Integrating the Error Function: Solving a Chemistry Research Question

agalliasthe
Messages
3
Reaction score
0

Homework Statement



This is a question I'm trying to solve for chemistry research - but is homework-like, so I thought it best fit in this category.

Homework Equations



I am trying to find the indefinite integral of:

F(x,y)=\int \int e^{-k_1x^2-k_2y^2+k_3xy} dx dy

k_1, k_2, k_3 are constants

The Attempt at a Solution



I realize that the solution involves the error function. When I reduce it to:

F(x,y)=\int e^{-k_2 y^2} dy \int e^{-k_1x^2+k_3xy} dx

I am not sure how to treat the terms in the dx integral.

When I ask WolphramAlpha, it tells me:

\int e^{-k_1 x^2+k_3xy} = \frac{\sqrt{\pi}exp(\frac{k_3^2y^2}{4k_1})}{2 \sqrt {k_1}}erf\left( \frac{2k_1x-k_3y}{2\sqrt{k_1}}\right )+c

but I'm not sure how they got this and I'm not sure how to proceed from here. I haven't been able to find a good erf integral table. Can you offer a suggestion on a way to solve this?Thanks.
 
Physics news on Phys.org
\int e^{-k_1 x^2+k_3xy} dx = \sqrt{\frac{\pi}{k_1}}e^{\frac{k_3^2 y^2}{4k_1}}erf\left( \frac{2k_1x-k_3y}{2\sqrt{k_1}}\right )+C

Making it look slightly better, may help out.
 
I'd start by completing the square on -k_1x^2++k_3xy, and then using an appropriate substitution...
 
Thanks - completing the square works beautifully!

Now I am wondering where to go from here:

F(x,y)=\int \int e^{-k_1x^2-k_2y^2+k_3 xy}dx dyF(x,y)=\int e^{-k_2y^2} \left ( \frac{\sqrt \pi}{2 \sqrt{k_1}} e^{\frac{k_3^2y^2}{4k_1}} erf \left (\frac{2k_1x-k_3y}{2\sqrt{k_1}} \right ) +c_1 \right )dy F(x,y)=\frac{\sqrt\pi}{2\sqrt{k_1}} \int e^{\frac{k_3^2y^2-4k_1k_2y}{4k_1}} erf \left (\frac{2k_1x-k_3y}{2\sqrt{k_1}} \right ) dy+c_1 \int e^{-k_2y^2}dythe second of these two integrals is just by definition, but I am struggling with the first. If I try integration by parts and let u=erf(~) and dv=exp(~)dy, then du becomes an exp(_) function and v is an erf(_)... so uv-\int vdu doesn't do much to simplify. If I let u=exp(~) and dv=erf(~)dy, then du becomes an exp(_) function and v becomes erf(_)+exp(_)... again, not simplifying the integral.

I'm not familiar enough with the error function to know what it can and can't do, but it seems like it should be agreeable to a Gaussian in some way and simplify somehow.

Any guidance? Thanks!
 
I figured it out with integration by parts twice - thanks for your help.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
Back
Top