Subject: problem with integration of gaussian distribution
From: Barton Willis
Date: Mon, 6 Jul 2009 11:20:24 -0500
-----maxima-bounces at math.utexas.edu wrote: -----
>I?tried?it?with?the?following?code:
>
>assume(?m?>?0,?d?>?0,?h?>?0,?A?>?0)$
>phi(p)?:=?A*exp(-(p-p_0)^2/(h/d)^2);
>psi(x,t)?:=
>integrate(phi(p)*(2*%pi*h)^(-1)*exp(%i/h*(p*x-p^2/(2*m)*t)),p,minf,inf);
>
>The?result?should?be?a?bell-shaped?curve?to.
>
>What?do?i?have?to?do?to?get?a?human-friendly?solution?(not?with?sin,?cos,
>but?exp)?of?this?integral?
To convert sin and cos to exponential form, use the function
exponentialize;
something like: (I'd guess that some of these questions aren't needed)
(%i24) psi(x,t)$
Is t positive, negative, or zero?pos;
Is x positive, negative, or zero?pos;
Is p_0 positive, negative, or zero?pos;
Is sqrt(sqrt(h^2*t^2+4*d^4*m^2)-2*d^2*m)*(sqrt(h^2*t^2
+4*d^4*m^2)+2*d^2*m)^(3/2)-h^2*t^2 positive, negative, or zero?pos;
(%i20) factor(exponentialize(%));
(%o20) (sqrt(m)*(sqrt(sqrt(h^2*t^2+4*d^4*m^2)+2*d^2*m) + ... <stuff>
Barton