Help required for logarithmic equation

[Delta11]

Need help solving a logarithmic equation:
(1/x) . exp(-x^2) = 10^-6.
Thanks.

 

[albbg]

If you just need a numerical result, you can use the free math tool Scilab. Code you have to write is:

deff('[y]=funct(x)','y=exp(-x^2)/x-1e-6');

[xres]=fsolve(1,funct)

from which you will obtain x=3.5426862

 

[_Eduardo_]

You may solve it by iteratives methods.

(1/x) . exp(-x^2) = 10^-6 --> x_n = SQRT(LN(10^6/x_n_1))

Starting with x = 1 converge in few steps to 3.542686172...

 

[Delta11]

Thanx for the help guys. Is there any other approach to solving this problem other than iterative method? Numerical iteration obviously fits but i am looking for a more conventional solution. It is a second order logarithmic equation after all!!!

 

[_Eduardo_]

It cannot be expressed by elementary functions.

You need special functions like the Lambert W-function (see Google).

If exp(-x^2)/x = a then x = sqrt( W(2/a^2) / 2 )

with W() the Lambert W-function

But... to obtain W() you need numerical methods or need software like Mathematica with the ProductLog[z] function...

 

[amriths04]

use first few terms(depending on the accuracy) of the taylor's series of the exponential and divide by x. Now it's just solving a polynomial.