Find complex roots from a nonlinear equation

2
votes

I need to find the roots from the equations as follows (Mathematica):

Sqrt[3]/2*x-(I-x*Sqrt[3]/2*c^2)*I/Sqrt[2*Pi]/d^3*Integrate[t*Exp[-t^2/2/d^2]/(Sqrt[3]/2*x+I*(t+b0)),{t,-Inf,Inf}]=0

i.e. as the picture shows:

enter image description here

where c, d, and b0 is given parameters, x is a complex root needs to find. I have tried several methods, including scanning the real and imagine part of x and the iteration approach, but non of them could resolve all the cases.

Are there any general approaches that could solve such kind of equation efficiently, or by MATLAB/Mathematica?

1
matlabwolfram-mathematica
Maybe differentiate it and use odesolve? - Jorge
Actually, command fsolve in MATLAB can solve some cases for b0 > 0, but failed at b0 < -1. - Tong
You have syntax errors (i->I, pi->Pi); once you fix them the integral can be done analytically and you can use FindRoot to solve the equation. - b.gatessucks
Thanks for the tips. However, does anyone know the general approaches besides the FindRoot or fsolve? - Tong
The integration looks doable analytically from a first look. Checking that now... - rubenvb

1 Answers

1
votes

did you try Matlab's mupad? it is a powerful symbolic tool, very similar to Maple wich gives really good results in non-numerical mathematics. Try there. declare the equation, give assumptions to the software ,i.e assume c real positive (don't copy this, I dont remember the proper syntax) and then solve! It will very likely find a solution if it exits (sometimes in some mathematical cases that you even don't know!)