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Erfi

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Erfi
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The erfi function, also called the "imaginary error function," is an entire function defined by

erfi(z)=-ierf(iz),
(1)

where erf(z) is the error function. It is implemented in the Wolfram Language as Erfi[z].

erfi(z) has derivative

d/(dz)erfi(z)=2/(sqrt(pi))e^(z^2),
(2)

and integral

interfi(z)dz=zerfi(z)-(e^(z^2))/(sqrt(pi)).
(3)

It has series about z=0 given by

erfi(z)=pi^(-1/2)(2z+2/3z^3+1/5z^5+1/(21)z^7+...)
(4)

(where the terms are OEIS A084253), and the asymptotic series for real z->+infty given by

erfi(z)∼(e^(z^2))/(sqrt(pi))(z^(-1)+1/2z^(-3)+3/4z^(-5)+(15)/8z^(-7)+...).
(5)

(OEIS A001147 and A000079).

See also

Dawson's Integral, Erf, Erfc

Related Wolfram sites

https://functions.wolfram.com/GammaBetaErf/Erfi/

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

  • erf
  • erfi x
  • erfi (0)

References

Sloane, N. J. A. Sequences A000079/M1129, A001147/M3002, and A084253 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Erfi

Cite this as:

Weisstein, Eric W. "Erfi." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Erfi.html

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