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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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Could you share the numerical values of a, b, c and d? That may help. –  drN Apr 20 at 22:13
    
yes:a=1.5, b=2, c=0.8, d=1 –  betatron Apr 20 at 22:49
    
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 –  Nasser Apr 20 at 22:57
    
therefore it is not possible to have a solution ? –  betatron Apr 20 at 23:06

1 Answer 1

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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Please, if x=m+In, how to find m=Re[x] and n=Im[x] –  betatron Apr 21 at 0:02
    
Many tanks Chris –  betatron Apr 21 at 0:02

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