Everything to know about Mellin-Barnes Integrals - Part 2: Live with the R&D Team

Good health to everybody who listen me. I will talk on this lecture about million bars integrals. Again this object which we discussed in previous my lecture is very general and it includes practically all mathematical functions which people use. In my previous lecture I gave definition this marine bounds integrals and explained why and how to to work with them. Right now I shortly repeat these aspects and present new one. So this integrals is defined by the Marin Barnes integral when we have integration by contouring. For simplicity, I took here vertical line which parallel to imagine your axis and inside of this integral we have ratio of products gamma functions and here you see linear combinations with respect S and here alpha car positive symbolical maybe in terms if it is one. For example, we come to major Z function and here also positive arbitrary symbolical values. I mentioned such inequalities here and we have restrictions that this should be positive in sense gamma function from positive value does not have poles and the same here it should be argument positive if we take as moving by vertical line. And history of this integrals started from work of Pincherli and then Merlin and Barnes introduced Mealing transform which is equivalent of Laplace of Maryland trans Fourier transformation. I mentioned about this in previous lecture Dickson Farrar also studied and at last in 40s years Meir introduced merger function. And in last year in last my lecture I give list of particular cases of this function. It includes more than 100 functions which has names like elementary functions arctan logarithm, exponential or BC special functions like BC chair, Bashev, Gauss and other one and inconvenience of definitions functions which was based on this million Barnes integral and has names folks and major G function is evidently if you look on construction gammas which looks very similar to those which Slater no which was used for Marin bars integral we see strange situation like here we have mayor arms. Here we have continuation this set and with where Z10 like 1 minus Bezita. It is some kind craziness and gives very inconvenience in studying and practically practical work with these functions. Here I shortly demonstrated situation. Now for example, if we take this set coefficients we see list and then continuation the same list here, but with minus. No, it's okay, but if we look on parameter where you see beginning normal looks normally and then one minus where And I suggested to return back to fundamental definition which was met in articles of Dickson Ferrari and this definition includes A, B, C, D. Here I show only companion with two this function two sets. So definition include A, B, B, C, D and lens capital A capital B capital C capital D and alpha beta, gamma delta it is coefficients and it is like the same like in Fox H but produce a lot of important more easy usable details. Believe me, because I have experience with something like started from 1970. I started to work with this function before even and here we will get a more powerful set of opportunities if we make such definition at least for our resource functions for repository and I will talk how this definition converted allow to write formulas in more easy visible format. Before continuation talk, I will mention that Tigran Iskanyan who is working in Wolfram research and who implemented this function Foxh made talk about function fox H function I shortly start his talk Sorry. Talking maybe no, in any case I will not continue but you can listen his report about fox age function using this link. Okay, let me return back. To my file. Okay, I gave name this function Slater H function and similar function I called Slater G. It was described in my first book published in 1978 and translated five years later. Here about this notebook about this book I talked a previous lecture and you already saw this product for Slater there and for me. We find this definition in our documentation in Mathematica and again we see here this May +1Q and here we don't have such effect and this effect like one minus set parameters also absent in this talk I can mention that we have conversions one function to other for example Slater H to fox H and back and I show rules which allow to do this. This is exact formulas, exact formula. And here what happening with gammas. If we made this set of changes and this Fox and Slater age functions have main characteristics I mentioned in previous my talk. They looks like these sums which converts to such sums. And it's important to have restriction on argument no, in Mathematica we always argument have argument between minus PI and PI including PI but in theoretical literature we see such type restrictions and we have can be written this restriction in form of Slater alpha. It is from Dixon Ferrari. It has I have it in this notebook. Okay, Below I present sets of examples for million Barnes integrals and as I mentioned in my previous talk, we have about 100 plus cases when we can convert functions into major g form. Here I used resource function but we have also major Z reduce which is rather broken and weak function and should be replaced by functionality from this from this new function I will repeat evaluation on your eyes and you see it's loading package and takes time loading package in a resource function. Let me show what we have loading login. It's interesting oh, talks something. But in any case, you see, we have already representations and then we need to make table form. You try to make stable form. Oh, mate. Okay, so we have function and then mayor Z form and the same here functions and so on. You already saw this operation in my previous talk. Then I can make a conditional expression goes to A and we have only measure without conditions and similar variations allow to make for example in million Barnes representation. So each of these functions representable through this million Barnes integrals. And then we can convert this into sums. No, this here I show example how it works for logarithmic cases. It's interesting to see something new in this area. Okay, I used fox reduce. Again. We take previous functions and additionally meta Cleffler function and some others like Kumar function 1, F1. And as a result we get fox representation through folks function. We can write in such format. Also here I added new construction which is important also in application to risk potential. Basically K it is a kernel of so called car transform, Meyer transform bsg it is kernel hunky transform. So basically, for example, this one it is kernel fractional integral transform. This function, which I called Slater H function has the following main characteristics and why we have these characteristics and how they influence on everything. No, inside of gamma functions we have if we look asymptotical on asymptotical aspects of these gamma functions we find by Sterling formula companion like Z in power Z exponential in power. No exponential function from Z. In this case it is S from S and also power function like S in degree atom some kind. And all these three guys are living inside exponent exponent inside of asymptotic expansions of this product and this companions can exist or can be in degree zero like disappear. And also can be integral, negative or positive. And this values if they equal 0 or positive or negative, provides regulation of behavior. This marine bands integrals and fox H Slater H functions. You should not forget that inside of this construction for concrete parameters A, B, C, D and so on, we have more than 100 particular cases of very different kind. You know, for example, we meet the exponential function. We meet the sinus cosine basis. It is some kind periodical type functions. We mean the functions like piecewise. I show you kernel in fractional derivative or in this transform. No, many different guys live in this house by name Slater H. Okay, and here I I'm talking started to talk about this more accurately. We can write exponential expansions of each companion. Here, for example, you see this expansion. Oh, sorry, I don't want to open something. We Verified this formula here. And then gamma function. This is sterling formula. And if we have a ratio of 2 gamma function it is just asymptotically exponent not an explain power power s in degree difference. If A equal B it is 1. Yes, but if we move at infinity in vertical or opposite direction for real X and Y we have such behavior this formula. This formula. And if we accurate it is coming from sterling formula. And if we make expansions, we after some exercises come to the formula which practically useful. But look at here we don't find companion Z in degree Z. Instead we find companion no why in degree iy. But it is restricted companion if we have real why? And so as a result in vertical direction we have exponential term or in main symptomatic expansion and power term term like Y in degree Y absent it makes very special moments in using. But if we have I applied this steering formulas for each of these gammas. And then we have opportunity to build asymptotic for gamma product. We substitute for each gamma main terms here make such operations. Make such operations. For example, as a result we come to very nice looking main term of this big big product. We find in this main term Z in the S in degree minus S with coefficient coefficient is this one. And also we find exponential term here something in degree minus S and find this power term with corresponding coefficients. And if we accurately combine this, like I made in my first book mentioned before. We can build a restriction for conversions of this which allow to converge this marine bands integral. No, Right now I will go to convolution and perceival equality of marine for marine transform. Here we have marine transform is from integral which is convolution for million transform. And we have result as a result marine transform of each of these. It's easier to understand if here we have for example five gamma functions. Here we have three gamma functions. Not. Not here, sorry, Not inside here. But this F1 it is a result of ovulation. Marian transform of each of these F1 and F2 includes for example here five gamma function. Here three gamma functions. Together we have eight gamma functions or something cancelled a little. And then we know eight gamma functions which is a result of marine transform of something. It means inverse lab, meaning transform from this result gives value of integral. Value of integral. And here I made. I show how to derive formula. If we have F1 and F2 Slater H functions. I put tilde on top of one of them. And this one you saw before. Then I made these operations accurately. And what we have, we have the following. No, at first. Integral from this expression. I will it accurately and integral integral from this one also done. And at last where it is? Sorry, I lost a little Marine transform of this one. Example of using Jack. Oh, sorry. Here. In my notebook I have to find formula. Formula is where it is. Maybe I understood what happened. A technological problem. Formula big formula disappeared during some manipulations. But here I gave result example evolution of integral from product of two functions. If we have one function equal. Equal. Equal. This one other function this one. Marine transform of this it. It should be formula here where it is. Ah, okay. Maybe here. So we have integral from such product. But for concrete values A and B beta and gamma. And formula should include arbitrary beta and gamma. It disappeared. It should be stay here. It can happen by technological reason because we made two widescreen the white screen. No formula. Basically here you. You see this integral. It is integral median transform from one function. Multiply million transform as a function and we have resulting formula. After my lecture, we make reductions of this notebook in in sense formula will return back. Okay, one more example. We know that if we have for example gamma functions we can make inverse transform from gamma function and get original function. In this case fox H which can be converted into mere Slater Slater H function. I did not do this yet. But here we have representation, Representation and integral from one Fox H Marine transform from fox H. It is this result because it's inversion from previous integral under corresponding conditions. I did not write these conditions here, but below they exist because as I mentioned, I did not give completely verified and version of this material. And here you see constructions for conditions. I know that Danny is working with this area and we hope to have soon in Mathematica realized this formula and particular case post later for mayor's Z function. No, below again I afraid. Maybe we have problem here or not? Let me show do we have problem? No, for now I don't see problem. Duck. So this integral from million from it is inverse Marine transform from product of marine transform of each function. And here I accurately derived this formula and we have a result this guy equal to this one. It is more generic function integral from product of box or Slater H Functions can be represented through pox or Slater H. This formula in such format is new formula. And. As a new formula which I demonstrate here is the following. Now we have for example Slater H function. It means we have this product, This product and we can do the following. Oh, I see. It looks like. No, it's not my mistake, but F1. Okay, F2 similar we can write a Product Product Original original function as this product and the rest. No. For example, we have here something like 5, not 5, 15 and here 3 multiply on 5, it is 15. In such style I divide it and inside of this group we can take any not only first in order in Gamma functions. So a lot of representations in such two factors. Yes, and each such representation it is can be written. It can be written as integral integral. Because here it is gammas which for one function gammas from for other function we have together original function. And so we can make a lot of integral representations for Slater H function. It's interesting to evaluate Laplace transform and other integral transforms using these ideas and technology. Okay, here I should mention that we may talk about compositional structure of classical integral transforms on conference half year ago. And here we have example when we apply Laplace transform to Slater H function. Look here it is generic formula, which you know from previous generic formula here. Yes, it is huge. But what to do? It's life. Huge Formula monstros. So we have integral convolution type from product of two Slater functions. And then again results later. H, if one function it is exponential, it means we have Laplace transform from other function equal to something which appeared from this one. And here we have examples. Examples of this procedure. I am sorry, Of course I should not make so long lines here. But we fixed problem to such style, which looks more but worse than previous one. At last moments. No, here we can see formula formula for Laplace transform. It is again Fox H I wrote here. Equal to this one Laplace transform of this later function. It is this one. And at last we can tell about particular cases. Cases of mere Z function are the cases of Fox H and of Slater H. Of course, and we have very specific case which are very popular for differential equations. Who studied differential equations? It is meta Kleffler function. If you see for example, for situation 10 here. No, not here. 011. If you put 1, 1, you get exponential function from right sum. And for this function we have transform million and Laplace transforms. And here I want to mention effect connected with this coefficient. How to convert Fox functions into mir G? Not only for cases, when we have everywhere one, one and so on. Don't forget that we have multiplication formula for gamma function. If we have this product and Z we can write like product with n factors. But here only Z and something like coefficients appeared. So here we have such construction. And if we Instead alpha right, for example 2 divided on 3, then we can change variable integration and get here 2s news differents and here 3s. Here 3s. Then we can instead of this 3s, write a product of gamma from S using above formula. Here also and here only 2 because coefficient 2. So what we have we have many gammas, 6 in numerator and 2 in denominator. Instead of this three gammas. But each gamma here has coefficient plus or minus S. So this can be used for building FOX a measure function. But how Many will be 3? Sum of 3 merger functions. You can image if you write Here something like 10 divided on 11. Here you get 11. Major functions from such companions. By this reason constructions through folks more convenient in applications. More compact applications than mere G constructions. Here I evaluated. Example. You see here. With respect to delta we have marine transform of meta clefler. Multiply on this one. With respect by pair we have Laplace transform of this function. And all this it is FOX function. And I present verification of such stuff. Now here you see formulas which allow. Which useful for deriving these operations. Question how to evaluate this later or fox H function. They can be evaluated by through sum of residues in poles corresponding poles. It is poles which you saw before. S equal to this form function. This is instead of such construction we have poles in point in these points. Then we can evaluate residues. But don't forget that we have cohesion alpha car. The same we can do for other residues in right poles. And we make. We should make summation by all poles. It is left poles, right poles. Infinite sum appeared and technologically it looks like this one. We should replace one. Singular situation value in into residues. It is poor formally, but we can do this absolutely accurately and with proofs and everything. But I show how formulas big formulas appear. No, Then we can try to manipulate with this gammas convert to and other manipulations can be done successfully only for situations when we have rational values for coefficients. As I mentioned before, Then it is convertible to hypergyometric prefq functions. No major in such style produce such representations. Generically we have constructions like Slater construction. It is. Sum of left residues in left poles and sum of residues in right poles 0. It is technological aspect in mathematical no 0 should be no. This formula I show in previous my lecture. Here I tried a placed material which is not done well, but we can be useful. Because people started to ask about asymptotics. Folks age about other moments and I collected information from my previous research. For example, here we have formulas like this one representable through FOX H I believe and tried such formula is not working in current FOX H cases. This is for people who wants to go more deeper in this area. Main term asymptotics, main term. New. For example, here if you go to Internet you find article in Wolfram function site about major G. What what is happening? You see number, we have number or formula. And you can find this formula, but you cannot find this formula, this formula and this formula you cannot find because this formulas appeared later. We stopped to develop visible version wall from function site maybe 15 years ago. But I did not stop my own developing. And right now we have partially implemented, but not completely implemented advanced more advanced versions for Wolfram function site in mathematical. And if somebody wants, I can present such type additional formulas we can place for fox not for measure and so on these formulas and people can use them. But now I attach portion of my research here in report. I think that next report I can devote to fractional integral differentiation or other questions. Now, for example, integral transforms. Because everything is under development and I'm still I am not machine to do so many formulas. In spite that I made already a lot and I'm too busy with other projects. And of course my attention devoted to this area to make jump huge jump in integrate and integral transforms directions in mathematical system involved from language. And also I want to teach people how to operate with these difficult objects. Because myself I started 50 years ago, started to work and it was not easy to follow people who did not care about teaching procedures for young people. Many things I opened myself recreated and will recreate in this area. And if somebody wants to work, we have a lot what to do in this area. And all should go to computers. You see how difficult how horrible formulas looking. And for example, I have formulas for relation arbitrary indefinite integrals from product of arbitrary rational functions in arbitrary degrees. But this needs in Luricella function, which we still don't have in Mathematica. And I hope we will have Luricella G function. Now we have only Apple F1 function as first step in this direction. And I have many formals in this area which still not implemented to mathematical. Thank you for attention and my lecture is over. Questions? No questions. Thank you.