Integral Transforms through Mellin-Barnes Integration

Hi, my name is Paco Jane and I'm here with Oleg Marachev. We're going to be talking today about integral transforms and how they can be computed using Mellon, what's called Mellon Barnes Integration, which is a technology that's been developed over many years by Oleg and others. The talk is divided into four parts here. I'm going to take you through this first part about how, what the point of this project is and what the result was. Oleg will take you into some advanced theory of the mathematics. There's some additional material in this section for those that are interested, but we won't be going over it. And then there's references in the final section, which you can view as a handout. Oleg's. Many of Oleg's books are mentioned. So to start out, what are integral transforms? Why are we interested in them? The motivation for this project in general. So integral transforms are a technique that's used in science, engineering and so forth to solve problems. And you can see how popular it is to do practical work. In terms of this is some data collected by my manager Michael Trott, showing different preprints articles and how many of them mentioned some something involving an integral transform. And these are the different types of transform. There are many different integral transforms with different names and these just show you how popular they are in, in the different papers available in the literature. What is an integral transform? Mathematically speaking, it's an integral of this type of form where you take. The basic idea is you take a function, input function, you multiply it by some kernel which is a function of the variable of your original function, and then a new variable tau and t in this case and maybe some parameters. And it's basically computing this integral. Just to make it a little less abstract, I want to take you through just a basic example application and this shows this is solving a differential equation in, to be exact, Laplace's equation in a infinite wedge. So you imagine this green shape is a wedge that goes infinitely into and out of your screen and it's charged, or rather it's held at a potential fee at the top and the bottom of the wedge. And electrostatic fee theory tells us that we use Laplace's equation to solve this type of problem. But this equation, when we, you know, the symmetry of this problem dictates that we should use some sort of angular coordinate instead of the x and Y coordinates here. So we're going to use theta and R to describe the system. And the Laplace equation in that coordinate system isn't so nice to look at. But what's nice, and this is a common theme you'll see when using interval transforms to solve differential equations, is that by applying the transform, the equation becomes a simpler form. So when we apply the Mellon transform in this case to the original function and write this differential equation in terms of the Mellon transform of the function, it becomes a simple equation. Those of you who have probably, you know, who've studied these equations know that the solution can just be read off as a linear combination of sine and cosine. Oh, I have a typo here, apologies. I think B was supposed to be the sine.

And.

Okay, so we can solve this really easily. The difficult part though is that our boundary conditions, if you know the problem, is specified with some boundary conditions of the potential on those angle on the edges of the wedge have to be also computed in sort of the melon transformed world. And so we need to be in, in other words, to, to solve this practical problem, we, we can simplify some of the differential equation solving, but we in the, the, the, the exchange for that is that we need to know how to do a Mellon transform. And so that's what the point of this project was. We developed a resource function highlighted here to compute Mellon transform and actually almost two dozen different types of integral transforms which are useful for solving problems like this. And you can read more of the details if you're interested. Just to quickly summarize the result again is this generic integral transform resource function. And you can see the different transforms here. If you just call it with zero arguments, you get a table showing what transforms are available and how they're defined in terms of an integral. The whole technology that allows us to solve these integrals involves two functions called Meier G and FOX H, which Oleg will talk to you in detail about in just a moment. I wanted to point out that a lot of what we've done so far relies on the Meierg technology, whereas the FOX H is still in development. And of importance to note there is that doing inverse transforms often relies on the FOX H as opposed to the Meier G. So there's some work remains to be done there. So come see our talk next year at the next wtc. But just. And then just to show you some like, concrete results, what I've done is I've taken the two functions one over Z. Excuse me, one over one plus Z and exp. Just the decaying exponential function and compute. Use this resource function that we've developed to compute for different transforms shown in this first column. All these different transforms. Hankel, Hartley, Hilbert, Kontorovich, Lebedev. Always trouble with that name. All these different transforms, we can actually compute them in Mathematica and you can see the results are returned in terms of Meierg.

But.

But we can get the results in terms of more basic functions by hitting that result with activate and function expand. And then you see that some of them become very simple, like just a rational function or you know, hyperbolic cosecant. And others are written in terms of hypergeometrics. But they're all. They're all something that Mathematica can actually work with. So with that I'm going to hand off to Oleg and he will continue the talk.

Okay, let me say more the following. The main transforms which the Most popular is 4 year transform which people study in all colleges, universities and so on. But equivalent to Fourier transform is mailing transform which not so known, but more convenient for evaluation integrals and why it was happened. Let's look on simple function A in degree B. And if you make notation X on first position or in second position you get power or exponent functions. And we can say that this is kernels of million transform you see here. And kernel Laplace transform and Fourier transform. And if we make combinations of this product of these two kernels and try to evaluate similar integral we come to gamma function which is very important. And if we have many gamma functions ratio of products of gamma functions and make similar integrals like here inverse marine transform. And here we have product of ratio of gamma functions. Many many gamma functions. We define super function which has name either Z or more generic fox H function. But I will show it below. Here you see just basic formulas and we for transforms. And also here we see formulas which demonstrate how interrelated Laplace transform with four year or million transform. And I'm sorry, let me open. And here I just evaluated this example for Laplace transform using our code which you see here. And similar result we can get from Mathematica code implemented already in Mathematica. But here we have more accurate value because if Y is integer positive real we have the same results. But our evaluation gives a more generic formula. It's just example. I can now I can go to definition of fox and major G functions. This is fox function as I told, we have integral by vertical line or by special loop L. And here we have product ratio of many gamma functions. So we have four groups. First group enumerator includes may function near function other group but coefficients alpha and beta positive for each. Here additional groups and if we have all coefficients equal one. In particular, we can write that this is meer Z function. I presented this equality here. And in Mathematica we have measure Z function during maybe 25 or more years. When I came here 33 years ago, it was my first big project to implement function Fox H appeared just recently three years ago it was implemented by is. What? What? Sorry, what is interesting here? If we try to evaluate how many functions representable through FOX or through mayor. Here you see 127 functions and if we load this collection function, we can evaluate for each of them major Z form representation through measurement. For this we have common major reform. But also we have reduce common which is not so powerful because implemented several years before our recent results. Here you see, all this is representable what it means? It means that we can create integral transforms with kernel from this, right? From these functions, each of this function can be kernel of integral transform. So we can have 127 transforms or. No, currently we don't have so much. Sorry. What is it going then? Okay, if we evaluate this set, We can apply similar conversion using kernel built major Z reduce. As a result, we don't have so many. You see here not everything can be convertible to mere reform. And you see here that only half of them convertible to measure. Currently we have also folks reduce but poor folks function. It is meta cleflayer function. Okay?

Yeah.

And here I demonstrate more generic transforms than laplace or four year or million transform and others you see here so called Z transform with the function in kernel. It is presented in a demonstration of Parker. In this zebra graphics you can find line with J transform and you cannot find line with ash H transform. Fox H transform it is this line will appear later. It's under development right now. And all such researchers well made in books. Some cocubus marriage of fractional integrals and derivatives. Right now Chinese publishing house translated it to China and we wrote special appendix with description fractional integral differentiation. And this FOX transform you can find in book Kilbus my course and Saigo and also in books Jakubovich Luchko. References to these books you find below at the end of this article. And below here you see representations through integrals. All these transforms. It's some kind as a variant of zebra representation which you saw before. What is it interesting, we can write kernels of this list of function transforms and then we can apply major form from kernels. And you will see that each of these kernel is major function. And here you see how this major function each function the function includes Gamma functions. You can see, you know, for example, here it is a Hilbert kernel. Hilbert kernel. And this is gamma functions, four of them with plus s plus s, with minus s and then with plus and minus. No, in my classification, if follow my books. Plus minus, plus minus. Here we have also kernel for usual indefinite integral. So indefinite integral. It is a particular case of this construction. And here you see minus and minus. This is kernel for Laplace. Laplace it is exponent. You see only +s here kernel or so called major transform. You see here. Plus S plus S. No such classification. We can provide for unassigned transform Fourier sign transform. Fourier you see plus s here and minus s here is this kernel for the Fourier transform. So I as I explained, in practically all classical transforms are particular cases of fox or mayor Z transforms. And what is remarkable that we have examples. No, for example, here I demonstrate how to derive formula for inverse siltest transformation. Still just transform. Let me show where it is here up still just. Still just stilt is where is this kernel. Somewhere? Ah, this one. It is still this transform kernel. Generalized stiltes transform. You see gamma + class S and gamma minus S and parameter rho somewhere. And if you apply this still this transform still this transform which we have already in our repository to such function fox H function. We get also fox H function and the condition sigma more than zero. It means that inverse tilt is transformed from this guy. It is this result. Oh, what is it? So it is way how we can derive formulas for inverse Laplace tiltes, inverse J transform, inverse fox H transform and so on. And I made such evaluations and presented these formulas. And here you see procedure how to derive this inverse the fox H transform. No, I will not go in details, it's too complicated. But this horrible construction. This horrible construction. Oh, again. Source. What is it?

I don't know.

Shows the following. If we have function F which looks like this big, big, big, big and many parameters. This function is inverse fox H transform from usual Fox function. And we have corresponding value here. Procedure evaluation. You see here. And. Let me conclude. Maybe here I have example. Example. When we have inverse series transform and formula for them under condition sigma more than zero or less than zero. So basically, this piecewise construction for inverse still just transform. Here you see only formulas. But we know how to convert to programs. And soon we will have such programs park of big master in this area and we will realize this. And you feel that we cover all such big area. Integration, definite integration, indefinite integration, all integral transforms. But right now I am talking and you are seeing Integral transforms of convolution type. There exist other types integral transforms by indexes. Here we also have examples you can find in this notebook. Kantarovich, Liebeter transform, mirror fog transform and more generic WIM transform. This area was developed by my former PhD student and now professor Portugal University, Simeon Barisich, Yakubovich and Luchko. They published corresponding books in this area and gave conditions for the the existence such transforms. Jacoboich published also book index transforms. You can find these books here you see Hypergymetric approach to integral transforms and Jacobovich index transform. No also I work with transforms and integrals whole. You see here five volumes integrals and series which published together with Prudnik of Briskov and recently published this book Savishenka Briskov Handbook Million transforms and many other results we have and I my dream is to convert this from text from my head into programs available for millions of people. Thank you for attention.

Yep, thank you all. Let me pull up the chat just while we have a few more minutes. We can see if there's any questions. Let's see. Public chat so Sotiris Michos asks is there some formal or theoretic method for discovering if a function is expressible as Meier G or is it done through some computational approach and the reverse from Meier G to a simpler closed form?

I will answer. Recently we published the measure form in our repository and this major reform tested with about 9,000 cases. And before this I published my first book. It was about how many 45 years ago with a huge table of particular cases function representable through Meerz form. And so I have biggest at the world table of functions convertible to this mayorg form.

Yes.

And right now it is on our programs.

Yes, the resource function Myerg form which is used by the function we presented today and you can find it in the function repository it can cover about 9,000 cases.

Yes. And we tested all this with fractional order integral differentiation. We made several talks about this with Sparkle. Yes.

Yeah, yeah, yeah. And past wtcs we have another question or a couple questions from Peter Burberry who asks is Fox H more general than Meyer G?

Yes, for the most part folks, more general. And meet the clear flare function is you only function which poor case of mayor of fox H But if you put coefficients in meta clear rational it will be convertible to fox to me to mirror.

Yeah. So what does it mean? This is another question from Peter Burberry. What does it mean? The generic integral transform is a provisional resource function. That's a Good question. So yeah, let me show that real quick because I didn't show show the link to. To the function repository documentation page, but I probably should have and you'll see a provisional market. That's where. So there's a documentation page. Do I have it up still? Let me just quickly do. Oh shoot. I'm only sharing. I don't know how to switch my share so I'm. I'm not going to show it. But there. Yeah. If you go to the web the WFR function repository and search for generic integral transform you'll see the documentation page for this function we've been discussing and also you'll see a red banner at the top that says provisional. What it means is you can try it out and use it. There are still some bugs that are known so I didn't want to fully put it out quite yet and like I said the will probably publish it in the very near future. I don't think that that should be a problem. But the inverse transforms which as I said before require some fox H technology that's still under development. We probably won't have that in until maybe early of 2024 so but hopefully it won't be provisional.

Yes. And to this notebook which viable for public you will make correction add this link, I believe.

Yeah, it may be in there.

You will add a link to mayor's form.

Ah, yeah, yeah.

You will type additional line.

Sure, sure, yeah. I don't know how we'll share that.

But not now after lecture.

Yeah. So officially our time is about up but we can continue to use the room is what our moderator told us. So we'll just continue to answer questions until. Until we have to leave which is at 4 o' clock but hopefully it won't be that long anyway. Jesus. I think is how you would pronounce it says how about developing discrete analysis. Can you give a comment? Well look, I'll let you.

Discrete analysis.

Discrete analysis.

Ah, let me tell it seem interesting. You don't know what I tell right now. I have several publications by mathematician Rico. You never heard from me a Russian mathematician who studied million discrete transform. So there exists different approaches to discrete transforms. There exists discrete Laplace mailing 4 year and some other transforms but it needs to have own investigators.

Yeah. Yes, I would say that yeah there's many discrete transforms that we didn't cover at all because. And they're used quite often. It's not that they're unimportant but it's just. It's just that our technology is a Different technology to do those calculations, so it's just not included. It would be as big a project probably as what we've done so far to do discrete analysis.

Yes, partially. We will try to add into repository collection transforms with summation instead Integration.

Yep. All right, so looks like that's about all the comments. Let me just check. Oh, there is another question here from Dave Galbi. I hope I'm saying that right. Does the using the G transform to compute an integral and then using activate function expand allow you to compute integrals for functions you couldn't compute otherwise the G transform. Interesting.

Maybe you can send a concrete example question. We cannot say generically.

Yeah, but I would say that we know having tested even just a simple transform like Fourier coast this what we presented today will handle cases that the built in Fourier coast transform the built in Mathematica one won't handle. So in that sense it does let you compute integrals you couldn't otherwise. I think it expands the class enhances the class of functions we can Currently.

We are working to demonstrate new abilities use based on folks and Meyer functions in comparison with those which realized in Mathematica. Because history of this you know development I'm in company 33 years and when came square root from X integrate 2 just started to be some independent objects because version 1 expanded to X as I remember version 2 was got new integrate. Then my former PhD student Victor Adamcik made code for definite integration. I made code for indefinite integration after Kelly Roach who worked here at that time. And then we worked many years to make war from function site and many other features to handle correctly branch cuts and many other so a computer mathematical is different from classical described in literature because argument for computers evaluation always on one complex plane. And in any case right now we need to make next level of our integration and integral transforms because at that time was done not so powerful. It was done not so powerful and new technology based on mere fox functions allowed in hundreds times speed that speed up integration hundreds times. You cannot believe but it's just substitution for a concrete parameter in to very generic formulas from product of two or even three major or fox H function. And if we have such formulas and we have already many such formulas just substitute the concrete parameters you will get concrete integral which current Mathematica try to evaluate several minutes or hanks sometimes and so on. And we're on the way to make to realize this ambassador project. Yes, no, we come I compare existing functionality with those which we are doing right now in repository functions.

Yeah, great. All right. I Don't see any more questions. Let me just pop over back. There's two places where questions can appear so I'm going to pop back over here. Ah, there is another question. Oh, actually there's a. There's a few. Danny has some comments. I won't read those. Sotiraismikos Sorry if I butchered the name. Are there any plans for including other big functions in the multivariable domain like Lauricello ones Also the Apple functions.

Okay, okay. I am working in this direction so several years Also and function F1 if you see history appeared about 25 years ago by my initiative because single not not domain no double integral representation exists for 4,4 IPLF 2,3 and 4 but Apple function F1 has representation through single integral and more generic function in this way is Lauricella FDF function and for Lorichella's ABCD functions it is extension of Apple functions. We have these formulas but not have don't have code for each other there and we know specialists who knows more than we in this area have contact with them. I hope we can implement Richella there function and also if we look on double hypergamytric series here first attention to Kumar functions, Humber functions and Horn functions We have collection double million bars integrals for these guys and I believe it will be next efforts to produce these functions to implementing resource for beginning and some of them can go to journal like Apple function why it is important? No, for example Riemann Green functions for partial differential equations if you try to solve such partial equations like area Poisson, Darpu or more generic we can use and should use such functions. By the way, I published book in this area it was my first book which presented for publication 40 more years ago and published after 30 years only in Russian exists.

Yeah, okay. There's still another question Peter Burberry oh no, wait, yeah, is that. That's Peter Burberry's Is there a way to generalize fox H to include include hoin G the documentation for fox H reduce states some advanced special functions are not represented in terms of fox H so. And then he shows an example in Mathematica code of trying to reduce hoin.

G coinc it is different type and class of functions because they have four singular points and and Gauss and other hypergamy have three maybe or two singular points. I'm talking about differential equations and some cases of course from coin functions can be converted to hypergeometric functions but you can ask this Tigran Ishkanyan who implemented this coin functions.

All right, great. Thanks, all who's still here? I'm just going to mark the chat as answered.