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I am trying to solve for x the integral equation :

a Integrate[Exp[-s^2]/(s - c x)^2, {s,-Infinity, Infinity}] + 
b Integrate[Exp[-s^2]/(s - d x)^2, {s,-Infinity, Infinity}] = 1

where a, b, c and d are numerical values. I have tried NIntegrate and some iteration but didn't get me too far... can anyone help me with a better idea?

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    $\begingroup$ Could you share the numerical values of a, b, c and d? That may help. $\endgroup$
    – dearN
    Commented Apr 20, 2014 at 22:13
  • $\begingroup$ yes:a=1.5, b=2, c=0.8, d=1 $\endgroup$
    – Betatron
    Commented Apr 20, 2014 at 22:49
  • $\begingroup$ The problem I think is that you have a pole, when s=c x even though x is unknown, it has to be between -Infinity and +infinitym right? So somewhere along the real line, is a pole. So does not converge $\endgroup$
    – Nasser
    Commented Apr 20, 2014 at 22:57
  • $\begingroup$ therefore it is not possible to have a solution ? $\endgroup$
    – Betatron
    Commented Apr 20, 2014 at 23:06

1 Answer 1

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Perhaps this will help:-

eqn = a Integrate[Exp[-s^2]/(s - c x)^2, {s, -Infinity, Infinity}, 
     Assumptions -> Im[c x] != 0] + 
   b Integrate[Exp[-s^2]/(s - d x)^2, {s, -Infinity, Infinity}, 
     Assumptions -> Im[d x] != 0];

a = 1.5; b = 2; c = 0.8; d = 1;

Plot[eqn, {x, -1000, 1000}]

enter image description here

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  • $\begingroup$ Please, if x=m+In, how to find m=Re[x] and n=Im[x] $\endgroup$
    – Betatron
    Commented Apr 21, 2014 at 0:02
  • $\begingroup$ Many tanks Chris $\endgroup$
    – Betatron
    Commented Apr 21, 2014 at 0:02

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