic, doesn't $\sqrt{(r1-u)}\sqrt{(r2-u)(-r3+u)}$ cancel out $\sqrt{(c+u^2(-1+2u)}$?

Or am I just misunderstanding the formulation?

Maybe I should try and find the derivative of the expression WRT u.

Regards and thanks for the reference.

TerryW

 

[renormalize]

Looking at the result from Mathematica, I wonder if it is actually correct? For example, the term $(r2-r3)^2$ on the denominator of the second square root cancels out the term $(r2-r3)$ at the beginning of the expression. Also, if r1, r2, and r3, are the roots of my cubic, doesn't $\sqrt{(r1-u)}\sqrt{(r2-u)(-r3+u)}$ cancel out $\sqrt{(c+u^2(-1+2u)}$?

Good observation! You're absolutely correct, up to a sign: depending on the specific values of the roots $r_2,r_3\,$, the two ratios you cite reduce to:$$\frac{r_{2}-r_{3}}{\sqrt{\left(r_{2}-r_{3}\right)^2}}=\pm1,\quad\frac{\sqrt{(r_1-u)}\sqrt{(r_2-u)(-r_3+u)}}{\sqrt{(c+u^{2}(-1+2u)}}=\frac{1}{\sqrt{2}}$$so the Mathematica expression for the integral becomes simply:$$\pm\,\frac{\sqrt{2}\,F\left(\sin^{-1}\left(\frac{u-r_{3}}{r_{2}-r_{3}}\right)\mid\frac{r_{2}-r_{3}}{r_{1}-r_{3}}\right)}{\sqrt{r_{1}-r_{3}}}$$You'll have to determine the overall sign by examining the roots of your particular cubic.