What is the analytical expression for the error function and its integrals?

[nanath]

I would like to know how the following integral for the error function gets derived (found by following the link):

http://www.wolframalpha.com/input/?i=integrate+exp(-x^2)+from+x0+to+inf

Note: this is not a homework question, merely a query.

 

[Stephen Tashi]

What constitutes an explanation will depend on your background. Do you know about integration techniques such a changing variables? And do you know that the definition of erfc(x) itself involves an integral?

 

[Char. Limit]

Well, the definition of erfc(z) is actually:

erfc(z) \equiv 1 - erf(z)

However, this can be pretty easily changed to the integral definition if we remember that

erf(z) \equiv \frac{2}{\sqrt{\pi}} \int_0^z exp\left(-t^2\right) dt

and furthermore that

\frac{2}{\sqrt{\pi}} \int_0^\infty exp\left(-t^2\right) dt = 1

By substituting these definitions in for 1 and erf(z), we get this expression:

erfc(z) = \frac{2}{\sqrt{\pi}} \int_0^\infty exp\left(-t^2\right) dt - \frac{2}{\sqrt{\pi}} \int_0^z exp\left(-t^2\right) dt

Which is easily proven using basic integral properties to be equal to the expression in the OP.

 

[JJacquelin]

I would like to know how the following integral for the error function gets derived (found by following the link):

http://www.wolframalpha.com/input/?i=integrate+exp(-x^2)+from+x0+to+inf

Note: this is not a homework question, merely a query.

By definition of erfc :
http://mathworld.wolfram.com/Erfc.html

 

[nanath]

Thanks for the derivation. My final question is how one would analyze this integral (from 0 to z as you have it). Is it along the same lines we go about to derive the value of the error function from -inf to +inf?

 

[Char. Limit]

We usually just find values numerically. The only value that I know of where erf(x) takes a simple, closed-form value is x=0. Other than that, it's all approximation.

 

[nanath]

Thanks for your timely response. Although knowing that it has to be found numerically makes me slightly unhappy I'm glad to find that I wasn't missing anything more fundamental.

 

[JJacquelin]

Thanks for your timely response. Although knowing that it has to be found numerically makes me slightly unhappy I'm glad to find that I wasn't missing anything more fundamental.

The erfc(x) is a function which is used exactly like the functions exp(x), ln(x), cos(x) or many others. All these functions have to be found numerically (except for a few particular values of x).
For example, what is the analytical expressiion of :
integrate (1/t)*dt from t=1 to t=x ?
Of course, the answer is ln(x). Then would you say "knowing that it has to be found numerically makes me slightly unhappy" ?

The only difference between erfc(x) and ln(x) is that one is familiar to you and the other not.

An almost similar question arised elsewhere about the integration of x^x. The consequence was a funny discussion reported in the paper "The Sophomores Dream Function" :
http://www.scribd.com/JJacquelin/documents