       Forcing a parameter to be integer when using 'Integrate'

• To: mathgroup at smc.vnet.net
• Subject: [mg67545] Forcing a parameter to be integer when using 'Integrate'
• From: Ian Linington <i.e.linington at sussex.ac.uk>
• Date: Fri, 30 Jun 2006 04:14:36 -0400 (EDT)
• Organisation: University of Sussex, IT Services
• Sender: owner-wri-mathgroup at wolfram.com

```Hello,

I am a mathematica novice, so please excuse me if this question is a
bit dumb...

I would like to solve the integral:

> Integrate[E^((n*I)*x)/Sqrt[1 + a^2*Sin[x]^2], {x,0,2*Pi}]

with n integer and a real and positive.
The problem is that I don't know how to tell Mathematica about these
conditions on a and n.

If I explicitly assign an integer value for n (i.e. 4 in this example),
Mathematica solves the integral:

> Integrate[E^((4*I)*x)/Sqrt[1 + a^2*Sin[x]^2], {x,0,2*Pi}]

(with a few conditions) and gives the output

> If[(Re[a^(-1)] != 0 || -Im[a^(-1)] < 0) &&
(Re[a^(-1)] != 0 || -Im[a^(-1)] >= 0) && ...
... (lots more)...,
(4*(-8*(2 + a^2)*EllipticE[-a^2] +
(4 + a^2)*(4 + 3*a^2)*EllipticK[-a^2]))/(3*a^4),
Integrate[E^((4*I)*x)/Sqrt[1 + a^2*Sin[x]^2],
{x, 0, 2*Pi},
Assumptions -> !((Re[a^(-1)] != 0 || -Im[a^(-1)] < 0)
&& (Re[a^(-1)] != 0 || -Im[a^(-1)] >= 0) && ...)
... (lots more) ...]]

Simlilarly if we pick other integer values for n.

The question is, how to get a generic formula in terms of a and n?