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Re: Is there a way to integrate and differentiate Erfi?
Posted:
Feb 21, 2013 5:46 AM


Works fine here
D[Erfi[x], x]
(2*E^x^2)/Sqrt[Pi]
% == Limit[(Erfi[x + d]  Erfi[x])/d, d > 0]
True
Integrate[Erfi[x], x]
(E^x^2/Sqrt[Pi]) + x*Erfi[x]
D[%, x] == Erfi[x]
True
Integrate[Erfi[x], {x, a, b}]
(a)*Erfi[a] + (E^a^2  E^b^2 + b*Sqrt[Pi]*Erfi[b])/ Sqrt[Pi]
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}]
Erfi[x]
Integrating termbyterm
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]
True
Bob Hanlon
On Wed, Feb 20, 2013 at 10:28 PM, <eagles.g11.teams@gmail.com> 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 >



