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Error Function   (28.3.00)

Becker, Thomas

Can anybody tell me the definition of the Error Function

2/sqrt(Pi) * int((exp(-t^2), t=0..x);

for complex arguments?

Willard, Daniel Dr   (29.3.00)

Abramowitz and Stegun: Error function of complex argument:

erf(x+i*y) = erf(x) + (2*Pi*x)^(-1)*exp(-x^2)*[(1-cos(2*x*y)) + i*sin(2*x*y)]
             + 2/Pi*exp(-x^2)*sum((exp(-4^(-1)*n^2))/(n^2+4*x^2)*[f[n](x,y) 
             + i*g[n](x,y)], n=1..infinity) + Epsilon(x,y)

where f[n]=2*x-2*x*cosh(n*y)*cos(2*x*y) +n*sinh(n*y)*sin(2*x*y);
      g[n]=2*x*cosh(n*y)*sin(2*x*y)+n*sinh(n*y)*cos(2*x*y);
      abs(Epsilon(x,y)) is roughly 10^(-16)*abs(erf(x+i*y))

David Holmgren   (29.3.00)

The complex error function (sometimes known as the plasma dispersion function) is defined as

     w(z) = exp(-z^2)*(1 + (2*I)/sqrt(Pi)*int(exp(t^2),t=0..z)

using Maple notation. Here, z is a complex number (as is t). The real part of w(z) is the Voigt line profile.

Robert Israel   (29.3.00)

But... that _is_ the definition. Since exp(-t^2) is entire, you can integrate on any path from 0 to x.

Well, if you want to avoid explicit path integrals, you could say

erf(x) = 2*x/sqrt(Pi) * int(exp(-x^2*t^2), t=0..1);

Joel A. SHAPIRO   (29.3.00)

The place to look at for questions like this is Abramowitz and Stegun, Handbook of Mathematical Functions, which has a chapter (7) on the error function and fresnel integrals. Another, higher-brow source on such questions is Whittaker and Watson, A Course in Modern Analysis, but looking up erf there leads to confluent hypergeometric functions, which might be a longer path to what you want.

Jurgen Barsuhn   (30.3.00)

The definition is the same ar for a real argument, i.e.

erf(z)=2/sqrt(Pi)*int(exp(-t^2,t=0..z)

for any complex z. As exp(-t^2) is a regular function, its integral is independent of the path of the integration. If you e.g. choose z=2+3*I, you may take the path from 0 to 2 along the real axis and then from 2 to 2+3*I parallel to the imaginary axis. Then you obtain only "real" integrals, if you handle the factor I just like any constant. To force Maple to this integration path you may define the function g(x,y)=exp(-(x+I*y)^2). Compare the results of the following lines:

> 2/sqrt(Pi)*int(exp(-t^2),t=0..z);
> erf(2+3*I);evalf(%);

> 2/sqrt(Pi)*int(exp(-t^2),t=0..2+3*I);evalf(%);

> 2/sqrt(Pi)*int(exp(-t^2),t=0..2)+2/sqrt(Pi)*int(exp(-t^2),t=2..2+3*I);
> normal(%);evalf(%);

> g:=(x,y)->exp(-(x+I*y)^2);
> g(x,0);g(2,y);evalc(%);

> 2/sqrt(Pi)*int(g(x,0),x=0..2)+2/sqrt(Pi)*int(I*g(2,y),y=0..3);
> normal(%);evalf(%);

The factor I for g(2,y) in the preceding integral results from the differential parallel to the imaginary axis which is I*dy

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