I was taking currently in a elementary calculus course where i found how to find arc lengths of a smooth continuous curve.
so here is how i started :
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\Rightarrow y=\pm \frac{b}{a}\sqrt{a^2-x^2}$$By applying the formula of the arc length of a function, we get:$$L=4\int_0^a\sqrt{1+\frac{b^2x^2}{a^2(a^2-x^2)}}dx=4\int_0^a\sqrt{\frac{a^4+(b^2-a^2)x^2}{a^2(a^2-x^2)}}dx$$Now I made a little subsitution recalling trigonometry: $$x=a\sin(u)\\dx=a\cos(u)du$$So the Integral now can be expressed as:$$L=4\int_0^{\frac{\pi}{2}}a\cos(u)\sqrt{\frac{a^4+(b^2-a^2)a^2\sin^2(u)}{a^2(a^2-a^2\sin^2(u))}}du=\\4\int_0^{\frac{\pi}{2}}a\cos(u)\sqrt{\frac{a^4+(b^2-a^2)a^2\sin^2(u)}{a^2(a^2\cos^2(u)+a^2\sin^2(u)-a^2\sin^2(u))}}du=\\4\int_0^{\frac{\pi}{2}}\sqrt{a^2+(b^2-a^2)\sin^2(u)}du$$So we have:$$L=4a\int_0^{\frac{\pi}{2}}\sqrt{1+\frac{(b^2-a^2)}{a^2}\sin^2(u)}du$$ Letting $m=\frac{(b^2-a^2)}{a^2}$ we finally get:$$L=4a\int_0^{\frac{\pi}{2}}\sqrt{1+m\sin^2(u)}du$$
BUT
later i found these type of integrals are known as elliptic integrals and can not be given in terms of elementary functions.
say elliptic integral of another type $$u( k)=\int_{0}^{\frac{\pi}{2}} \frac{1}{\sqrt{1-k^2\sin(\theta)}} \, d\theta $$
expanding by binomial theorem or mclaurian expansion will lead to :
$$u(k)=\frac{\pi}{2}\Biggl[{1+\Biggl(\frac{1}{2}\Biggl)^2k^2+\Biggl(\frac{1.3}{2.4}\Biggl)^2k^4+\Biggl(\frac{1.3.5}{2.4.6}\Biggl)^2k^6+....}\Biggl]$$
Now
I am very much interested
in finding the values of these series rather than the arc length of ellipse.
I just found on wikipedia where AGM method is used as : $$u(k)=\frac{\pi}{2M(1,\sqrt{1-k^2})}$$
But
At first glance
it is easy to see the following as a hypergeometric series : i.e. $$_2F_1\bigg[\frac{1}{2},\frac{1}{2};1;k^2\bigg]=\frac{1}{2}u(k)$$
$$LHS=\sum_{n=0}^{\infty}\frac{\bigg[(\frac{1}{2})_n\bigg]^2 k^{2n}}{(1)_n n!}$$ where $(a)_n=\frac{\Gamma_{a+n}}{\Gamma_a}$ simplify a bit $$=\frac{1}{\pi}\sum_{n=0}^{\infty} \bigg(\frac{\Gamma_{\frac{1}{2}+n}}{n!}\bigg)^2 k^{2n}$$
But i really do not know how the values of hypergeometric series are evaluated.I have seen(of course in a basic way) some special cases of $_3F_2$ and $_7F_6$ but do not know what to do with those generated by ellptic integral.
Being unexperienced i will admire any suggestion from anyone.
please tell me how these elliptic integrals and hypergeometric series are evaluated.