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 RUclips. 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!
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.
I think the most general version of these functions in that field would be the GKZ hypergeometric functions. Have to check again tho, just telling from what I remember..
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.
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!!
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 :)
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.
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.
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! ;-)
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.
Just chiming in to share/answer the question at the start. All the folks I've worked with, mostly talking about basic hyoergeometric functions, read the function name (including subscripts) from left to right, e.g., 2-F-1 or 3-ϕ-2.
My first encounter with hypergeometric series and elliptic functions was when I got interested in solving quintic polynomials. Still struggling to understand them and how they come about from trying to solve certain quintics.
I love the way you introduce these new topics and I take these videos as introductions without following as closely as usual. For me, video is not yhe best format for topics involving so many new definitions. As a student, it takes some practice and manual "fiddling around" for me to develop a feel, and books or monographs seem to work better. I wonder if other viewers feel this way.
Yes I am the same. I often listen in the background and flick back to check a calculation overall makes sense without following along super closely. My goal is just to gain some sort of familiarity and demystify these functions so that when I encounter them in my own work, they don't seem alien from the onset. Funnily enough, my thesis involves the 2d Ising model, which is parameterised in terms of Jacobi theta functions, and has an elliptic integral appearing in one stage and a hypergeometric function with half integer parameters appearing at another stage. Thanks to this video and the next one in the series I can really see the link between these two results!
hmmm earlier its said that the elliptic integral has no closed form, but it turns out to be equal to an ordinary hypergeometric function. Does that mean integrals with no closed-form solutions can still be written down, even if we have to use infinite series?
I did hypergeometric functions in Part II Complex Methods and for the life of me I cannot even remember how to say the 2F1(a,b,c) out loud. I actually liked the course too but hypergeometric functions were the big "Wait... What?" topic in the course.
In Conkwright, "Introduction to the Theory of Equations", p. 85, Conkwright states "The general quintic equation has been solved, though not, of course, by radicals. The solution is obtained by means of elliptic functions. The general equation of degree n has been solved in terms of Fuchsian functions." As best as an amateur such as myself can understand these Fuchsian functions are some kind of generalization of elliptic functions. I've seen some of the work involved with the solution of the quintic involving hypergeometric functions and symmetries of the icosahedron... way too deep for me! I have seen geometric interpretations of the solutions of certain classes of quadratic and cubic equations for equations with real coefficients, like completing the square or completing the cube. Are there any geometric interpretations of the solutions of quartic, quintic, or higher degree equations that help illuminate how these solutions are determined? Do hypergeometric, elliptic, or Fuchsian functions have any geometric interpretations that might illuminate how these solutions are obtained for those of us who are not advanced pure theoretical mathematicians?
I think they are related with each other somehow in a way like arc length of a 3-d curve, if you are interested i think we can discuss about this more closely
Can you explain please why we can interchange the integral and the sum in 8:32 ? I guest its because the series is uniformly convergence but I can't proof it.
If you want to prove the fact just use integration by parts: integral of (sin(teta))^m from 0 to pi/2=integral of (-cos(teta))' (sin(teta))^(m-1) from 0 to pi/2.
It refers to how many parameters are in the function. 2F1 has 2 top parameters and 1 bottom parameter. A 3F3 function in x with top parameters a1,a2,a3 and bottom parameters b1,b2,b3 would have it's terms in the series as a product of rising factorials to n from a1,a2,a3 divided by a product of rising factorials to n from b1,b2,b3, times x^n over n!. You'd write the function like 3F3(a1,a2,a3; b1,b2,b3; x), but other general notations for pFq exist where a is a p-vector and b is a q-vector.
Didn't watch the whole video yet so I'm jumping the gun a bit here, but what's the usefulness of defining a rising factorial? With some re-indexing, it can be expressed as a ratio of regular factorials
Let a,b and c real positives numbers satifying: abc= a²b² + b²c² +c²a² Find the greatest value of real number m where: m= a² +b² +c² can you do this please
could a way to define, or at least conceptualize, the rising factorial be something akin to \prod_{i=0}^{i=n-1}\(a+i\)? sorry for the plaintext latex but I'd hope it wont be hard to picture
the semicolons separate the parameters in the numerator from the parameters in the denominator, and then the variable. Example: 4F3(a,b,c,d;e,f,g;x) Since you already have the numbers explicitly mentioned in the "name" of the function, the semicolons are kind of redundant, but it is tidy.
In quantum field theory, when calculating so-called "quantum loop corrections" to physical processes ("scattering amplitudes"), you end up with these functions, because they are constitutents of the class of solutions of integrals of rational functions.
Fore example, some hypergeometric functions are used as fundamental solutions to second-order ODEs, i.e. there are analytical solutions to these kind of ODEs which kind of simplifies the quality analysis of solutions.
A generalisation of the hypergeometric function, the Heun function, appears when you study paths of charged particles around Kerr black holes in a De Sitter background
Idk why this was in my suggestions since I was a C student at best in math, but uhh... let me get my pen and paper, because this is interesting and I'm not sure why.
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 RUclips. 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!
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.
I think the most general version of these functions in that field would be the GKZ hypergeometric functions. Have to check again tho, just telling from what I remember..
@@rabbit-ku1bn You may be quite right, actually, if they generalize to any number of arguments x,y,z,... besides the parameters themselves.
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.
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!!
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 :)
this sounds like an epic video so i clicked on it, and it was
nice
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.
I've been waiting for elliptical integrals. Thank you!
agree
That was very nice.
Thank you for opening up new avenues for me, professor!
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.
Nice!!!
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! ;-)
Oh my god, this is my JAM right here
I've used Hypergometric functions (and their generalisation, Heun Functions) and call them two-eff-one, etc.
It is incredible how this integral appears in the formula for the period of a real pendulum without dissipation!!
Oh this is my happy place, especially when Bessel functions and Bessel-Hankel integrals join in with some potential theory problems.
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.
Love this topic, thanks for covering this!!
totally agree!
Hello Michael can you make a video on some equivalent functions? Thank you for all you've done!
this would be great to see!
Just chiming in to share/answer the question at the start. All the folks I've worked with, mostly talking about basic hyoergeometric functions, read the function name (including subscripts) from left to right, e.g., 2-F-1 or 3-ϕ-2.
This function even appeared in my Master’s research project on analytic fluid optimisation!
Cool!!!
15:04
Please please please do more on this subject and provide references. I love your channel but this is abobe and beyond interesting.
totally agree!!!
My first encounter with hypergeometric series and elliptic functions was when I got interested in solving quintic polynomials. Still struggling to understand them and how they come about from trying to solve certain quintics.
That was so good!
Are there going to be anymore videos from your number theory playlist?
I love the way you introduce these new topics and I take these videos as introductions without following as closely as usual. For me, video is not yhe best format for topics involving so many new definitions. As a student, it takes some practice and manual "fiddling around" for me to develop a feel, and books or monographs seem to work better. I wonder if other viewers feel this way.
Yes I am the same. I often listen in the background and flick back to check a calculation overall makes sense without following along super closely. My goal is just to gain some sort of familiarity and demystify these functions so that when I encounter them in my own work, they don't seem alien from the onset.
Funnily enough, my thesis involves the 2d Ising model, which is parameterised in terms of Jacobi theta functions, and has an elliptic integral appearing in one stage and a hypergeometric function with half integer parameters appearing at another stage. Thanks to this video and the next one in the series I can really see the link between these two results!
Was just going to comment on one of your recent videos that I wanted to see a video on hypergeometric functions!
Great! I realise this is not the usual fare but I should very much like to see more on this kind of topic
agree
For the next part you should to another application to the hypergeometric summation like the Barnes integral
Im a physicist but I really enjoy these math videos. Keep up the outstanding work
hmmm earlier its said that the elliptic integral has no closed form, but it turns out to be equal to an ordinary hypergeometric function. Does that mean integrals with no closed-form solutions can still be written down, even if we have to use infinite series?
I did hypergeometric functions in Part II Complex Methods and for the life of me I cannot even remember how to say the 2F1(a,b,c) out loud.
I actually liked the course too but hypergeometric functions were the big "Wait... What?" topic in the course.
In Conkwright, "Introduction to the Theory of Equations", p. 85, Conkwright states "The general quintic equation has been solved, though not, of course, by radicals. The solution is obtained by means of elliptic functions. The general equation of degree n has been solved in terms of Fuchsian functions." As best as an amateur such as myself can understand these Fuchsian functions are some kind of generalization of elliptic functions. I've seen some of the work involved with the solution of the quintic involving hypergeometric functions and symmetries of the icosahedron... way too deep for me! I have seen geometric interpretations of the solutions of certain classes of quadratic and cubic equations for equations with real coefficients, like completing the square or completing the cube. Are there any geometric interpretations of the solutions of quartic, quintic, or higher degree equations that help illuminate how these solutions are determined? Do hypergeometric, elliptic, or Fuchsian functions have any geometric interpretations that might illuminate how these solutions are obtained for those of us who are not advanced pure theoretical mathematicians?
*Reads title*
"I like your funny words, magic man"
This is Gauss hypergeometric function. And you can call it 2 F 1 also. And I will call (a)n Pochammer symbols. I saw that name in many books.
I'm hyped
I think they are related with each other somehow in a way like arc length of a 3-d curve, if you are interested i think we can discuss about this more closely
8:32 Why can we change the order?
Can you explain please why we can interchange the integral and the sum in 8:32 ? I guest its because the series is uniformly convergence but I can't proof it.
Cant believe it, I have just learned about them a couple of days ago
Random question, but why in the notation is the a and b separated by a comma but the rest of the parameters separated by semicolons?
If you want to prove the fact just use integration by parts: integral of (sin(teta))^m from 0 to pi/2=integral of (-cos(teta))' (sin(teta))^(m-1) from 0 to pi/2.
does it have logarithmic singularity at k=1. can you suggest me book
Please prove the generalised binomial theorem
Is the rising factorial is just `(n + a - 1)! / (a - 1)!` ?
Yes, usually written Gamma(a+n)/Gamma(a).
Sorry for the noob question, but what do the 2 and 1 mean in 2F1? What changes if you change those numbers?
It refers to how many parameters are in the function. 2F1 has 2 top parameters and 1 bottom parameter.
A 3F3 function in x with top parameters a1,a2,a3 and bottom parameters b1,b2,b3 would have it's terms in the series as a product of rising factorials to n from a1,a2,a3 divided by a product of rising factorials to n from b1,b2,b3, times x^n over n!. You'd write the function like 3F3(a1,a2,a3; b1,b2,b3; x), but other general notations for pFq exist where a is a p-vector and b is a q-vector.
@@cinnabun-ysera ohhhh i see. Thank you dragon waifu!
Didn't watch the whole video yet so I'm jumping the gun a bit here, but what's the usefulness of defining a rising factorial? With some re-indexing, it can be expressed as a ratio of regular factorials
Let a,b and c real positives numbers satifying: abc= a²b² + b²c² +c²a²
Find the greatest value of real number m where: m= a² +b² +c²
can you do this please
Is easy with some Lagrande multipliers.
We're allowed to do that in this case @8:39 👍👍👍
could a way to define, or at least conceptualize, the rising factorial be something akin to \prod_{i=0}^{i=n-1}\(a+i\)? sorry for the plaintext latex but I'd hope it wont be hard to picture
additionally, (a-1+n)!/(a-1)!
What's the difference in using commas and semicolons?
the semicolons separate the parameters in the numerator from the parameters in the denominator, and then the variable.
Example: 4F3(a,b,c,d;e,f,g;x)
Since you already have the numbers explicitly mentioned in the "name" of the function, the semicolons are kind of redundant, but it is tidy.
@@sergiokorochinsky49 do you mean numerator instead of "nominator"?
Never heard of "nominator".
@@calvinjackson8110 ...yes, I meant "numerator" (I was half-asleep when I wrote it). I will fix it to avoid embarrassment.
I like that! But how on Earth is this useful, sometimes I lose myself in all these math courses..
In quantum field theory, when calculating so-called "quantum loop corrections" to physical processes ("scattering amplitudes"), you end up with these functions, because they are constitutents of the class of solutions of integrals of rational functions.
Fore example, some hypergeometric functions are used as fundamental solutions to second-order ODEs, i.e. there are analytical solutions to these kind of ODEs which kind of simplifies the quality analysis of solutions.
A generalisation of the hypergeometric function, the Heun function, appears when you study paths of charged particles around Kerr black holes in a De Sitter background
Now show their relationship with the AGM fixed point
Put the stress on the "F" syllable:
[two-EFF-one]
Outstanding work! Hoping for more such videos - really, love it.
Idk why this was in my suggestions since I was a C student at best in math, but uhh... let me get my pen and paper, because this is interesting and I'm not sure why.
Cube
Experts pronounce _2 F_1 as two-eff-one.
die musik im coca cola spot in der werbung war russisch, zumindest bemerkenswert
Dur, dur !
What going on with this channel lol. From pre-school maths to crazy hypergeometric maths????