Complete Elliptic Integral of the First Kind

In this video, I evaluate a special integral that relates to ellipses.
#mathematics #calculus #integrals #definiteintegrals #elliptical #firstkind #pi #sine #trigonometry #unitcircle #gamma
#drpeyam #blackpenredpen #flammablemaths #3blue1brown #michaelpenn #khanacademy #tibees #andrewdotson #stevemould #vibingmaths #polarpi

[@prabalbaishya6179]

nice. I guess the elliptic integral of the 1st kind can also be represented in terms of hypergeometric series/functions.

[@gutentag1752]

Yes there is a representation in terms of hypergeometric series. Michael Penn and Flammable Maths both have videos on this topic if you are interested. It is also not too complicated to come up with this yourself. You have to know some properties of the Beta and Gamma function as well as the Binomial theorem and rising factorials. Some steps might not be super obvious but keeping in mind you want to arrive at a representation in terms of a hypergeometric series it is quiet doable. It is also somewhat a good exercise I would say.