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

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

- Thread starter nanath
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- #1

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

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

- #2

Stephen Tashi

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Char. Limit

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[tex]erfc(z) \equiv 1 - erf(z)[/tex]

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

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

and furthermore that

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

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

[tex]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[/tex]

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

- #4

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By definition of erfc :

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

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

http://mathworld.wolfram.com/Erfc.html

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- #6

Char. Limit

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- #8

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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

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