Incorrect definite integral
- To: mathgroup at smc.vnet.net
- Subject: [mg9468] Incorrect definite integral
- From: Josip Loncaric <josip at icase.edu>
- Date: Thu, 6 Nov 1997 02:40:12 -0500
- Organization: ICASE
- Sender: owner-wri-mathgroup at wolfram.com
Mathematica 3.0 makes a curious mistake in the following evaluation.
f[k_,x_,y_] := 2*Sinh[x]*Cos[k*y]/(Cosh[x]-Cos[y])
d[k_,x_] := Integrate[f[k,x,y],{y,0,Pi}]/(2*Pi)
i[k_,x_,y_] := Integrate[f[k,x,y],y]/(2*Pi)
For positive x, the definite integral d[k,x] should evaluate to an
expression equivalent to E^(-Abs[k]*x). Mathematica obtains the
correct result EXCEPT when Abs[k]=1, when d[1,x] and d[-1,x] return
-Sinh[x]
which is quite wrong. By contrast, the indefinite integral i[1,x,y] is
(2*ArcTan[Coth[x/2]*Tan[y/2]]*Cosh[x] - y*Sinh[x])/Pi
which smoothly varies from 0 at y=0 to E^(-x) as y->Pi from below, as it
should.
Mathematica's definite integral gets a wrong answer because it
apparently selects the wrong direction in taking the limit y->Pi. Why
anyone would approach the upper limit from ABOVE when the integrand
need not even be defined outside the bounds of integration is puzzling.
--
Dr. Josip Loncaric, Senior Staff Scientist ICASE, M/S 403, NASA Langley
Research Center, Hampton, VA 23681-0001 Phone: (757) 864-2192
mailto:josip at icase.edu Fax: (757) 864-6134
http://www.icase.edu/~josip/