# [Maxima] problem with integration of gaussian distribution

Barton Willis willisb at unk.edu
Mon Jul 6 11:20:24 CDT 2009

```-----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

```