Definite integral of exp and error function

[petru]

Hi,

I've been trying to evaluate the following integral

[tex] \int_{z}^{\infty}\exp\left(-y^{2}\right)\mathrm{erf}\left(b\left(y-c\right)\right)\,\mathrm{d}y [/tex]

or equivalently

[tex] \int_{z}^{\infty}\exp\left(-y^{2}\right)\mathrm{erfc}\left(b\left(y-c\right)\right)\,\mathrm{d}y [/tex]

[tex]\mathrm{erf}\left(x\right)=\frac{2}{\sqrt{\pi}} \int_{0}^{x}\exp\left(-u^{2}\right)\,\mathrm{d}u, \quad\quad \mathrm{erfc}\left(x\right)=1-\mathrm{erf}\left(x\right)=\frac{2}{\sqrt{\pi}} \int_{x}^{+\infty}\exp\left(-u^{2}\right)\,\mathrm{d}u[/tex]

I guess I tried to employ all techniques I'm familiar with but with no result.
Can anyone help me with this one, please?
Thank you!

 

[mathman]

To the best of my knowledge, it can't be done analytically. I suggest you start with the erf representation and see if the two exponentials might be combined into one, so that you might have an erf for the integral.

 

[petru]

Thanks mathman for your reply. I guess I'm not able to deal with this integral. I have a question though. I'm not a mathematician nor a math student so I was wondering if anyone could explain to me why the integral

[tex]\int_{-\infty}^{\infty}\exp\left(-y^{2}\right) \mathrm{erf}\left(b\left(y-c\right)\right)\,\mathrm{d}y=-\sqrt{\pi}\,\mathrm{erf}\left(\frac{bc}{\sqrt{1+b^{2}}}\right)[/tex]

can be evaluated quite easily (using differentiation under integral sign method) and the integral from my original post seems to have no analytical solution?

Thanks!

 

[mathman]

I haven't looked at it in detail, but it looks like the problem is analogous to integrating the Gaussian. When you integrate over the entire real line you get a neat analytic solution, but integrating over part of the line ends up with erf.

 

[petru]

Ok, I guess I know what you mean. Thanks again!

 

[JJacquelin]

Hi !

in attachment, a method for solving the definite integral.

 

[petru]

Hi JJacquelin! Your post helped me with showing that the constant of integration [tex]C=0[/tex] in a more general formula:

[tex]
\int_{-\infty}^{\infty}\exp\left(-b^{2}(x-c)^{2}\right)\mathrm{erf}\left(a\left(x-d\right)\right)\,\mathrm{d}x=\frac{\sqrt{\pi}}{b} \mathrm{erf}\left(\frac{ab\left(c-d\right)}{\sqrt{a{}^{2}+b{}^{2}}}\right),\quad b>0
[/tex]

Thank you!