• Sep 17th 2008, 07:05 AM
minivan15
[integral from 0 to pi/2] dx/SQRT(a - cosx)

(a >1)

I need to write this in terms of the elliptic integrals F

the elliptic integral of the first kind is F(k,q) = [integral from 0 to q]
$
du/SQRT(1 - (ksinu)^2)
$

I do not see a starting point on this! :(

I know elliptic integrals are tricky and ambiguous and all that...I merely need to REWRITE the given integral in terms of F

• Sep 17th 2008, 12:45 PM
shawsend
If $\int_{0}^{\pi/2}\frac{dx}{\sqrt{a-\cos(x)}}=\frac{2}{\sqrt{a-1}}\int_0^{\pi/4}\frac{du}{\sqrt{1-\frac{2}{a-1}\sin^2(u)}}$

can you use the identity $\sin^2(x/2)=\frac{1-\cos(x)}{2}$ to get there? Just make the substitution for $cos(x)$, and another one $u=x/2$.
• Sep 17th 2008, 09:01 PM
minivan15

SQRT(a -(1 - 2(sinx/2)^2) = SQRT( a - 1 + 2(sinx/2)^2)

then factoring out a - 1 gives SQRT(a-1) * SQRT(1 + 2/(a-1)*(sinx/2)^2)

where k^2 is both negative and not necessarily less than one...the conditions for k in elliptic integrals are 0< k <1

did I do something wrong, or is the idea slightly off? I tried playing around with the concept but didn't get anywhere. Is there another answer or is this one correct but I'm just doing it incorrectly?
• Sep 18th 2008, 03:09 AM
shawsend
I'm not sure. Sorry. Looks then to me $a$ needs to be greater than $3$.
• Feb 18th 2014, 12:32 AM
prash
I want to solve this eqn using the elliptic integral
int(1/sqrt(1-ksin(phi)))
• Feb 18th 2014, 01:03 AM
romsek
Quote:

Originally Posted by prash
I want to solve this eqn using the elliptic integral
int(1/sqrt(1-ksin(phi)))

do you mean

$$\int \frac{1}{\sqrt{1- k\sin(\phi)}}d\phi$$

if so it's

$$-\frac{2 F\left(\frac{1}{4}(\pi-2\phi),\frac{2k}{k-1}\right) \sqrt{\frac{k\sin(\phi)-1}{k-1}}}{\sqrt{1-k\sin(\phi)}}$$

$$F(\phi,m)\mbox{ is the elliptic integral of the first kind.}$$
• Feb 18th 2014, 01:06 AM
prash
I don understand about the reversal of polarity statement.I am new to elliptic integrals,i do have one doubt.whether i can get a variable solution when i use elliptical integral or only numerical solution is possible?
• Feb 18th 2014, 01:07 AM
romsek
my mistake, getting the tex correct was a pain, it's corrected.
• Feb 18th 2014, 01:09 AM
prash
@Romsek,Can u elaborate me with the proof for the solution
• Feb 18th 2014, 01:23 AM
romsek