       Re: Help to solve an integral by using Mathematica Integrate[Sqrt[t

• To: mathgroup at smc.vnet.net
• Subject: [mg112852] Re: Help to solve an integral by using Mathematica Integrate[Sqrt[t
• From: Valeri Astanoff <astanoff at gmail.com>
• Date: Sun, 3 Oct 2010 05:41:08 -0400 (EDT)
• References: <i84adn\$gvn\$1@smc.vnet.net> <i84bkk\$hqk\$1@smc.vnet.net> <i86v14\$hqv\$1@smc.vnet.net>

```On 2 oct, 11:46, Hugo <hpe650... at gmail.com> wrote:
> Valeri,
>
> Suggested solution is very close to solution found in book. How did
> you get it? Please, let me know how did you implement this integral.
>

Good day,

Leonid's solution :
Re[-(2/15)
I (2 (1 + (-1 + z) z) EllipticE[1 - z] -
2 (1 + (-1 + z) z) EllipticE[I ArcCoth[Sqrt[z]], 1 - z] +
z (1 + z) (EllipticF[I ArcCoth[Sqrt[z]], 1 - z] -
EllipticK[1 - z]))];

and my own :

(1/15)*(4*(1 + (z-1)*z)*EllipticE[z] - 2*(2 + (z-3)*z)*EllipticK[z])

are numerically identical when z lies between 0 and 1.

I observed that for most rational values of z,
the solution was a linear combination of EllipticE
and EllipticK. Then I performed a polynomial fitting
of the coefficients.
Well, I confess I sort of guessed it...
--
Valeri

```

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