Hypergeometric functions and Elliptic Integrals -- Part 1

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[QuargCooper]

Hi, PhD student who has been working with basic and generalised Hypergeometric functions a bit recently, and at least in our research group (in the context of Integrable Systems), the convention we use is the first you suggest, i.e. Two-F-One, three-phi-two etc.

Really enjoying these "higher-level" videos, they're well explained, and nice to see some accessible advanced topics on YouTube. Seeing these objects related to my research popping up on my subscriptions list was such an unexpected and pleasant surprise. Looking forward to what you do next!

[hbm]

I encounter them in loop calculations in quantum field theory, and usually as soon as one wants to calculate 2-loop (or more) integrals, one ends up with using specific variations from a generalization of these hypergeometric functions, called the Kampé-de-Fériet functions; as always, solutions of integrals of rational functions.

[JustPassingBy_]

Really enjoying the channel exploring higher level topics, and I prefer them over the constest type stuff. Overall amazing channel keep up with the amazing work!!

[phonon]

I would always refer to pFq(.) as the "Hypergeometric p F q function"

Many times in the literature, 2F1() is simply written as F() (without the subscripts) and is referred to as "the Hypergeometric function" perhaps because it's the most common one.

[surem]

Speaking of fun integrals involving sinus that are connected to the hypergeometric function:

The integral of sin(sin(x))/x from 0 to infinity is equal to:
pi/2*(1F2(1/2; 1, 3/2 ; -1/4))
or, more explicitly:
pi/2*(infinite sum from n=0 to infinity of (-1)^n/((2n+1)*4^n*(n!)^2).

Pretty cool :)

[weonlygoupfromhere]

this sounds like an epic video so i clicked on it, and it was

[ZetaGirlPower]

I've been waiting for elliptical integrals. Thank you!

[synaestheziac]

This is great, thanks! I’ve always wanted to learn about these topics but they didn’t come up in my undergrad classes and I ended up going to grad school for philosophy, so I’ve been learning advanced math on my own through yours and others’ videos.

[manucitomx]

That was very nice.
Thank you for opening up new avenues for me, professor!

[bencheesecake]

Thanks so much for the approachable introduction. These came up in reading for a thesis, but I had no frame of reference to begin and really struggled.

[physicsatroeper]

For what it's worth, I always pronounced it "Two F One". I suspect that is the standard (like the binomial coefficients (n, k) is "n chose k"). This is some neat stuff. I look forward to seeing what you talk about next. For elliptic integrals, perhaps something about Jacobi Elliptic Functions (sn, cn and dn): I always thought that was a very cool subset of mathematical physics (a sort of generalization of trigonometry). Also the relationship of other "hypergeometric" functions to other famous "special functions". Nice video! ;-)

[MathHammer]

Oh this is my happy place, especially when Bessel functions and Bessel-Hankel integrals join in with some potential theory problems.

[pedroricardomartinscasella]

It is incredible how this integral appears in the formula for the period of a real pendulum without dissipation!!

[jimskea]

I've used Hypergometric functions (and their generalisation, Heun Functions) and call them two-eff-one, etc.

[TheNiTeMaR]

This function even appeared in my Master’s research project on analytic fluid optimisation!

[mariocortes]

Im a physicist but I really enjoy these math videos. Keep up the outstanding work

[thomaschristophe]

Hello Michael can you make a video on some equivalent functions? Thank you for all you've done!

[speeshers]

Love this topic, thanks for covering this!!

[habermasnyc]

Please please please do more on this subject and provide references. I love your channel but this is abobe and beyond interesting.

[ThatGuyWithDiabetes]

Learned HG functions when in the field of differential equations. And oh boy does it excite me to know that this proof directly implies that the elliptic integral is a solution to a certain ODE plugging in 1/2, 1/2, and 1 to the differential equation.