Re: This integral blows my PC (Maple 9.5) Help !

From: Carl Devore (devore_at_math.udel.edu)
Date: 08/30/04 

Date: Mon, 30 Aug 2004 18:02:34 -0400

On Mon, 30 Aug 2004, aran wrote:
> While integration p1 from 0 to 1 (int(p1(lambda),lambda=0..1)) the PC works
> and works.. And memory usage increasing and increasing until the whole
> stops.. Tried on 2 different PCs (one w/ w2k one with XP). Same
> result...

When I try, Maple can get the antiderivative, which is a sum of polylogs,
but Maple cannot compute the limit at 0. Ignoring the constants, you can
observe this in
int(1/x^5/(exp(1/x)-1), x= 0..1)
(same integral as yours except the constants are different).

> Can someone help? What do I wrong?

You did nothing wrong.

> May be there are other ways to integrate?

In this case you can make a substitution in the integral with the
student[changevar] command.

> the function to integrate:
>
> p1:=lambda-> c1/(lambda^5*(exp(beta/(lambda*T))-1));
> where
> beta:=Constant(h)*Constant(c)/Constant(k);
> c1:=2*Pi*Constant(h)*(Constant(c))^2;
> T:=3200

Rather than use the ScientifcConstants package, I'll just use symbolic h,
c, and k, and assume they are positive. You can substitute in the actual
constants at the end.

> restart;
> assume(h>0, k>0, c>0);
> beta:= h*c/k; T:= 3200; c1:= 2*Pi*h*c^2;
> p1:= lambda-> c1/lambda^5/(exp(beta/lambda/T)-1);
> J:= Int(p1(lambda), lambda= 0..1); # Note capital I
> student[changevar](u= exp(beta/lambda/T), J, u);
> value(%);

-6400*Pi*c*k*ln(A-1) -6400*I*Pi^2*c*k -Pi*h*c^2/2
+83886080000000*Pi^5*k^4/3/c^2/h^3
-61440000*Pi*k^2/h*polylog(2,A)
+393216000000*Pi*k^3/c/h^2*polylog(3,A)
-1258291200000000*Pi*k^4/c^2/h^3*polylog(4,A)

where

A = exp(c*h/3200/k);

The imaginary "I" is cancelled by the imaginary components of the
polylogs.

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