integral_transforms.pdf

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transform —libraryforintegraltransforms
Tableofcontents
Preface..................................... ii
HelpPages
transform::fourier,transform::invfourier —Fourierandinverse
Fouriertransform.............................. 1
transform::laplace,transform::invlaplace —Laplaceandinverse
Laplacetransform.............................. 3
Introduction
The transform libraryprovidessomeintegraltransformations.
Thepackagefunctionsarecalledusingthepackagename transform and
thenameofthefunction.E.g.,use
>>transform::fourier(exp(-t^2),t,s)
tocomputetheFouriertransformofe −t 2 withrespectto t atthepoint s .
Thismechanismavoidsnamingconflictswithotherlibraryfunctions.Ifthisis
foundtobeinconvenient,thentheroutinesofthe transform packagemaybe
exportedvia export .E.g.,aftercalling
>>export(transform,fourier)
thefunction transform::fourier maybecalleddirectly:
>>fourier(exp(-t^2),t,s)
Allroutinesofthe transform packageareexportedsimultaneouslyby
>>export(transform)
Thefunctionsavailableinthe transform librarycanbelistedwith:
>>info(transform)
ii
transform::fourier,transform::invfourier– Fourierandinverse
Fouriertransform
−1 fe ist dt
oftheexpressionf=f(t)withrespecttothevariabletatthepoints.
transform::invfourier(F,S,T) computestheinverseFouriertransform 1 2
oftheexpressionF=F(S)withrespecttothevariableSatthepointT.
R 1 −1 Fe −iST dS
Call(s):
transform::fourier(f,t,s)
transform::invfourier(F,S,T)
Parameters:
f,F —arithmeticalexpressions
t,S —identifiers(thetransformationvariables)
s,T —arithmeticalexpressions(theevaluationpoints)
ReturnValue:anarithmeticalexpression
Overloadableby: f,F
RelatedFunctions: numeric::fft , numeric::invfft
Details:
A
Anunevaluatedfunctioncallisreturned,ifnoexplicitrepresentationof
thetransformisfound.
transform::invfourier(F,S,T) iscomputedas
transform :: fourier ( F , S , −T )/ 2 / PI .
Thisresultisreturned,ifnoexplicitrepresentationofthetransformation
isfound.
ThediscreteFouriertransformisimplementedbythefunctions numeric::fft
and numeric::invfft .
Example1.ThefollowingcallproducestheFouriertransformasanexpres-
sioninthevariable s :
>>transform::fourier(exp(-t^2),t,s)
1
transform::fourier(f,t,s) computestheFouriertransform R 1
A
A
A
A
390095853.001.png
/ 2\
1/2 | s|
PI exp|---|
\ 4/
>>transform::invfourier(%,s,t)
2
exp(-t)
NotethattheFouriertransformcanbeevaluateddirectlyataspecificpoint
suchass=2aors=5:
>>transform::fourier(t*exp(-a*t^2),t,s),
transform::fourier(t*exp(-a*t^2),t,2*a),
transform::fourier(t*exp(-a*t^2),t,2)
/ 2\
1/2 | s| 1/2 / 1\
1/2IsPI exp|----| 1/2 IPI exp|--|
\ 4a/IPI exp(-a) \ a/
--------------------------,---------------,------------------
3/2 1/2 3/2
a a a
Example2.Anunevaluatedcallisreturned,ifnoexplicitrepresentationof
thetransformisfound:
>>transform::fourier(besselJ(0,1/(1+t^2)),t,s)
/ / 1 \ \
transform::fourier|besselJ|0,------|,t,s|
| | 2 | |
\ \ t+1/ /
>>transform::invfourier(%,s,t)
/ 1 \
besselJ|0,------|
| 2 |
\ t+1/
Notethattheinversetransformisrelatedtothedirecttransform:
>>transform::invfourier(unknown(s),s,t)
transform::fourier(unknown(s),s,-t)
-------------------------------------
2PI
2
Example3.Thedistribution dirac ishandled:
>>transform::fourier(t^3,t,s)
2IPIdirac(s,3)
>>transform::invfourier(%,s,t)
3
t
>>transform::fourier(heaviside(t-t0),t,s)
/ I\
exp(Ist0)|PIdirac(s)+-|
\ s/
Example4.TheFouriertransformofafunctionisrelatedtotheFourier
transformofitsderivative:
A
>>transform::fourier(diff(f(t),t),t,s)
-Istransform::fourier(f(t),t,s)
Background:
Reference:F.Oberhettinger,“TablesofFourierTransformsandFourier
TransformsofDistributions”,Springer,1990.
transform::laplace,transform::invlaplace– Laplaceandinverse
Laplacetransform
0 fe −st dt
oftheexpressionf=f(t)withrespecttothevariabletatthepoints.
A
transform::invlaplace(F,S,T) computestheinverseLaplacetransformof
theexpressionF=F(S)withrespecttothevariableSatthepointT.
Call(s):
transform::laplace(f,t,s)
transform::invlaplace(F,S,T)
3
transform::laplace(f,t,s) computestheLaplacetransform R 1
A

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