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Re: elimination of a real variable from a complex function
*To*: mathgroup at smc.vnet.net
*Subject*: [mg75452] Re: elimination of a real variable from a complex function
*From*: dh <dh at metrohm.ch>
*Date*: Tue, 1 May 2007 03:25:11 -0400 (EDT)
*References*: <f146hb$m05$1@smc.vnet.net>
HI Gautam,
First, note that there is a syntax error:... * Cos*(k*z + ...
Further, the integral must give -I \[lambda]. Therefore, if you
calculate the series of the integral around \[lamda]==0, all
coefficients except the linear one are zero. If you are lucky, this may
give you equations that allow you to determine the constants.
hope this helps, Daniel
Gautam Sethia wrote:
> hi..
> I am doing the stability analysis of the solutions of a complex integral
> equation. In this context I am facing a problem which is detailed below.
> Could some body help me resolve the problem.
> Thanks
> Gautam C Sethia
> ***************************************
> The following complex integral equation is in a single real variable
> \[Lambda] and a number of real parameters viz. A, \[CapitalOmega], v,
> \[Alpha], k and n. The last two parameters i.e. k and n can have only
> integer values. My objective is to eliminate \[Lambda] from this equation
> and have an equation only in terms of the parameters. This should in
> principle be possible as the complex equation can be reduced to a system of
> two real equations and a single real variable \[Lambda] and that should
> enable us to eliminate \[Lambda] from these two equations.
>
> \[ImaginaryI]\[Lambda] + Integrate[((A*Cos[z] + 1)*Cos*(k*z + \[Alpha] +
> (\[CapitalOmega]*Abs[z])/v)*
> (1 - E^((-I)*(n + (\[Lambda]*Abs[z])/v))))/(2*Pi), {z, -Pi, Pi}] = 0
>
> I tried the following:
> 1. I wrote the above equation as a set of two equations by separating into
> real and imaginary parts. That is facilitated by the fact that variable
> \[Lambda] as well as all other parameters are real.
> 2. In each of these two equations, the integration could be done separately
> from -\[Pi] to 0 and 0 to \[Pi] so that we can get rid of the Abs function
> on z.
> 3. The right hand side of these two equations can then be integrated in
> mathematica (it takes about 10-15 minutes on my laptop) giving horrendous
> looking analytical expressions.
> 4. I am at a loss at this point and have no clue how to eliminate \[Lambda]
> from these two equations involving those two expressions.
>
> Could you please suggest me a way to go about achieving my objective using
> mathematica. Thanks...Gautam C Sethia
>
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