Transforming an elliptic integral with arbitrary coefficients and imaginary roots

Question

I'm trying to solve the integral $$\int_0^\pi \dfrac{\text{d}\phi}{\sqrt{\cos^2(\phi) - a \cos(\phi) + b}} = \int_{-1}^1 \dfrac{\text{d}x}{\sqrt{1-x^2}\sqrt{x^2-ax+b}} $$ where $a$ and $b$ are arbitrary positive real numbers, and $a^2-4b<0$.

I know that in theory the solution can be written in terms of elementary functions and standard elliptic integrals, but I'm struggling to put it in that form. I've tried working out a change of variables like $$x = \dfrac{rt+s}{ut+v}$$ to simplify the radicand for comparing against tables (Byrd and Friedman or Gradshteyn and Ryzhik), but since my polynomial has imaginary roots I keep getting stuck. Computer algebra like Wolfram Alpha also seems to fail.

Is this just impossible, or am I missing something? Thanks.

Possibly my 2-part answer to How to integrate $ \int \frac{x}{\sqrt{x^4+10x^2-96x-71}}dx$? could be of use, or at least the references at the end of the 2nd part. However, I haven't looked at this stuff since then, and it was all done in a couple of weeks (and I'd never worked with elliptic integrals before then), so I can't offer anything more specific about your question without spending several hours getting back into this subject. — Dave L. Renfro, Feb 26 '21 at 18:27
@DaveL.Renfro Wow !!!. You already wrote a book in the above link. It was marvelous. Thanks. — Felix Marin, Mar 12 '21 at 21:03

score 1 · Accepted Answer · answered Mar 12 '21 at 19:01

1

You didn't look hard enough. The relevant formula is Byrd & Friedman 259.00.

Since $x^2-ax+b$ has complex roots, rewrite it as $(x-c)^2+d$. Define $$A^2=(1-c)^2+d\qquad B^2=(1+c)^2+d\qquad g=\frac2{\sqrt{AB}}\qquad m=\frac{4-(A-B)^2}{4AB}$$ Then the given integral evaluates to $gK(m)$ where $K$ is the complete first-kind elliptic integral and $m$ is the parameter.

answered Mar 12 '21 at 19:01

Parcly Taxel

103,344

I didn't understand your reference: "Byrd & Friedman 259.00". I couldn't find any thing about it. – Felix Marin Mar 12 '21 at 21:26
@FelixMarin The OP mentioned that book in the question. The actual title is Handboook of Elliptic Integrals for Engineers and Physicists - the standard reference for these objects. – Parcly Taxel Mar 13 '21 at 03:57
(It also has an A&S-like numbering scheme, which is where the 259.00 comes from.) – Parcly Taxel Mar 13 '21 at 04:07
Thanks. I appreciated. – Felix Marin Mar 14 '21 at 00:29

Transforming an elliptic integral with arbitrary coefficients and imaginary roots

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