Fourier Cosine transform - Maple Help

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inttrans[fouriercos] - Fourier Cosine transform

Calling Sequence

fouriercos(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 fouriercos function computes the Fourier Cosine transform (F(s)) of expr (f(t)), a linear transformation from C0,C0, defined by:

Fs=20ftcosstⅆtπ

• 

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 be transformed.

• 

The Fourier Cosine transform is self-inverting.

• 

The fouriercos function recognizes derivatives (diff or Diff) of functions of rapid descent and can be used to solve ODEs and PDEs.

• 

The fouriercos 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(fouriercos, 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.

• 

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

• 

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

Examples

withinttrans

addtable,fourier,fouriercos,fouriersin,hankel,hilbert,invfourier,invhilbert,invlaplace,invmellin,laplace,mellin,savetable

(1)

fouriercos3t+a,t,w

32sinawSsiawcosawCiawπ

(2)

fouriercos1t2+1,t,s

122πⅇs

(3)

fouriercos,s,x

1x2+1

(4)

fouriercosⅆⅆxfx,x,w

wfouriersinfx,x,wπ2f0π

(5)

fouriercosft,t,u

fouriercosft,t,u

(6)

fouriercos,u,s

fs

(7)

F:=∫0∞gxcosxsⅆx:

fouriercosF+1s,s,t

122πgt+1t

(8)

fouriercostⅇ3tsint,t,w

623w4+16w2100πw22w+102w2+2w+102

(9)

de:=ⅆ2ⅆt2ytyt=cos2t:

fouriercosde,t,s

s2fouriercosyt,t,sπ+2Dy0πfouriercosyt,t,s=122πDiracs2

(10)

solve,'fouriercos'yt,t,s

122πDiracs2+2Dy0πs2+1

(11)

fouriercos,s,t

15cos2tDy0ⅇt

(12)

subsyt=,de

ⅆ2ⅆt215cos2tDy0ⅇt+15cos2t+Dy0ⅇt=cos2t

(13)

simplify

cos2t=cos2t

(14)

fouriercosBesselJ0,4t,t,s3

2Heaviside7sπs32+16

(15)

Add an entry to the table

addtablefouriercos,ht,Hs,t,s:

fouriercosht,t,s

Hs

(16)

See Also

dsolve, inttrans, inttrans[addtable]


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