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[MUG] Re: Error Function
| [MUG] Re: Error Function |
|
Author: Maple Group
Posted: 31/03/2000 15:10:02 GDT
|
>> From: Maple Group
>> From: "Becker, Thomas "
| Can anybody tell me the definition of the Error Function
| for complex arguments?
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From: "Willard, Daniel Dr DUSA-OR"
To:
Subject: Error Function
Date: Wed, 29 Mar 2000 13:02:48 -0500
From 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))
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From:
Date: Wed, 29 Mar 2000 13:43:36 -0500 (CDT)
Subject: Error Function
To:
Dear Dr. Becker,
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.
Sincerely,
David Holmgren
Brandon University
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Date: Thu, 30 Mar 2000 09:18:20 +1200 (NZST)
From: John Harper
To:
Subject: Error Function
Same definition. Same power series. It's an entire function.
See Abramowitz&Stegun p297.
John Harper, School of Mathematical and Computing Sciences,
Victoria University, Wellington, New Zealand
e-mail phone (+64)(4)463 5341 fax (+64)(4)463 5045
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Date: Wed, 29 Mar 2000 22:19:48 GMT
From: J H Davenport
To:
Subject: Error Function
The power series for erf, obtained by term-bu-term integration,
is still valid:
erf(z)=2/sqrt(Pi) * sum (-1)^nz^{2n+1)/n!(2n+1)
James Davenport
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Date: Wed, 29 Mar 2000 14:24:45 -0800 (PST)
From: Robert Israel
To:
Subject: Error Function
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);
Robert Israel
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
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Date: Wed, 29 Mar 2000 18:06:15 -0500 (EST)
From: "Joel A. SHAPIRO"
To:
Subject: Error Function
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.
Joel Shapiro, Prof. of Physics, Rutgers University
(732) 445-3886 Fax: (732) 445-4343
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Date: Thu, 30 Mar 2000 11:13:25 +0200
From: Barsuhn
Subject: Error Function
To:
Dear Thomas,
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 paralle to the imaginary axis which is I*dy
All the best Jurgen
--
-------------------
Prof. Dr. Jurgen Barsuhn
Fachhochschule Bielefeld
University of Applied Sciences
Fachbereich Elektrotechnik und Informationstechnik
D-33511 Bielefeld
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