Elliptic Integral Homework: Expanding for Large k^2

In summary, the student attempted to approximate the elliptic integral E(k) by truncating at the first order term, but found that the expansion was not valid for large k. After rewriting the expansion as a power series, it was found that the approximation converged up to a multiplicative constant on the resulting ln(m) term.
  • #1
G01
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Homework Statement



Sub problem from a much larger HW problem:

From previous steps we arrive at a complete elliptic integral of the second kind:

[tex]E(k)=\int_0^{\pi/2} dx \sqrt{1+k^2\sin^2x}[/tex]

In the next part of the problem, I need to expand this integral and approximate it by truncating at the first order term. (k is large)

Homework Equations



[tex]E(k) = \frac{\pi}{2} \sum_{n=0}^{\infty} \left[\frac{(2n)!}{2^{2 n} (n!)^2}\right]^2 \frac{k^{2n}}{1-2 n}[/tex]

The Attempt at a Solution



I believe I should use the expansion quoted above.

Here is my question. Based of the previous steps I know that k^2 has to be large. Also, the sign of k^2 is opposite of what is is in the standard form of E(k).

So, 1. Does this expansion truncated at first order approximate the integral well if k^2 is large?

I think not. Is there another expansion, one for large k^2, that I can potentially use?

2. Can I just change the sign in the odd terms of the expansion to account for the sign change of k?

I think this should work.
 
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  • #2
The expansion above is not valid for large [tex]k[/tex]. Simply rewrite

[tex]
E(k)=\int_0^{\pi/2} dx \sqrt{1+k^2\sin^2x} = \pm \int_0^{\pi/2} dx k \sin x\sqrt{1+\frac{1}{k^2\sin^2x}}
[/tex]

and expand this in [tex]1/k[/tex] yourself. Note that taking the square root introduces a sign ambiguity that should be chosen to give the correct sign to [tex]E(k)[/tex].
 
  • #3
fzero said:
The expansion above is not valid for large [tex]k[/tex]. Simply rewrite

[tex]
E(k)=\int_0^{\pi/2} dx \sqrt{1+k^2\sin^2x} = \pm \int_0^{\pi/2} dx k \sin x\sqrt{1+\frac{1}{k^2\sin^2x}}
[/tex]

and expand this in [tex]1/k[/tex] yourself. Note that taking the square root introduces a sign ambiguity that should be chosen to give the correct sign to [tex]E(k)[/tex].

Hmm. This makes sense, however, it is only a good approximation when sin(x) is larger than 1/m.

However, the contributions to the integral in [0,1/m] are small, so can I fix the approximation by changing the lower bound on the integral to 1/m?
 
Last edited:
  • #4
G01 said:
Hmm. This makes sense, however, it is only a good approximation when sin(x) is larger than 1/m.

However, the contributions to the integral in [0,1/m] are small, so can I fix the approximation by changing the lower bound on the integral to 1/m?

I'd write the expansion as a power series, do the integrals and then check convergence.
 
  • #5
Yeah you were right. The higher order terms don't necessarily converge with a lower bound of 1/m.

Checked with my prof. Looks like if you use 1/m as the lower bound and expand to first order, you get the correct result up to a multiplicative constant on the resulting ln(m) term.

I think can finish the rest of the problem now. Thanks fzero!
 

1. What is an elliptic integral?

An elliptic integral is a type of mathematical function that is used to calculate the area under an elliptic curve. It is often used in physics and engineering to solve problems related to motion and energy.

2. How is an elliptic integral expanded for large k^2?

An elliptic integral can be expanded for large k^2 by using a series of approximations and mathematical techniques such as the binomial theorem and Euler's formula. This allows for a more simplified and accurate calculation of the integral.

3. What is the significance of expanding for large k^2?

Expanding an elliptic integral for large k^2 allows for a more efficient and accurate way of solving complex problems in physics and engineering. It also allows for a better understanding of the behavior of elliptic curves and their applications.

4. Can the expansion of an elliptic integral be applied to any problem?

The expansion of an elliptic integral is most commonly used in problems involving motion and energy, but it can also be applied to other areas such as optics and electromagnetism. However, its applicability may vary depending on the specific problem at hand.

5. Are there any limitations to expanding for large k^2?

While expanding an elliptic integral for large k^2 can provide more accurate results, it may also introduce some error due to the use of approximations. Additionally, the expansion may not be feasible for some particularly complex problems, and in those cases, other methods may be necessary.

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