addtable (inttrans Package)
The integral transforms package contains functionality that allows great flexibility in the patterns that can be entered into the tables, as well as the ability to match these patterns quickly and efficiently.
restart
withinttrans:
Basic Functionality
addtablefourier,ft,Ffs,t,s
fourierfx,x,y
Ffy
addtablefouriercos,ft,Fcs,t,s
fouriercosfx,x,y
Fcy
addtablefouriersin,ft,Fss,t,s
fouriersinfx,x,y
Fsy
addtablehankel,ft,Fhas,ν,t,s,hankel=ν::Range−∞,∞
hankelfx,x,y,μ
Fhay,μ
addtablehilbert,ft,Fhis,t,s
hilbertfx,x,y
Fhiy
addtableinvlaplace,ft,Fils,t,s
invlaplacefx,x,y
Fily
addtableinvmellin,ft,Fims,t,s,invmellin=−∞..∞
invmellinfx,x,y,−∞..∞
Fimy
addtablelaplace,ft,Fls,t,s
laplacefx,x,y
Fly
addtablemellin,ft,Fms,t,s
mellinfx,x,y
Fmy
Functions with Several Parameters
This functionality distinguishes between user-defined symbolic constants and variable parameters.
In the following case, a is a variable parameter:
addtablefourier,gta,Gfs−a,t,s,a
fouriergx,x,y;
Gfy−1
Whereas, in the following situation, alpha is a user-defined symbolic constant, and it must be matched exactly by "alpha" for the pattern to be successfully returned. The first call to fouriercos will not return an answer for this reason--because 4 is not the same as alpha--while the second will.
addtablefouriercos,gat,α,Gcs−α+a,t,s,a
fouriercosg3x,4,x,y
fouriercosg3x,α,x,y
Gcy−α+3
The rest of these examples use variable parameters as opposed to symbolic constants:
addtablefouriersin,gat+b,Gsas−b,t,s,a,b
fouriersingαx+β,x,y
Gsαy−β
addtablehankel,gta,Ghas,ν−a,t,s,a,hankel=ν::Range−∞,∞
hankelgx,x,y,μ
Ghay,μ−1
addtablehilbert,gt+b,Ghis+sinb,t,s,b
hilbertgx+b,x,y
Ghiy+sinb
addtableinvlaplace,gt+a,Gils1−a,t,s,a
invlaplacegx+3,x,y
−12Gily
addtableinvmellin,gta,Gims−a,t,s,a,invmellin=−∞..∞
invmellingx,x,y,−∞..∞
Gimy−1
addtablelaplace,gat,Glas+a,t,s,a
laplacegx,x,y
Gl11+y
addtablemellin,gtb,Gmsb,t,s,b
melling6x,x,y
Gmy6
You can specify conditions on the parameters to be matched:
addtablehankel,hta,Hhas,ν−a,t,s,a,a::Range−3,5,hankel=ν::Range−∞,∞
hankelhx,x,y,μ
Hhay,μ−1
addtableinvlaplace,ht+a,Hil2s1−a,t,s,a,a::integer
addtableinvlaplace,ht+a,Hil1s1−a,t,s,a,a::Notinteger
invlaplacehx+32,x,y
−2Hil1y
invlaplacehx+3,x,y
−12Hil2y
addtablemellin,ht,b,Hm1sb,t,s,b,_testeq_signumb+1
addtablemellin,ht,b,Hm2sb,t,s,b,_testeq_signumb−1
mellinhx,6,x,y
Hm2y6
mellinhx,−6,x,y
1Hm1y6
invmellin and hankel
With invmellin and hankel, arguments are added to the addtable function, as arguments are added to the transforms themselves. These arguments can be varied, introducing greater flexibility in the information that can be added to user-defined tables.
addtablehankel,ita,Iha2s,ν−a,t,s,a,a::Range−3,5,hankel=ν::Range0,∞
addtablehankel,ita,Iha1s,ν−a,t,s,a,a::Range−3,5,hankel=ν::Range−∞,0
assume0<μ
hankelix,x,y,μ
Iha2y,μ~−1
assumeμ<0
Iha1y,μ~−1
addtableinvmellin,it,a,Iim1s−a,t,s,a,_testeq_signuma+1,invmellin=0..∞
addtableinvmellin,it,a,Iim2s−a,t,s,a,_testeq_signuma+1,invmellin=−∞..0
addtableinvmellin,it,a,Iim3s−a,t,s,a,_testeq_signuma−1,invmellin=0..∞
addtableinvmellin,it,a,Iim4s−a,t,s,a,_testeq_signuma−1,invmellin=−∞..0
invmellinix,7,x,y,3..7
Iim3y−7
invmellinix,−7,x,y,3..7
Iim1y+7
invmellinix,7,x,y,−7..−3
Iim4y−7
invmellinix,−7,x,y,−∞..0
Iim2y+7
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