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Topic: Is there a way to integrate and differentiate Erfi?
Replies: 1   Last Post: Feb 21, 2013 5:46 AM

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Bob Hanlon

Posts: 906
Registered: 10/29/11
Re: Is there a way to integrate and differentiate Erfi?
Posted: Feb 21, 2013 5:46 AM
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Works fine here

D[Erfi[x], x]


% == Limit[(Erfi[x + d] - Erfi[x])/d, d -> 0]


Integrate[Erfi[x], x]

-(E^x^2/Sqrt[Pi]) + x*Erfi[x]

D[%, x] == Erfi[x]


Integrate[Erfi[x], {x, a, b}]

(-a)*Erfi[a] + (E^a^2 - E^b^2 + b*Sqrt[Pi]*Erfi[b])/

SeriesCoefficient[Erfi[x], {x, 0, n}]

Piecewise[{{2/(n*Sqrt[Pi]*((1/2)*(-1 + n))!),
Mod[n, 2] == 1 && n >= 0}}, 0]

The series expansion for Erfi[x] is then

Sum[2/((2 n + 1) Sqrt[Pi] n!) x^(2 n + 1),
{n, 0, Infinity}]


Integrating term-by-term

Sum[2/((2 n + 1) Sqrt[Pi] n!)*
Integrate[x^(2 n + 1), x],
{n, 0, Infinity}] // Simplify

-((-1 + E^x^2)/Sqrt[Pi]) + x*Erfi[x]

Note that this differs from earlier result by an arbitrary constant of
integration but its derivative is still Erfi[x]

D[%, x] == Erfi[x]


Bob Hanlon

On Wed, Feb 20, 2013 at 10:28 PM, <> wrote:
> It appears that Mathematica does not know how to integrate or differentiate the Erfi function. Am I correct? I am able to use Limit[(f(t+d)-f(t))/d, d -> 0] to get the derivative, but are there reasonable approaches to finding Integrate[Erfi]?
> Thanks!
> NS

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