# How to solve this integral equation?

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?

• Could you share the numerical values of a, b, c and d? That may help. – dearN Apr 20 '14 at 22:13
• yes:a=1.5, b=2, c=0.8, d=1 – Betatron Apr 20 '14 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 '14 at 22:57
• therefore it is not possible to have a solution ? – Betatron Apr 20 '14 at 23:06

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}]


• Please, if x=m+In, how to find m=Re[x] and n=Im[x] – Betatron Apr 21 '14 at 0:02
• Many tanks Chris – Betatron Apr 21 '14 at 0:02