New in Generic Integral Transforms

Paco Jain & Oleg Marichev, Wolfram Research

(RowDefault[, (Spacer1), , (Spacer1), ])  #WolframTechConf

Abstract

Practically all algebraic functions and named functions of the hypergeometric type occur as special cases of the Fox H-function. Furthermore, the convolution integral of the product of two or more FoxH functions is, in a large class of cases, itself a FoxH function, known in terms of the parameters of the two Fox H factors. In particular, about 80% of all definite integrals evaluated in handbooks and articles are the cases of this type of generic Fox H integration. Following up on previous talks that highlighted the use of this technology for symbolically computing fractional diffeo-integrals and generic integral transforms, our talk will demonstrate recent advancements in GenericIntegralTransform.

Today, following on several years of work, we will present developments to the SpecialFunctionsAndCalculus paclet.

Load the paclet:

Quit[]

PacletInstall["WolframAlphaMath/SpecialFunctionsAndCalculus"]
Needs["WolframAlphaMath`SpecialFunctionsAndCalculus`"]
PacletObject["WolframAlphaMath/SpecialFunctionsAndCalculus"]["Location"]

SystemOpen[%]

Primer: Mathematical functions as cases of superfunctions MeijerG and FoxH

MeijerG and FoxH functions can be defined by formulas (see  links)

MeijerG[{{(Subscript[a, 1]),…,(Subscript[a, n])},{(Subscript[a, n+1]),…,(Subscript[a, p])}},{{(Subscript[b, 1]),…,(Subscript[b, m])},{(Subscript[b, m+1]),…,(Subscript[b, q])}},z]==(1/(2 π I))ContourIntegrate[((((∏_(k=1)^m)Gamma[(Subscript[b, k])+s]) (∏_(k=1)^n)Gamma[1-(Subscript[a, k])-s] )/(((∏_(k=n+1)^p)Gamma[(Subscript[a, k])+s]) (∏_(k=m+1)^q)Gamma[1-(Subscript[b, k])-s]))(z^(-s)),{s,ℒ}]

(Subscript[G, p,q]^(m,n))(z\[VerticalSeparator]({{(Subscript[a, 1]),…,(Subscript[a, n]),(Subscript[a, n+1]),…,(Subscript[a, p])}, {(Subscript[b, 1]),…,(Subscript[b, m]),(Subscript[b, m+1]),…,(Subscript[b, q])}}))==(1/(2 π I))(∫_ℒ)((((∏_(k=1)^m)Γ(s+(Subscript[b, k]))) (∏_(k=1)^n)Γ(1-(Subscript[a, k])-s))/(((∏_(k=n+1)^p)Γ(s+(Subscript[a, k])) )(∏_(k=m+1)^q)Γ(1-(Subscript[b, k])-s)))(z^(-s))\[DifferentialD]s

FoxH[{{{(Subscript[a, 1]),(Subscript[α, 1])},…,{(Subscript[a, n]),(Subscript[α, n])}},{{(Subscript[a, n+1]),(Subscript[α, n+1])},…,{(Subscript[a, p]),(Subscript[α, p])}}},{{{(Subscript[b, 1]),(Subscript[β, 1])},…,{(Subscript[b, m]),(Subscript[β, m])}},{{(Subscript[b, m+1]),(Subscript[β, m+1])},…,{(Subscript[b, q]),(Subscript[β, q])}}},z]==(1/(2 π I))ContourIntegrate[((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])s]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) s] )/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) s]) (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) s]))(z^(-s)),{s,ℒ}]

==(1/(2 π I))(∫_ℒ)((((∏_(j=1)^m)(Gamma[(Subscript[b, j])+(Subscript[β, j])s]))(∏_(i=1)^n)(Gamma[1-(Subscript[a, i])-(Subscript[α, i]) s]) )/(((∏_(i=n+1)^p)(Gamma[(Subscript[a, i])+s (Subscript[α, i])])) (∏_(j=m+1)^q)(Gamma[1-(Subscript[b, j])-(Subscript[β, j]) s])))(z^(-s)) \[DifferentialD]s

It is evidently to see that MeijerG is the case of FoxH for unite coefficients (Subscript[α, i])==(Subscript[β, j])==1:

MeijerG[{Table[(Subscript[a, i]),{i,1,n}],Table[(Subscript[a, i]),{i,n+1,p}]},{Table[(Subscript[b, j]),{j,1,m}],Table[(Subscript[b, j]),{j,m+1,q}]},z]==FoxH[{Table[{(Subscript[a, i]),1},{i,1,n}],Table[{(Subscript[a, i]),1},{i,n+1,p}]},{Table[{(Subscript[b, j]),1},{j,1,m}],Table[{(Subscript[b, j]),1},{j,m+1,q}]},z]

In the subsubsections below we demonstrate lists FoxMeijer170 with 170 named mathematical functions from Mathematica which can be represented as cases MeijerG and FoxH functions:

MeijerGForm[f[z],z]      
FoxHForm[f[z],z]

Also we have a large test set with 10148 compositions of these functions, which after 278.49 Seconds was transformed into MeijerG and FoxH forms. For example,

{#,MeijerGForm[#,z],FoxHForm[#,z]}&/@{(E^(-z)),((Sin[z])^n),HypergeometricPFQ[Table[(Subscript[a, i]),{i,1,p}],Table[(Subscript[b, j]),{j,1,q}],z]}

FoxMeijer170

FoxMeijer170={(a^z),(E^z),(1/(1-z)),(Sqrt[z]),(z^b),(((1+z))^a),((Abs[1-z])^a),AiryAi[z],AiryAiPrime[z],AiryBi[z],AiryBiPrime[z],AngerJ[a,z],AngerJ[a,b,z],ArcCos[z],ArcCosh[z],ArcCot[z],ArcCoth[z],ArcCsc[z],ArcCsch[z],ArcSec[z],ArcSech[z],ArcSin[z],ArcSinh[z],ArcTan[z],ArcTan[a,z],ArcTan[z,a],ArcTanh[z],BesselI[a,z],BesselJ[a,z],BesselK[a,z],BesselY[a,z],Beta[z,a,b],Beta[c,z,a,b],Beta[z,c,a,b],BetaRegularized[z,a,b],BetaRegularized[c,z,a,b],BetaRegularized[z,c,a,b],BilateralHypergeometricPFQ[{(Subscript[a, 1])},{(Subscript[b, 1])},z],BilateralHypergeometricPFQ[{(Subscript[a, 1]),(Subscript[a, 2])},{(Subscript[b, 1]),(Subscript[b, 2])},z],BilateralHypergeometricPFQ[{(Subscript[a, 1]),(Subscript[a, 2])},Table[(Subscript[b, i]),{i,1,q}],z],BilateralHypergeometricPFQ[{(Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3])},{(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3])},z],BilateralHypergeometricPFQ[Table[(Subscript[a, i]),{i,1,p}],{b1,b2,b3},z],BilateralHypergeometricPFQ[Table[(Subscript[a, i]),{i,1,p}],Table[(Subscript[b, i]),{i,1,q}],z],CarlsonRC[x,z],CarlsonRC[z,y],CarlsonRE[y,z],CarlsonRK[y,z],ChebyshevT[a,z],ChebyshevU[a,z],Cos[z],Cosh[z],CoshIntegral[z],CosIntegral[z],DawsonF[z],EllipticE[z],EllipticK[z],Erf[z],Erf[a,z],Erf[z,b],Erfc[z],Erfi[z],ExpIntegralE[a,z],ExpIntegralEi[z],Fibonacci[z],Fibonacci[a,z],FresnelC[z],FresnelF[z],FresnelG[z],FresnelS[z],Gamma[a,z],Gamma[a,b,z],Gamma[a,z,b],GammaRegularized[a,z],GammaRegularized[a,b,z],GammaRegularized[a,z,b],GegenbauerC[a,b,z],HankelH1[a,z],HankelH2[a,z],Haversine[z],(((1-z))^a) HeavisideTheta[1-Abs[z]],(((-1+z))^a) HeavisideTheta[-1+Abs[z]],HermiteH[a,z],Hypergeometric0F1[a,z],Hypergeometric0F1Regularized[a,z],Hypergeometric1F1[a,b,z],Hypergeometric1F1Regularized[a,b,z],Hypergeometric2F1[a,b,c,z],Hypergeometric2F1Regularized[a,b,c,z],HypergeometricPFQ[{(Subscript[a, 1]),(Subscript[a, 2])},Table[(Subscript[b, j]),{j,1,q}],z],HypergeometricPFQ[Table[(Subscript[a, j]),{j,1,p}],{b1,b2,b3},z],HypergeometricPFQ[Table[(Subscript[a, j]),{j,1,p}],Table[(Subscript[b, j]),{j,1,q}],z],HypergeometricPFQRegularized[{(Subscript[a, 1]),(Subscript[a, 2])},Table[(Subscript[b, j]),{j,1,q}],z],HypergeometricPFQRegularized[Table[(Subscript[a, j]),{j,1,p}],{b1,b2,b3},z],HypergeometricPFQRegularized[Table[(Subscript[a, j]),{j,1,p}],Table[(Subscript[b, j]),{j,1,q}],z],HypergeometricU[a,b,z],InverseHaversine[z],KelvinBei[0,z],KelvinBei[a,z],KelvinBer[0,z],KelvinBer[a,z],KelvinKei[0,z],KelvinKei[a,z],KelvinKer[0,z],KelvinKer[a,z],LaguerreL[a,z],LaguerreL[a,b,z],LegendreP[ν,z],LegendreP[a,b,z],LegendreP[a,b,2,z],LegendreP[a,b,3,z],LegendreQ[ν,z],LegendreQ[a,b,z],LegendreQ[a,b,2,z],LegendreQ[a,b,3,z],Log[z],LucasL[z],LucasL[a,z],ParabolicCylinderD[a,z],({{\[Piecewise], ({{1, Abs[z]>1}, {0, True}})}}),({{\[Piecewise], ({{1, Abs[z]<1}, {0, True}})}}),({{\[Piecewise], ({{(((1-z))^a), Abs[z]<1}, {0, True}})}}),AppellF3[a,a1,b,b1,c,1-(1/z),1-z] (({{\[Piecewise], ({{(((1-z))^(-1+c)), Abs[z]<1}, {0, True}})}})),AppellF3[a,a1,b,b1,c,1-z,1-(1/z)] (({{\[Piecewise], ({{(((1-z))^(-1+c)), Abs[z]<1}, {0, True}})}})),Hypergeometric2F1[a,b,c,1-(1/z)] (({{\[Piecewise], ({{(((1-z))^(-1+c)), Abs[z]<1}, {0, True}})}})),Hypergeometric2F1[a,b,c,1-z] (({{\[Piecewise], ({{(((1-z))^(-1+c)), Abs[z]<1}, {0, True}})}})),({{\[Piecewise], ({{(((-1+z))^a), Abs[z]>1}, {0, True}})}}),AppellF3[a,a1,b,b1,c,1-(1/z),1-z] (({{\[Piecewise], ({{(((-1+z))^(-1+c)), Abs[z]>1}, {0, True}})}})),AppellF3[a,a1,b,b1,c,1-z,1-(1/z)] (({{\[Piecewise], ({{(((-1+z))^(-1+c)), Abs[z]>1}, {0, True}})}})),Hypergeometric2F1[a,b,c,1-(1/z)] (({{\[Piecewise], ({{(((-1+z))^(-1+c)), Abs[z]>1}, {0, True}})}})),Hypergeometric2F1[a,b,c,1-z] (({{\[Piecewise], ({{(((-1+z))^(-1+c)), Abs[z]>1}, {0, True}})}})),PolyLog[2,z],PolyLog[a,z],ScorerGi[z],ScorerGiPrime[z],ScorerHi[z],ScorerHiPrime[z],((Abs[1-z])^a) Sign[1-z],((Abs[1-z])^a) Sign[-1+z],Sin[z],Sinc[z],Sinh[z],SinhIntegral[z],SinIntegral[z],SphericalBesselJ[a,z],SphericalBesselY[a,z],SphericalHankelH1[a,z],SphericalHankelH2[a,z],StruveH[ν,z],StruveL[ν,z],UnitStep[z],UnitStep[1-Abs[z]],(((1-z))^a) UnitStep[1-Abs[z]],(((1-z))^(-1+c)) AppellF3[a,a1,b,b1,c,1-(1/z),1-z] UnitStep[1-Abs[z]],(((1-z))^(-1+c)) AppellF3[a,a1,b,b1,c,1-z,1-(1/z)] UnitStep[1-Abs[z]],UnitStep[-1+Abs[z]],(((-1+z))^a) UnitStep[-1+Abs[z]],(((-1+z))^(-1+c)) AppellF3[a,a1,b,b1,c,1-(1/z),1-z] UnitStep[-1+Abs[z]],(((-1+z))^(-1+c)) AppellF3[a,a1,b,b1,c,1-z,1-(1/z)] UnitStep[-1+Abs[z]],WeberE[ν,z],WeberE[ν,a,z],WhittakerM[a,b,z],(E^(-z/2)) WhittakerM[a,b,z],(E^(z/2)) WhittakerM[a,b,z],WhittakerW[a,b,z],(E^(-z/2)) WhittakerW[a,b,z],(E^(z/2)) WhittakerW[a,b,z],MittagLefflerE[(3/4), z],MittagLefflerE[ρ, z],MittagLefflerE[(2/7), μ,z],MittagLefflerE[ρ, μ,z]};

Exploring the SpecialFunctionsAndCalculus paclet

The paclet defines the following six Wolfram Language functions:

SampleUnivariateFunctions:   produce mathematical expressions to use as inputs to the other functions

(*SeedRandom[4]*)
in=SampleUnivariateFunctions[]

MeijerGForm/FoxHForm:   convert functions to MeijerG/FoxH

meijerGForm=MeijerGForm[in, \[FormalZ]]

foxHForm=FoxHForm[in, \[FormalZ]]

MarichevIntegrate:   like Integrate, using MeijerG/FoxH technology

integrate=MarichevIntegrate[in, {\[FormalZ], 1, 2}]

integrate//Activate//FunctionExpand

FractionalOrderD:   compute (indefinite) fractional-order diffeo-integrals

FractionalOrderD[in, {\[FormalZ], 3/2}]

GenericIntegralTranform:   compute integral transforms for a variety of kernels

GenericIntegralTransform[in,\[FormalZ],\[FormalP], "Laplace"]

Generic integral transforms

For other functions, see our previous years' (2020-2023) WTC talks; today I will focus on GenericIntegralTransform. What is an integral transform?

F[t]:= Integrate[f[x]K[x, t], {x, a, b}]

The choice of kernel function K[x,t] defines the particular transform in question.

GenericIntegralTransform[]

Transform Examples (14)

Fourier

GenericIntegralTransform[x/(x^2-1),x,z,"Fourier"]

FourierInverse

GenericIntegralTransform[x/(x^2-1),x,z,"FourierInverse"]

H

GenericIntegralTransform[Sin[x], x, t, {"H", {{},{}},{{{0,1}},{}}}]

HInverse

GenericIntegralTransform[Sin[x], x, t, {"HInverse", {{},{}},{{{0,1}},{}}}]

Hankel

GenericIntegralTransform[Sin[x],x,z,{"Hankel", ν}]

Hilbert

GenericIntegralTransform[Sin[x],x,z,"Hilbert"]

KontorovichLebedev

GenericIntegralTransform[Sin[x],x,z,"KontorovichLebedev"]

Laplace

GenericIntegralTransform[Sin[x],x,z,"Laplace"]

LaplaceInverse

GenericIntegralTransform[Sin[x],x,z,"LaplaceInverse"]

Mellin

GenericIntegralTransform[Exp[-x],x,b,"Mellin"]

Neumann

GenericIntegralTransform[Sin[x],x,z,{"Neumann", ν}]

Stieltjes

GenericIntegralTransform[Sin[x],x,z,{"Stieltjes", ρ}]

StieltjesInverse

GenericIntegralTransform[Sin[x],x,z,{"StieltjesInverse", ρ}]

Struve

GenericIntegralTransform[Sin[x],x,z,{"Struve", ν}]

Weyl

GenericIntegralTransform[Sin[x],x,z,{"Weyl",α}]

Focusing on Laplace

report=AnalyzeTestSuite["LaplaceFoxH-10146"]

report[["results","with result",1]]//RandomChoice

round-tripping example

in=((AiryBi[((((3/2)))^(2/3)) (\[FormalZ]^(2/3))])/(E^\[FormalZ]))

laplace=GenericIntegralTransform[in,\[FormalZ],\[FormalS],"Laplace"]
laplace//Activate//FunctionExpand

laplaceInverse= GenericIntegralTransform[laplace, \[FormalS],\[FormalZ], "LaplaceInverse"]
laplaceInverseActive=laplaceInverse//Activate//FunctionExpand

{in, laplaceInverseActive//Simplify}

comparison with System`LaplaceTransform

ints=RandomSample[SampleUnivariateFunctions[All, "FunctionType"->"MediumTestSet"],100]

ints[[1]]

systemLaplace={#,AbsoluteTiming@TimeConstrained[LaplaceTransform[#,\[FormalZ], \[FormalT]],15]}&/@ints;

Grid[systemLaplace, Frame->All]

avgTime=Total@systemLaplace[[All,2,1]]/100

Length@Select[systemLaplace[[All,2,2]],#===$Aborted&]

Length@Select[systemLaplace[[All,2,2]],!FreeQ[#,LaplaceTransform]&]

numerical confirmation

in=SampleUnivariateFunctions["TrueUnivariate"->True]

With[{\[FormalS]=s=RandomComplex[]},
integrateLaplace=NIntegrate[in Exp[-\[FormalZ] \[FormalS]],{\[FormalZ],0,Infinity}]//EchoTiming
]

laplace=GenericIntegralTransform[in,\[FormalZ],\[FormalS],"Laplace"]//EchoTiming
laplaceActivate=%//Activate//FunctionExpand

{integrateLaplace, laplaceActivate}/.\[FormalS]->s

Next steps

conditions of convergence

handling of distributions (generalized functions), e.g. DiracDelta

additional testing/bench-marking

better documentation

Citations

Biography

(HyperlinkTemplate[https://en.wikipedia.org/wiki/Oleg_Marichev, https://en.wikipedia.org/wiki/Oleg_Marichev, https://en.wikipedia.org/wiki/Oleg_Marichev, HyperlinkActionRecycled, HyperlinkActive, , ])

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Works of Oleg Marichev

Marichev O. I. A method of calculating of integrals of special functions. (Theory and tables of formulas) (in  Russian), Nauka i Tekhnika, Minsk, 1978.

Marichev, O. I. (Маричев, Олег  Игоревич)  (1983) . Handbook  of  integral  transforms  of  higher  transcendental  functions : theory  and  algorithmic  tables .       
                              Ellis  Horwood  Series  in  Mathematics  and  its  Applications  (in  Russian) . Translated  by  Longdon, L . W . Chichester : Ellis  Horwood  Ltd .

Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.); Prudnikov, A.  P.  (Прудников, А.  П.)  (1986) . Tables  of  indefinite  integrals  (in  Russian) .  
                              Translated  by  Gould, G . G . Moscow : Nauka  (Наука) .

Prudnikov, A. P.  (Прудников, А. П.); Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.). Integraly  i  ryady  Интегралы  и  ряды [Integrals  and   
                              series]  (in  Russian) . Vol . 1–5  (1  ed .) . Nauka  (Наука) . 1981 − 1986.

Prudnikov, A. P.  (Прудников, А. П.); Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.)  (2002) [1986]. Integrals  and  Series. Vol. 1 : 
                             Elementary  Functions . Translated  by  Queen, N. M. (1  ed.). Gordon &  Breach  Science  Publishers/CRC  Press.  (798 pages.)

Prudnikov, A. P.  (Прудников, А. П.); Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.)   (2002)[1986] . Integrals  and  Series . Vol . 2 : Special  
                              functions . Translated  by  Queen, N . M . (1  ed .) . Gordon &  Breach  Science  Publishers/CRC  Press. (750 pages.)

Prudnikov, A. P., Brychkov, Yu. A. and  Marichev, 0. I. Evaluation  of  integrals  and  Mellin  transform. Itogi  Nauki  i  Techniki. Math. Anal. VINITI. 27, 3 - 146.     
                              (Russian). (1989).

Prudnikov, A. P.  (Прудников, А. П.); Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.)  (2002)[1990] . Integrals  and  Series . Vol . 3 : More  
                              special  functions . Translated  by  Gould, G . G . (1  ed .) . Gordon &  Breach  Science  Publishers/CRC  Press . (800 pages.)

Prudnikov, A. P.  (Прудников, А. П.); Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.) . Integrals  and  Series . Vol . 4 : Direct  Laplace  
                              Transforms  (1  ed .) . Gordon &  Breach  Science  Publishers/CRC  Press . (618  pages .)

Prudnikov, A. P.  (Прудников, А. П.); Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.). Integrals  and  Series . Vol . 5 : Inverse  Laplace  
                              Transforms  (1  ed .) . Gordon &  Breach  Science  Publishers/CRC  Press .(595  pages .)

Prudnikov, A. P.  (Прудников, А. П.); Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.)  (2003) . Integraly  i  ryady  Интегралы  и  ряды    
                            [Integrals  and  series]  (in  Russian) . Vol . Set  1 - 3  (2 nd  revised  ed .) . Fiziko - Matematicheskaya  Literatura, Fizmatlit[ru]  (Физматлит) . (reprint   
                            2013  by  Let  Me  Print)

Prudnikov, A. P.  (Прудников, А. П.); Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.) (2003) . Integraly  i  ryady  Интегралы  и  ряды  
                           [Integrals  and  series : Elementary  functions]  (in  Russian) . Vol . 1 : Elementarnye  funktsii  (Элементарные  функции)  (2 nd  revised  ed .) . Fiziko -  
                            Matematicheskaya  Literatura, Fizmatlit[ru]  (Физматлит) . (reprint  2013  by  Let  Me  Print)

Prudnikov, A. P.  (Прудников, А. П.); Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.) (2003). Integraly i ryady Интегралы и ряды [Integrals 
                           and series: Special functions] (in Russian). Vol. 2: Spetsialnye funktsii (Специальные функции) (2nd revised ed.). Fiziko-Matematicheskaya 
                           Literatura, Fizmatlit [ru] (Физматлит). (reprint 2013 by Let Me Print)

Prudnikov, A. P.  (Прудников, А. П.); Brychkov, Yu.  A.  (Брычков, Ю.  А.); Marichev, O.  I.  (Маричев, О.  И.) (2003). Integraly i ryady Интегралы и ряды [Integrals 
                          and series: Special functions. Further chapters.] (in Russian). Vol. 3: Spetsialnye funktsii. Dopolnitelnye glavy (2nd revised ed.). Fiziko-
                          Matematicheskaya Literatura, Fizmatlit [ru] (Физматлит). (710 pages) (reprint 2013 by Let Me Print)

Brychkov Yu. A., Marichev O. I., Savischenko N. V. Handbook of Mellin transforms, Advances in Applied Mathematics, CRC Press, Boca Raton, FL, 2019. (587   
                          pages.)

Samko, S . G ., Kilbas, A . A . and  Marichev, 0.1 . (1987) . Fractional  integrals  and  derivatives  and  some  of  their  applications . Minsk . Nauka  i  Tekhnika .      
                         English  translation : Fractional  integrals  and  derivatives : Theory  and  Applications . Gordon  and  Breach . 1993.  London . New  York .  (976 pages.)

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                (Marichev Oleg, Compositional Structure of Classical Integral Transforms, Wolfram Technology conference conference of 2023)

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Other literature (useful for development)

Bateman, H., Erdelyi, A. , Tables of Integral Transforms,  Vols  I  and  II . McGraw - Hill, New  York. (1954). (391 & 451 pages).

Braaksma, B . L . J . Asymptotic   expansions   and   analytic   continuations   for  a   class   of   Barnes - integrals . Compositio   Math . 15, 3, 239 - 341. (1964).

Brychkov, Yu . A . and   Prudnikov, A . P . Integral   transforms   of   generalized  functions . Moscow : Nauka . (1977).

Brychkov, Yu. A., Glaeske, H.-Yu., Prudnikov  A. P. and  Vu  Kim  Tuan.  Multidimensional  integral  transforms. Gordon  and  Breach. New  York. London. (1992).

Davis, B. Integral  transforms  and  their  applications . Springer - Verlag . Berlin . New  York. (1978) .

Erdelyi, A ., Magnus, W ., Oberhettinger, F . and  Tricomi, F . G. Higher  Transcendental  Functions, Vols  I  and  II . McGraw - Hill, New  York. (1953).

Gradshteyn, I . S . and  Ryzhik, I . M . Table   of   integrals, series   and   products . New - York . London . Toronto . Sydney . San  Francisco. (1980).

Kilbas  A . A ., Saigo  M ., H - Transforms : Theory  and  Applications, Chapman &  Hill/CRC, (2004), (389  pages .)

Kiryakova V.,A Guide to Special Functions in Fractional Calculus (2021).  (HyperlinkTemplate[https://www.mdpi.com/2227-7390/9/1/106, https://www.mdpi.com/2227-7390/9/1/106, https://www.mdpi.com/2227-7390/9/1/106, HyperlinkActionRecycled, HyperlinkActive, , ])

Luke, Y . L .  The  Special  Functions  and  Their  Approximations . Vols . ! and  II . Academic  Press . New  York. (1969).

Marichev, O., & Shishkina, E. Overview of fractional calculus and its computer implementation in Wolfram Mathematica, Fractional Calculus and Applied Analysis. (2024). p. 1 - 68. https://doi.org/10.1007/s13540-024-00332-x

Mathai, A . M . and  Saxena, R . K .  The  H - functions  with  applications  in  statistics  and  other  disciplines . New  York; London; Sydney : John  Wiley. (1978) .

Mathai, A.M., Saxena, R.K. and Haubold, H.J., The   H - function, Berlin, New   York : Springer - Verlag, (2010).

Nguyen  Thanh  Hai  and  Yakubovich, S . B .  The  double  Mellin - Barnes  type  integrals  and  their  applications  to  convolution  theory. World  Scientific   
         Intern . Pub! . Series  on  Soviet  and  East  European  Mathematics . 6.  Singapore, New  Jersey, London  and  Hong  Kong . (1992). (295 pages).

Oberhettinger, F . and  Higgins, T. P. Tables  of  Lebedev, Mehler  and  generalized  Mehler  transform. Boeing  Sc. Res . Laboratories, Seattle . Washington. (1961).

Oberhettinger, F .  Tables  of  Mellin  transforms . New - York - HeidelbergBerlin : Springer - Verlag . (1974).

Paris,  R. B. and  Kaminski  D.  Asymptotics  and  Mellin - Barnes  Integrals, Cambridge  Univ . Press, (2001), (422  pages .)

Sasiela,  R. J. Electromagnetic  Wave  Propagation  in  Turbulence, Evaluation  and  Application  of  Mellin  Transforms, Springer - Verlag, (1994). (300 pages).

Slater Lucy Joan, Generalized  hypergeometric  functions, Cambridge  univ. press, 1966, (273  pages) .

Sneddon, I. N.  Fourier  transform . New  York . McGray  Hill. (1951).

Sneddon, I. N. The  use  of  integral  transform . New  York . McGray  Hill,  (1972) .

Srivastava, H. M. and  Buschman, R. G. Convolution  integral  equations  with  special  function  kernels . New  Delhi, Bangalore : Wiley  Eastern  Ltd . (1977).

Srivastava, H . M ., Gupta, K . C . and  Goyal, S . P .  The  H - functions  of  One  and  Two  Variables  with  Applications . South  Asian  Publishers . India. (1982) .

Srivastava, H . M . and  Buschman, R . G .  Theory  and  applications  of  convolution  integral  equations . Kluwer  Acad . Pub! . Ser . Math . and  App! . 79.    
         Dordrecht . Boston . London. (1992) .

Titchmarsh  E . C .  Introduction  to  the  theory  of  Fourier  integrals . Oxford  Univ . Press, Oxford. (1937).

Yakubovich S.B., Luchko Yu.F., The hypergeometric Approach to Integral Transforms and Convolutions, Kluwer Academic Publishers, (1994). (324 pages).

Yakubovich S.B., Index Transforms, World Scientific, (1996). (248 pages).

Zwillinger D. Table of integrals, series and products. Academic Press; 8th edition. New - York. London. Toronto. Sydney. San  Francisco. (2014). (1133 pages).