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How do I Show Equal

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  How do I Show Equal "Thomas D. Dean" <tomdean@speakeasy.org> - 2012-01-13 12:14 -0800
    Re: How do I Show Equal Axel Vogt <&noreply@axelvogt.de> - 2012-01-14 09:31 +0100

#306 — How do I Show Equal

I am trying to duplicate a Maxima proof.

#######################################
# Theorum
# for any real k,a,b
# integrate(exp(x)*x^k,x=a..b)
#    =exp(1)^(-I*Pi*k)*(gamma_incomplete(k+1,-b)
#     - gamma_incomplete(k+1,-a))
# maxima proof
# S:'integrate(exp(x)*x^k,x)';
# S1:ev(S, nouns);
# subst(-x=exp(%i*pi)*x,S1);
# /* From Newton-Leibnitz formula S is equal */
# subst(x=b,%)-subst(x=a,%);
# radcan(%);
# subst(pi=%pi,%);
restart;
S:=integrate(exp(x)*x^k,x);  # this seems to include the ev(S, nouns)
soln:=exp(1)^(-I*Pi*k)*(GAMMA(k+1,-b)-GAMMA(k+1,-a));
# from I should be able to reduce to soln
subs(x=b,S) - subs(x=a,S);

But, my lack of Maple expertise prevents reduction of this expression to 
soln.

How do I do this?

Tom Dean

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#310

On 13.01.2012 21:14, Thomas D. Dean wrote:
> I am trying to duplicate a Maxima proof.
>
> #######################################
> # Theorum
> # for any real k,a,b
> # integrate(exp(x)*x^k,x=a..b)
> # =exp(1)^(-I*Pi*k)*(gamma_incomplete(k+1,-b)
> # - gamma_incomplete(k+1,-a))
> # maxima proof
> # S:'integrate(exp(x)*x^k,x)';
> # S1:ev(S, nouns);
> # subst(-x=exp(%i*pi)*x,S1);
> # /* From Newton-Leibnitz formula S is equal */
> # subst(x=b,%)-subst(x=a,%);
> # radcan(%);
> # subst(pi=%pi,%);
> restart;
> S:=integrate(exp(x)*x^k,x); # this seems to include the ev(S, nouns)
> soln:=exp(1)^(-I*Pi*k)*(GAMMA(k+1,-b)-GAMMA(k+1,-a));
> # from I should be able to reduce to soln
> subs(x=b,S) - subs(x=a,S);
>
> But, my lack of Maple expertise prevents reduction of this expression to soln.
>
> How do I do this?
>
> Tom Dean

Int(exp(x)*x^k,x); value(%);
eval(%, x=b) - eval(%,x=a); # need not be correct to do that!
simplify(%, symbolic);
simplify(%, size);

gives

-k*(GAMMA(k,-a)-GAMMA(k,-b))*(-1)^(-k)+b^k*exp(b)-a^k*exp(a);

The 2-argument GAMMA is Maple's incomplete Gamma function
and you have to compare the definition in both systems to
check why there

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