# Help with an integral

1. Mar 29, 2012

### phsopher

I was reading a paper where the following integral appears:

$$I = \int_0 ^{\pi}dt\sqrt{k^2 + \sin^2t}$$

In the limit $k^2 \ll 1$ the authors present the following approximation

$$I \approx 2 + \frac{k^2}{2}\left(\ln{\frac{1}{k^2}} + 4\ln 2 + 1\right).$$

I'm trying to reproduce this result but with no luck. Any idea how it should go? I've plotted both expressions as a function of k and they indeed agree for small k.

Last edited: Mar 29, 2012
2. Mar 29, 2012

### conquest

Did you try expanding the square root near k^2=0? then dropping terms of higher order than 1 in k^2.

I'm a bit confused why k should be a lot smaller than 2 and not 1

3. Mar 29, 2012

### phsopher

Sorry that was a typo on my part, it should of course be $k^2 \ll 1$.

I don't see how to expand it around k^2=0 since the derivative with respect to k^2 diverges at k^2=0. In any case the result I want to get is not a polynomial in k^2 so I need a different approach.

Last edited: Mar 29, 2012
4. Mar 29, 2012

### JJacquelin

Hi !

This is a complete elliptic integral of the second kind.
If one knows the properties of these kind of special functions, the expansion leads to the expected approximation.
In attachment, more terms of the expansion are provided.

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• ###### Eliptic integral.JPG
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5. Mar 30, 2012

### phsopher

Thank you, that was very helpful. And now that I know what it's called I can plot using the inbuilt Mathematica functions; plotting the actual integral took ages.

However, I would still like to know how to obtain the expansion of E(p) in terms of q. Do you happen to know where it comes from?

6. Mar 30, 2012

### JJacquelin

The properties of special functions such as E(p) were studied one century ago and earlier.
They are gathered in the mathematical handbooks. For example :
M.Abramowitz, I.A.Stegun, "Handbook of Mathematical Functions", Dover Publications, N.-Y., 1972
J.Spanier, K.B.Oldham, "An Atlas of Functions", Hemisphere Pubishing Corporation, Springer-Verlag, 1987.
Good luck if you want to find the original manuscripts or books.

7. Mar 30, 2012