FourierCosTransform

FourierCosTransform[expr,t,ω]

gives the symbolic Fourier cosine transform of expr.

FourierCosTransform[expr,{t1,t2,…},{ω1,ω2,…}]

gives the multidimensional Fourier cosine transform of expr.

Details and Options

  • The Fourier cosine transform of a function is by default defined to be .
  • The multidimensional Fourier cosine transform of a function is by default defined to be .
  • Other definitions are used in some scientific and technical fields.
  • Different choices of definitions can be specified using the option FourierParameters.
  • With the setting TemplateBox[{FourierParameters, paclet:ref/FourierParameters}, RefLink, BaseStyle -> {InlineFormula}]->{a,b} the Fourier cosine transform computed by FourierCosTransform is .
  • Assumptions and other options to Integrate can also be given in FourierCosTransform.

Examples

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Basic Examples  (3)

Scope  (5)

Elementary functions:

Special functions:

Generalized functions:

Periodic functions:

Multivariate transform:

Options  (3)

Assumptions  (1)

Fourier cosine transform of BesselJ is a piecewise function:

FourierParameters  (1)

The default setting for FourierParameters is {0,1}:

Use a nondefault setting for a different definition of transform:

To get the inverse, use the same FourierParameters setting:

GenerateConditions  (1)

Use GenerateConditions->True to get parameter conditions for when a result is valid:

Properties & Relations  (3)

Use Asymptotic to compute an asymptotic approximation:

FourierCosTransform and InverseFourierCosTransform are mutual inverses:

Results from FourierCosTransform and FourierTransform agree for even functions:

The results agree for ω>0:

Possible Issues  (1)

Fourier cosine transform may be given in terms of generalized functions such as DiracDelta:

Neat Examples  (1)

Wolfram Research (1999), FourierCosTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierCosTransform.html.

Text

Wolfram Research (1999), FourierCosTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierCosTransform.html.

CMS

Wolfram Language. 1999. "FourierCosTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FourierCosTransform.html.

APA

Wolfram Language. (1999). FourierCosTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierCosTransform.html

BibTeX

@misc{reference.wolfram_2024_fouriercostransform, author="Wolfram Research", title="{FourierCosTransform}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/FourierCosTransform.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_fouriercostransform, organization={Wolfram Research}, title={FourierCosTransform}, year={1999}, url={https://reference.wolfram.com/language/ref/FourierCosTransform.html}, note=[Accessed: 22-December-2024 ]}