3.5.12 Integral Transforms and Related OperationsLaplace Transforms
| LaplaceTransform[expr, t, s] | the Laplace transform of expr |
InverseLaplaceTransform[expr, s, t]
| | the inverse Laplace transform of expr |
One-dimensional Laplace transforms. The Laplace transform of a function  is given by  . The inverse Laplace transform of  is given for suitable  by  . | Here is a simple Laplace transform. | |
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LaplaceTransform[t^4 Sin[t], t, s]
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| Here is the inverse. | |
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InverseLaplaceTransform[%, s, t]
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| Even simple transforms often involve special functions. | |
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LaplaceTransform[1/(1 + t^2), t, s]
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| Here the result involves a Meijer G function. | |
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LaplaceTransform[1/(1 + t^3), t, s]
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| The Laplace transform of this Bessel function just involves elementary functions. | |
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LaplaceTransform[BesselJ[n, t], t, s]
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Laplace transforms have the property that they turn integration and differentiation into essentially algebraic operations. They are therefore commonly used in studying systems governed by differential equations. Integration becomes multiplication by  when one does a Laplace transform. | |
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LaplaceTransform[Integrate[f[u], {u, 0, t}], t, s]
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LaplaceTransform[expr, { ,  , ... }, { ,  , ... }]
| | the multidimensional Laplace transform of expr |
InverseLaplaceTransform[expr, { ,  , ... }, { ,  , ... }]
| | the multidimensional inverse Laplace transform of expr |
Multidimensional Laplace transforms. Fourier Transforms
FourierTransform[expr, t,  ] | the Fourier transform of expr |
InverseFourierTransform[expr,  , t]
| | the inverse Fourier transform of expr |
One-dimensional Fourier transforms. | Here is a Fourier transform. | |
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FourierTransform[1/(1 + t^4), t,  ]
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| This finds the inverse. | |
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InverseFourierTransform[%, , t]
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In Mathematica the Fourier transform of a function  is by default defined to be  . The inverse Fourier transform of  is similarly defined as  . In different scientific and technical fields different conventions are often used for defining Fourier transforms. The option FourierParameters in Mathematica allows you to choose any of these conventions you want.
| common convention | setting | Fourier transform | inverse Fourier transform | | Mathematica default | {0, 1} |  |  | | pure mathematics | {1, -1} |  |  | | classical physics | {-1, 1} |  |  | | modern physics | {0, 1} |  |  | | systems engineering | {1, -1} |  |  | | signal processing | {0, -2 Pi} |  |  | | general case | {a, b} |  |  |
Typical settings for FourierParameters with various conventions. | Here is a Fourier transform with the default choice of parameters. | |
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FourierTransform[Exp[-t^2], t, ]
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| Here is the same Fourier transform with the choice of parameters typically used in signal processing. | |
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FourierTransform[Exp[-t^2], t, ,
FourierParameters->{0, -2 Pi}]
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FourierSinTransform[expr, t,  ] | Fourier sine transform | FourierCosTransform[expr, t,  ] | Fourier cosine transform |
InverseFourierSinTransform[expr,  , t]
| | inverse Fourier sine transform |
InverseFourierCosTransform[expr,  , t]
| | inverse Fourier cosine transform |
Fourier sine and cosine transforms. In some applications of Fourier transforms, it is convenient to avoid ever introducing complex exponentials. Fourier sine and cosine transforms correspond to integrating respectively with  and  instead of  , and using limits 0 and  rather than  and  .
FourierTransform[expr, { ,  , ... }, { ,  , ... }]
| | the multidimensional Fourier transform of expr |
InverseFourierTransform[expr, { ,  , ... }, { ,  , ... }]
| | the multidimensional inverse Fourier transform of expr |
FourierSinTransform[expr, { ,  , ... }, { ,  , ... }], FourierCosTransform[expr, { ,  , ... }, { ,  , ... }]
| | the multidimensional sine and cosine Fourier transforms of expr |
InverseFourierSinTransform[expr, { ,  , ... }, { ,  , ... }], InverseFourierCosTransform[expr, { ,  , ... }, { ,  , ... }]
| | the multidimensional inverse Fourier sine and cosine transforms of expr |
Multidimensional Fourier transforms. | This evaluates a two-dimensional Fourier transform. | |
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FourierTransform[(u v)^2 Exp[-u^2-v^2], {u, v}, {a, b}]
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| This inverts the transform. | |
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InverseFourierTransform[%, {a, b}, {u, v}]
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Z Transforms
| ZTransform[expr, n, z] | Z transform of expr | | InverseZTransform[expr, z, n] | inverse Z transform of expr |
Z transforms. The Z transform of a function  is given by  . The inverse Z transform of  is given by the contour integral  . Z transforms are effectively discrete analogs of Laplace transforms. They are widely used for solving difference equations, especially in digital signal processing and control theory. They can be thought of as producing generating functions, of the kind commonly used in combinatorics and number theory. This computes the Z transform of  . | |
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ZTransform[2^-n, n, z]
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| Here is the inverse Z transform. | |
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InverseZTransform[%, z, n]
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The generating function for  is an exponential function. | |
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ZTransform[1/n!, n, z]
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