How can we represent F(x,m) in the infinte polynominal of x,m? (Note that F(x,m) is the incomplete elliptical integral of the first kind, and I used its representation in the wikipedia) More specifically, what is the value of [F(x/2,(cos m)^-0.5)-F(m,(cos m)^-0.5)]*(cos m)^-0.5 when terms of O(m^3),O(x^3) are ignored? I searched about it for a while, but there was not so much information about the incomplete one actually.
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The series expansion in powers of $k$ of the incomplete elliptic integral of the first kind $$F(\varphi,k)= \int_0^\varphi \frac {d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}$$ can be simply obtained by expanding the integrand, $$F(\varphi,k)=\varphi+\frac{1}{4} k^2 (\varphi-\sin \varphi \cos \varphi)+\frac{3}{256} k^4 (12 \varphi-8 \sin 2 \varphi+\sin 4 \varphi)+{\cal O}(k^6),$$ and then if you wish you can further expand in powers of $\varphi$.
answered Sep 6, 2022 at 15:26