inttrans

 add entry to transform lookup table

Parameters

 tname - name of transform for which patt is to be added to the lookup table patt - pattern to be added to table expr - transform of patt t - independent variable in patt s - independent variable in expr parameter - (optional) list or set of parameters in patt and expr condition - (optional) conditions that the parameters must satisfy additional - additional argument for hankel and invmellin tables. This parameter is required for hankel and invmellin transform

Description

 • Adds an entry to the lookup table for the integral transform . After this function is executed, any call to with argument will result in being returned.
 • If you wish this information to be saved across sessions, a facility exists, savetable, that will save the information of a particular table to a particular file.
 • The expression may include any number of parameters, which may also be used in the transform expression .  Conditions may be placed on the parameters, by using the argument. The statement must be an unevaluated operator which evaluates to type boolean.  Unevaluated operators include Range, _testeq, _signum, and _evalb.
 • The hankel and invmellin transforms take additional arguments in the transform.  For this reason, they also take additional arguments within the .
 • The format for hankel is of the form hankel=mu::Range(-1,infinity), to specify that the transform can be performed only if the additional argument to hankel is within the range -1 to infinity.
 • The format for invmellin is of the form $\mathrm{invmellin}=3..5$, to specify that the transform can only be performed if the additional argument is contained within the range 3 to 5.

Examples

 > $\mathrm{with}\left(\mathrm{inttrans}\right):$
 > $\mathrm{fourier}\left(f\left(t\right),t,s\right)$
 ${\mathrm{fourier}}{}\left({f}{}\left({t}\right){,}{t}{,}{s}\right)$ (1)
 > $\mathrm{addtable}\left(\mathrm{fourier},f\left(t\right),F\left(s\right),t,s\right)$
 > $\mathrm{fourier}\left(f\left(x\right),x,z\right)$
 ${F}{}\left({z}\right)$ (2)

Functions with parameters

 > $\mathrm{laplace}\left(g\left(pa+b\right),p,x\right)$
 ${\mathrm{laplace}}{}\left({g}{}\left({p}{}{a}{+}{b}\right){,}{p}{,}{x}\right)$ (3)
 > $\mathrm{addtable}\left(\mathrm{laplace},g\left(xa+b\right),\frac{G\left(s+a\right)}{b-a},x,s,\left\{a,b\right\}\right):$
 > $\mathrm{laplace}\left(g\left(-p\right),p,x\right)$
 ${G}{}\left({x}{-}{1}\right)$ (4)
 > $\mathrm{laplace}\left(g\left(3p+2\right),p,x\right)$
 ${-}{G}{}\left({x}{+}{3}\right)$ (5)

Functions with conditional parameters

 > $\mathrm{hilbert}\left(f\left(at\right),t,s\right)$
 ${\mathrm{hilbert}}{}\left({f}{}\left({a}{}{t}\right){,}{t}{,}{s}\right)$ (6)
 > $\mathrm{addtable}\left(\mathrm{hilbert},f\left(a,t\right),F\left(s-a\right),t,s,\left\{a\right\},a::\left(\mathrm{Range}\left(3,7\right)\right)\right):$
 > $\mathrm{hilbert}\left(f\left(a,t\right),t,s\right)$
 ${\mathrm{hilbert}}{}\left({f}{}\left({a}{,}{t}\right){,}{t}{,}{s}\right)$ (7)
 > $\mathrm{assume}\left(3
 > $\mathrm{hilbert}\left(f\left(a,t\right),t,s\right)$
 ${F}{}\left({s}{-}{\mathrm{a~}}\right)$ (8)
 > $\mathrm{addtable}\left(\mathrm{mellin},h\left(a,t\right),F\left(s-a\right),t,s,\left\{a\right\},\mathrm{_evalb}\left(a=\mathrm{Pi}\right)\right)$
 > $\mathrm{mellin}\left(h\left(a,x\right),x,s\right)$
 ${\mathrm{mellin}}{}\left({h}{}\left({\mathrm{a~}}{,}{x}\right){,}{x}{,}{s}\right)$ (9)
 > $\mathrm{mellin}\left(h\left(\mathrm{Pi},x\right),x,s\right)$
 ${F}{}\left({s}{-}{\mathrm{\pi }}\right)$ (10)

Hankel and invmellin transform

 > $\mathrm{hankel}\left(f\left(t\right),t,s,\mathrm{ν}\right)$
 ${\mathrm{hankel}}{}\left({f}{}\left({t}\right){,}{t}{,}{s}{,}{\mathrm{\nu }}\right)$ (11)
 > $\mathrm{addtable}\left(\mathrm{hankel},f\left(t\right),F\left(s,\mathrm{μ}\right),t,s,\mathrm{hankel}=\mathrm{μ}::\left(\mathrm{Range}\left(-\mathrm{∞},\mathrm{∞}\right)\right)\right):$
 > $\mathrm{hankel}\left(f\left(t\right),t,s,\mathrm{ν}\right)$
 ${F}{}\left({s}{,}{\mathrm{\nu }}\right)$ (12)
 > $\mathrm{addtable}\left(\mathrm{hankel},g\left(ta\right),G\left(s-a,\mathrm{μ}\right),t,s,\left\{a\right\},\mathrm{hankel}=\mathrm{μ}::\left(\mathrm{Range}\left(-3,3\right)\right)\right):$
 > $\mathrm{hankel}\left(g\left(2t\right),t,s,\mathrm{ν}\right)$
 ${\mathrm{hankel}}{}\left({g}{}\left({2}{}{t}\right){,}{t}{,}{s}{,}{\mathrm{\nu }}\right)$ (13)
 > $\mathrm{assume}\left(1<\mathrm{ν},\mathrm{ν}<2\right):$
 > $\mathrm{hankel}\left(g\left(2t\right),t,s,\mathrm{ν}\right)$
 ${G}{}\left({s}{-}{2}{,}{\mathrm{ν~}}\right)$ (14)
 > $\mathrm{invmellin}\left(f\left(t\right),t,s,1..2\right)$
 ${\mathrm{invmellin}}{}\left({f}{}\left({t}\right){,}{t}{,}{s}{,}{1}{..}{2}\right)$ (15)
 > $\mathrm{addtable}\left(\mathrm{invmellin},f\left(ta\right),F\left(t-a\right),t,s,\left\{a\right\},a::\left(\mathrm{Range}\left(-3,3\right)\right),\mathrm{invmellin}=0..3\right):$
 > $\mathrm{invmellin}\left(f\left(t\right),t,s,1..2\right)$
 ${F}{}\left({t}{-}{1}\right)$ (16)