Ask Your Question
0

Strange behviour when trying to integrate gaussian function. bug?

asked 2011-02-14 14:36:02 -0500

Shashank gravatar image

updated 2011-02-14 16:57:25 -0500

I was trying integrate the following function

y=var('y')
integrate(x*exp(-(x-y)*(x-y)*2.0),x)

$$\frac{1}{8} \, {\left(\frac{2 \, {\left(\text{erf}\left(\sqrt{2} \sqrt{{\left(x - y\right)}^{2}}\right) - 1\right)} {\left(x - y\right)} \sqrt{\pi} y}{\sqrt{{\left(x - y\right)}^{2}}} - \sqrt{2} e^{\left(-2 \, {\left(x - y\right)}^{2}\right)}\right)} \sqrt{2}$$ The integral can easily be done by redefining variables and does not have to be expressed in terms of error functions. Also the behavior of sage becomes even strange when I change the coefficient from 2.0 to 2.1. Sage just refuses to do the integral

integrate(x*exp(-(x-y)*(x-y)*2.1),x)

$$\int x e^{\left(-2.1 \, {\left(x - y\right)}^{2}\right)}\,{d x}$$

Any ideas on how to make sage give an answer in the usual exponential form?

Edit: Sorry. My bad. It is an error function. However, I expect error function even for the second example, when I replace 2 by 2.1. Any ideas why that is the case?

edit retag flag offensive close merge delete

Comments

Could you be specific about the transformation you have in mind? As usual, I'm missing something-- I can't see how to do it without an erf popping up. (And you should expect this comment to vanish after I realize how stupid I was. :^)

DSM gravatar imageDSM ( 2011-02-14 16:44:46 -0500 )edit

@DSM - no need to make comments vanish, that's part of the open development process, to keep a record of everyone's thoughts. "Stupid" comments may later turn out to be prescient.

kcrisman gravatar imagekcrisman ( 2011-02-15 02:13:43 -0500 )edit

1 answer

Sort by ยป oldest newest most voted
1

answered 2011-02-14 15:06:59 -0500

kcrisman gravatar image

updated 2011-02-15 02:12:52 -0500

I assume this is still not what you want, as it doesn't use any substitution as you suggest, but it does seem to work:

sage: integrate(x*exp(-(x-y)*(x-y)*2.1),x)
-1/882*I*(21*I*(erf(sqrt(21/10)*sqrt((x - y)^2)) - 1)*(x - y)*sqrt(pi)*y/sqrt((x - y)^2) - I*sqrt(10)*sqrt(21)*e^(-21/10*(x - y)^2))*sqrt(10)*sqrt(21)

This is coming from Maxima, of course.

$$-\frac{1}{882} i {\left(\frac{21 i {\left(\text{erf}\left(\sqrt{\frac{21}{10}} \sqrt{{\left(x - y\right)}^{2}}\right) - 1\right)} {\left(x - y\right)} \sqrt{\pi} y}{\sqrt{{\left(x - y\right)}^{2}}} - i \sqrt{10} \sqrt{21} e^{\left(-\frac{21}{10} \, {\left(x - y\right)}^{2}\right)}\right)} \sqrt{10} \sqrt{21}$$

If I do simplify_full() to the answer, I get something that looks like

$$-\frac{1}{882} {\left(10 \sqrt{3} \sqrt{7} e^{\left(\frac{21}{5} x y\right)} + 21 {\left(\sqrt{2} \sqrt{5} y e^{\left(\frac{21}{10} x^{2} + \frac{21}{10} y^{2}\right)} \text{erf}\left(-\frac{1}{10} {\left(\sqrt{3} \sqrt{7} x - \sqrt{3} \sqrt{7} y\right)} \sqrt{2} \sqrt{5}\right) - \sqrt{2} \sqrt{5} y e^{\left(\frac{21}{10} x^{2} + \frac{21}{10} y^{2}\right)}\right)} \sqrt{\pi}\right)} \sqrt{3} \sqrt{7} e^{\left(-\frac{21}{10}x^{2} - \frac{21}{10} y^{2}\right)}$$

which is still messy but at least doesn't have any imaginary components. Is this not sufficient?

edit flag offensive delete link more

Comments

Can you elaborate on how you got the quoted answer? I am still getting this as the output \int x e^{\left(-2.1 \, {\left(x - y\right)}^{2}\right)}\,{d x}

Shashank gravatar imageShashank ( 2011-02-15 16:35:49 -0500 )edit

I literally typed integrate(x*exp(-(x-y)*(x-y)*2.1),x). If you do A=integrate(x*exp(-(x-y)*(x-y)*2.1),x), and then A.simplify_full(), you get the last thing. What version of Sage are you using? Maxima has steadily improved its integration ability, and we try to keep up with it.

kcrisman gravatar imagekcrisman ( 2011-02-16 00:54:10 -0500 )edit

Your Answer

Please start posting anonymously - your entry will be published after you log in or create a new account.

Add Answer

Question Tools

Stats

Asked: 2011-02-14 14:36:02 -0500

Seen: 831 times

Last updated: Feb 15 '11