I know that $$E(\theta|k^2)=\int_0^\theta{\sqrt{1-k^2\sin^2\phi}}\,d\phi.$$
I want to find $E(k)$ where $k=0.36$.
With a website I computed it to be approximately $1.518$.
How can I find the answer without computer?
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You (famously) cannot compute it explicitly – Paul Apr 01 '21 at 16:03
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1You could calculate Riemann sums. $\sin\theta$ and square roots can be calculated by Taylor series. – Kenta S Apr 01 '21 at 16:16
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1You want $,E(k):=\int_0^{\frac{\pi}2}\sqrt{1-k^2\sin^2\theta},d\theta.$ See Wikipedia article Elliptic integral. – Somos Apr 01 '21 at 17:40
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1You can use descending Landen transformation to relate $E(k) $ with $E(l) $ where $l=(1-\sqrt{1-k^2})/(1+\sqrt{1-k^2})$ and $l<k$. Using this transformation a few times you will reach a value of $l$ which is small enough and then you can evaluate using the series obtained from term by term integration. – Paramanand Singh Apr 09 '21 at 03:01
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1Wikipedia under computation gives a series. The terms seem to decrease as powers of $2$ so it might not take too many. The $c_n$ are also decreasing, so it may be better yet. – Ross Millikan Jun 14 '21 at 13:31
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1The Wikipedia article also gives an algorithm using the arithmetic-geometric mean which converges quite quickly, unless $k$ is very close to 1. Beware that some people (like Wolfram Alpha) use the convention that $$E(k)=\int_0^{\pi/2}\sqrt{1-k\sin^2(\theta)};d\theta$$ – PM 2Ring Jun 14 '21 at 14:58
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1Here is a Python version of the AGM algorithm, running on the SageMathCell server. – PM 2Ring Jun 14 '21 at 15:11
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1FWIW, SageMath has the EllipticE function, but it uses the same convention as Wolfram Alpha. – PM 2Ring Jun 14 '21 at 15:11