Help Solve Integral: I = \int_0 ^{\pi}dt\sqrt{k^2 + \sin^2t}

[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.

 

[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

 

[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.

 

[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.

 

[phsopher]

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.

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?

 

[JJacquelin]

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?

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.

 

[phsopher]

With a couple of degrees of separation from Gradshteyn and Ryzhik found "An elementary treatise on elliptic functions" from 1876. The series expansion (page 54) is derived starting on page 46 if anyone's interested.