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inttrans

  

fouriersin

  

Fourier Sine transform

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

fouriersin(expr, t, s)

Parameters

expr

-

expression, equation, or set of expressions and/or equations to be transformed

t

-

variable expr is transformed with respect to t

s

-

parameter of transform

opt

-

option to run this under (optional)

Description

• 

The fouriersin function computes the a Fourier Sine transform (F(s)) of expr (f(t)), a linear transformation C0,C0, defined by:

Fs=20ftsinstⅆtπ

• 

The function Fs returned is defined on the positive real axis only.

• 

Expressions involving exponentials, rational polynomials, trigonometrics (sin, cos) with linear arguments, and a variety of other functions can all be transformed.

• 

The Fourier Sine transform is self-inverting.

• 

The fouriersin function transforms derivatives (diff or Diff) of functions of rapid descent, and can be used to solve differential equations.

• 

The fouriersin function attempts to simplify an expression according to a set of heuristics and then match the result with a table of patterns. Entries can be added to this table by addtable(fouriersin, f(t), F(s), t, s), where F(s) is the transform of f(t), which may have an arbitrary number of parameters.

• 

If the option opt is set to 'NO_INT', then the program will not resort to integration of the original problem if all other methods fail.  This will increase the speed at which the transform will run.

• 

fouriersin  recognizes the Dirac-delta (or unit-impulse) function as Dirac(t) and Heaviside's unit step function as Heaviside(t).

• 

The command with(inttrans,fouriersin) allows the use of the abbreviated form of this command.

Examples

withinttrans:

fouriersin3t+a,t,w

32cosawSsiaw+sinawCiawπ

(1)

fouriersintt2+1,t,s

122πⅇs

(2)

fouriersin,s,x

xx2+1

(3)

fouriersinⅆⅆxfx,x,w

wfouriercosfx,x,w

(4)

fouriersinft,t,u

fouriersinft,t,u

(5)

fouriersin,u,s

fs

(6)

F∫0∞gxsinxsⅆx:

fouriersin3F,s,t

322πgt

(7)

fouriersintⅇ3tHeavisidet,t,w

62wπw2+92

(8)

fouriersinⅆ2ⅆt2ytyt=sin2t,t,s

ssfouriersinyt,t,sπ+2y0πfouriersinyt,t,s=122πDiracs2

(9)

solve,'fouriersin'yt,t,s

122πDiracs22sy0πs2+1

(10)

fouriersin,s,t

15sin2t+y0ⅇt

(11)

2t2

sin2t

(12)

fouriersinBesselJ0,4t,t,3+s

2Heaviside1+sπ3+s216

(13)

Adding to the table

addtablefouriersin,ht,Hs,t,s:

fouriersinht,t,s

Hs

(14)

See Also

dsolve

inttrans

inttrans[addtable]

 


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