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gives the symbolic Fourier transform of expr.
gives the multidimensional Fourier transform of expr.
  • The Fourier 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.
  • Some common choices for are (default; modern physics), (pure mathematics; systems engineering), (classical physics), and {0, -2Pi} (signal processing).
  • The following options can be given:
Assumptions$Assumptionsassumptions to make about parameters
FourierParameters{0,1}parameters to define the Fourier transform
GenerateConditionsFalsewhether to generate answers that involve conditions on parameters
  • FourierTransform yields an expression depending on the continuous variable that represents the symbolic Fourier transform of expr with respect to the continuous variable t. Fourier[list] takes a finite list of numbers as input, and yields as output a list representing the discrete Fourier transform of the input.
Click for copyable input
Click for copyable input
Click for copyable input
Elementary functions:
Special functions:
Piecewise functions and distributions:
TraditionalForm formatting:
Multidimensional Fourier transform:
The Fourier transform of BesselJ is a piecewise function:
Default modern physics convention:
Convention for pure mathematics, systems engineering:
Convention for classical physics:
Convention for signal processing:
Use GenerateConditions->True to get parameter conditions for when a result is valid:
The power spectrum of a damped sinusoid:
FourierTransform and InverseFourierTransform are mutual inverses:
FourierTransform and FourierCosTransform are equal for even functions:
FourierTransform and FourierSinTransform differ by for odd functions:
The result from an inverse Fourier transform may not have the same form as the original:
The Fourier transforms of weighted Hermite polynomials have a very simple form:
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