FourierSinTransform
✖
FourierSinTransform
gives the multidimensional Fourier sine transform of expr.
Details and Options
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- The Fourier sine transform is a particular way of viewing the Fourier transform without the need for complex numbers or negative frequencies.
- Joseph Fourier designed his famous transform using this and the Fourier cosine transform, and they are still used in applications like signal processing, statistics and image and video compression.
- The Fourier sine transform of the time domain function
is the frequency domain function
for
:
- The Fourier sine transform of a function
is by default defined to be
.
- The multidimensional Fourier sine transform of a function
is by default defined to be
or when using vector notation,
.
- Different choices of definitions can be specified using the option FourierParameters.
- The integral is computed using numerical methods if the third argument,
, is given a numerical value.
- The asymptotic Fourier sine transform can be computed using Asymptotic.
- There are several related Fourier transformations:
-
FourierTransform infinite continuous-time functions (FT) FourierSequenceTransform infinite discrete-time functions (DTFT) FourierCoefficient finite continuous-time functions (FS) Fourier finite discrete-time functions (DFT) - The Fourier sine transform is an automorphism in the Schwartz vector space of functions whose derivatives are rapidly decreasing and thus induces an automorphism in its dual: the space of tempered distributions. These include absolutely integrable functions, well-behaved functions of polynomial growth and compactly supported distributions.
- Hence, FourierSinTransform not only works with absolutely integrable functions on
, but it can also handle a variety of tempered distributions such as DiracDelta to enlarge the pool of functions or generalized functions it can effectively transform.
- The following options can be given:
-
AccuracyGoal Automatic digits of absolute accuracy sought Assumptions $Assumptions assumptions to make about parameters FourierParameters {0,1} parameters to define the Fourier sine transform GenerateConditions False whether to generate answers that involve conditions on parameters PerformanceGoal $PerformanceGoal aspects of performance to optimize PrecisionGoal Automatic digits of precision sought WorkingPrecision Automatic the precision used in internal computations - Common settings for FourierParameters include:
-
{0,1} {1,1} {-1,1} {0,2Pi} {a,b}
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Examples
open allclose allBasic Examples (6)Summary of the most common use cases
Compute the Fourier sine transform of a function:
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https://wolfram.com/xid/0yv4747dycuq-23b8wo
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Plot the function and its Fourier sine transform:
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https://wolfram.com/xid/0yv4747dycuq-7v3k9i
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Fourier sine transform of an exponential function:
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https://wolfram.com/xid/0yv4747dycuq-6zrucp
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For a different convention, change the parameters:
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https://wolfram.com/xid/0yv4747dycuq-y0sapp
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Fourier sine transform of the reciprocal of a square root:
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https://wolfram.com/xid/0yv4747dycuq-mg0u2w
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Compute the Fourier sine transform of a multivariate function:
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https://wolfram.com/xid/0yv4747dycuq-iun92t
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Plot the function and its transform:
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https://wolfram.com/xid/0yv4747dycuq-r7glj0
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Compute the transform at a single point:
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https://wolfram.com/xid/0yv4747dycuq-imjbib
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Scope (37)Survey of the scope of standard use cases
Basic Uses (3)
Fourier sine transform of a function for a symbolic parameter :
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https://wolfram.com/xid/0yv4747dycuq-4ynz91
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https://wolfram.com/xid/0yv4747dycuq-21w99q
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Fourier sine transforms involving trigonometric functions:
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https://wolfram.com/xid/0yv4747dycuq-37i4a9
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https://wolfram.com/xid/0yv4747dycuq-ylueav
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https://wolfram.com/xid/0yv4747dycuq-n6cguf
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https://wolfram.com/xid/0yv4747dycuq-9m2bjy
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Evaluate the Fourier sine transform for a numerical value of the parameter :
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https://wolfram.com/xid/0yv4747dycuq-7s4bdl
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Algebraic Functions (3)
Fourier sine transform of power functions:
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https://wolfram.com/xid/0yv4747dycuq-34yl88
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Sine transform of rational functions:
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https://wolfram.com/xid/0yv4747dycuq-e0m0lo
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https://wolfram.com/xid/0yv4747dycuq-fpdfnk
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https://wolfram.com/xid/0yv4747dycuq-4mwsj1
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https://wolfram.com/xid/0yv4747dycuq-3p3hty
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https://wolfram.com/xid/0yv4747dycuq-x4savd
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https://wolfram.com/xid/0yv4747dycuq-ci0tju
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https://wolfram.com/xid/0yv4747dycuq-c0ysf5
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https://wolfram.com/xid/0yv4747dycuq-d9kc2j
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https://wolfram.com/xid/0yv4747dycuq-k3l3y7
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https://wolfram.com/xid/0yv4747dycuq-vamovl
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Fourier sine transform of a quotient of two polynomials:
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https://wolfram.com/xid/0yv4747dycuq-y5uzc3
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https://wolfram.com/xid/0yv4747dycuq-thzach
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Exponential and Logarithmic Functions (3)
Fourier sine transforms for exponential functions:
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https://wolfram.com/xid/0yv4747dycuq-u1kxbo
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https://wolfram.com/xid/0yv4747dycuq-da2xmy
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https://wolfram.com/xid/0yv4747dycuq-el4sge
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https://wolfram.com/xid/0yv4747dycuq-guvbpy
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https://wolfram.com/xid/0yv4747dycuq-c33cpq
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https://wolfram.com/xid/0yv4747dycuq-nxe6en
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Fourier sine transform of a Gaussian:
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https://wolfram.com/xid/0yv4747dycuq-g6o6mg
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https://wolfram.com/xid/0yv4747dycuq-519vo9
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https://wolfram.com/xid/0yv4747dycuq-0jv9wn
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https://wolfram.com/xid/0yv4747dycuq-hymiq7
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https://wolfram.com/xid/0yv4747dycuq-up5x46
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Sine transforms of logarithmic functions:
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https://wolfram.com/xid/0yv4747dycuq-evmn0t
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https://wolfram.com/xid/0yv4747dycuq-4exxuo
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https://wolfram.com/xid/0yv4747dycuq-nf71us
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https://wolfram.com/xid/0yv4747dycuq-h4wyh6
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https://wolfram.com/xid/0yv4747dycuq-l0ukce
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https://wolfram.com/xid/0yv4747dycuq-kh8443
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https://wolfram.com/xid/0yv4747dycuq-6y7tru
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https://wolfram.com/xid/0yv4747dycuq-6rfwo
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Trigonometric Functions (3)
Composition of elementary functions:
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https://wolfram.com/xid/0yv4747dycuq-jttj4i
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https://wolfram.com/xid/0yv4747dycuq-xf6870
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https://wolfram.com/xid/0yv4747dycuq-gfk34x
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https://wolfram.com/xid/0yv4747dycuq-k0xhvg
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https://wolfram.com/xid/0yv4747dycuq-by62s2
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https://wolfram.com/xid/0yv4747dycuq-o44udk
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https://wolfram.com/xid/0yv4747dycuq-1raban
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https://wolfram.com/xid/0yv4747dycuq-dzakc7
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Fourier sine transform of the product of exponential and trigonometric functions:
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https://wolfram.com/xid/0yv4747dycuq-mpau2u
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https://wolfram.com/xid/0yv4747dycuq-nlr0o8
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https://wolfram.com/xid/0yv4747dycuq-6dnrw0
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https://wolfram.com/xid/0yv4747dycuq-229x0h
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Fourier sine transforms of arctangent functions:
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https://wolfram.com/xid/0yv4747dycuq-1lpb4
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https://wolfram.com/xid/0yv4747dycuq-zaxnsl
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https://wolfram.com/xid/0yv4747dycuq-tigy9
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https://wolfram.com/xid/0yv4747dycuq-f59a5a
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Special Functions (8)
Fourier sine transforms of expressions involving the Sinc function:
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https://wolfram.com/xid/0yv4747dycuq-geedsn
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https://wolfram.com/xid/0yv4747dycuq-sun6r0
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https://wolfram.com/xid/0yv4747dycuq-xq2xg3
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https://wolfram.com/xid/0yv4747dycuq-ksuktw
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Fourier sine transform of ExpIntegralEi:
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https://wolfram.com/xid/0yv4747dycuq-paksr2
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https://wolfram.com/xid/0yv4747dycuq-qhk5hj
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Transform of Erf:
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https://wolfram.com/xid/0yv4747dycuq-e4ovt2
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https://wolfram.com/xid/0yv4747dycuq-i3eyn
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Transform of Erfc:
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https://wolfram.com/xid/0yv4747dycuq-83z1yo
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https://wolfram.com/xid/0yv4747dycuq-bl2a36
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Expression involving the SinIntegral:
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https://wolfram.com/xid/0yv4747dycuq-ifruz9
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https://wolfram.com/xid/0yv4747dycuq-ojs53e
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https://wolfram.com/xid/0yv4747dycuq-lbgyzq
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https://wolfram.com/xid/0yv4747dycuq-tq0qk4
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Sine transforms for BesselJ functions:
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https://wolfram.com/xid/0yv4747dycuq-sheqd6
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https://wolfram.com/xid/0yv4747dycuq-v57fbv
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https://wolfram.com/xid/0yv4747dycuq-0po19w
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https://wolfram.com/xid/0yv4747dycuq-i8mbty
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https://wolfram.com/xid/0yv4747dycuq-htktsp
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https://wolfram.com/xid/0yv4747dycuq-hsl6h0
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Sine transforms for BesselY functions:
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https://wolfram.com/xid/0yv4747dycuq-p811xa
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https://wolfram.com/xid/0yv4747dycuq-s0a2t0
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https://wolfram.com/xid/0yv4747dycuq-ykk7rt
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https://wolfram.com/xid/0yv4747dycuq-ev7j53
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Piecewise Functions and Distributions (4)
Fourier sine transform of a piecewise function:
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https://wolfram.com/xid/0yv4747dycuq-8sy1xc
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https://wolfram.com/xid/0yv4747dycuq-zl93cr
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Restriction of a sine function to a half-period:
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https://wolfram.com/xid/0yv4747dycuq-ubddcp
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https://wolfram.com/xid/0yv4747dycuq-bfynq8
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https://wolfram.com/xid/0yv4747dycuq-oann5g
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https://wolfram.com/xid/0yv4747dycuq-2l2h82
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Transforms in terms of FresnelS:
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https://wolfram.com/xid/0yv4747dycuq-5lhjyp
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https://wolfram.com/xid/0yv4747dycuq-2bb6ob
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https://wolfram.com/xid/0yv4747dycuq-sb1j2z
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https://wolfram.com/xid/0yv4747dycuq-29gotf
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Periodic Functions (2)
Fourier sine transform of sine:
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https://wolfram.com/xid/0yv4747dycuq-u2x9am
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Fourier sine transform of SquareWave:
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https://wolfram.com/xid/0yv4747dycuq-ffviam
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https://wolfram.com/xid/0yv4747dycuq-rrfprq
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Generalized Functions (4)
Fourier sine transforms of expressions involving HeavisideTheta:
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https://wolfram.com/xid/0yv4747dycuq-60mbhv
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https://wolfram.com/xid/0yv4747dycuq-qs90mg
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https://wolfram.com/xid/0yv4747dycuq-jbyyj6
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https://wolfram.com/xid/0yv4747dycuq-v1jvk5
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Fourier sine transforms involving DiracDelta:
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https://wolfram.com/xid/0yv4747dycuq-uvvh37
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https://wolfram.com/xid/0yv4747dycuq-b16dou
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https://wolfram.com/xid/0yv4747dycuq-xxxbkb
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https://wolfram.com/xid/0yv4747dycuq-e0aopw
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Fourier sine transform involving HeavisideLambda:
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https://wolfram.com/xid/0yv4747dycuq-6mp5hw
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https://wolfram.com/xid/0yv4747dycuq-ffj755
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Fourier sine transform involving HeavisidePi:
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https://wolfram.com/xid/0yv4747dycuq-8ghc31
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https://wolfram.com/xid/0yv4747dycuq-xju3an
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Multivariate Functions (2)
Fourier sine transform of an exponential functions in two variables:
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https://wolfram.com/xid/0yv4747dycuq-9cug3a
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https://wolfram.com/xid/0yv4747dycuq-8yb0o2
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https://wolfram.com/xid/0yv4747dycuq-75byoz
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https://wolfram.com/xid/0yv4747dycuq-s8c5jc
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Fourier sine transform of product of exponential and SquareWave:
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https://wolfram.com/xid/0yv4747dycuq-cczu71
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https://wolfram.com/xid/0yv4747dycuq-24pq4p
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Formal Properties (3)
Fourier sine transform of a first-order derivative:
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https://wolfram.com/xid/0yv4747dycuq-cgv9xz
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Fourier sine transform of a second-order derivative:
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https://wolfram.com/xid/0yv4747dycuq-7cahab
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Fourier sine transform threads itself over equations:
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https://wolfram.com/xid/0yv4747dycuq-u3a9a5
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Numerical Evaluation (2)
Calculate the Fourier sine transform at a single point:
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https://wolfram.com/xid/0yv4747dycuq-saipi6
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Alternatively, calculate the Fourier sine transform symbolically:
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https://wolfram.com/xid/0yv4747dycuq-vt3ppm
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Then evaluate it for specific value of :
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https://wolfram.com/xid/0yv4747dycuq-czsagq
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Options (8)Common values & functionality for each option
AccuracyGoal (1)
The option AccuracyGoal sets the number of digits of accuracy:
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https://wolfram.com/xid/0yv4747dycuq-5ymjb
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https://wolfram.com/xid/0yv4747dycuq-f6u3ua
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https://wolfram.com/xid/0yv4747dycuq-fhbr
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Assumptions (1)
Fourier sine transform of BesselJ is a piecewise function:
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https://wolfram.com/xid/0yv4747dycuq-nagwk1
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https://wolfram.com/xid/0yv4747dycuq-fk9tc3
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FourierParameters (3)
Fourier sine transform for the unit box function with different parameters:
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Use a nondefault setting for a different definition of the transform:
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https://wolfram.com/xid/0yv4747dycuq-ejscab
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To get the inverse, use the same FourierParameters setting:
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https://wolfram.com/xid/0yv4747dycuq-j5l0np
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Set up your particular global choice of parameters once per session:
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https://wolfram.com/xid/0yv4747dycuq-tcyt2a
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https://wolfram.com/xid/0yv4747dycuq-kb7shy
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https://wolfram.com/xid/0yv4747dycuq-uc9bv5
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GenerateConditions (1)
Use GenerateConditions True to get the parameter conditions necessary for the result to be valid:
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https://wolfram.com/xid/0yv4747dycuq-dosv41
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PrecisionGoal (1)
The option PrecisionGoal sets the relative tolerance in the integration:
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https://wolfram.com/xid/0yv4747dycuq-n3x5d0
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https://wolfram.com/xid/0yv4747dycuq-9119x
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https://wolfram.com/xid/0yv4747dycuq-b3dg8e
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WorkingPrecision (1)
If a WorkingPrecision is specified, the computation is done at that working precision:
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https://wolfram.com/xid/0yv4747dycuq-d1i4yb
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https://wolfram.com/xid/0yv4747dycuq-duvdjv
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https://wolfram.com/xid/0yv4747dycuq-cx1m7u
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Applications (4)Sample problems that can be solved with this function
Ordinary Differential Equations (1)
Consider the following ODE with initial condition :
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https://wolfram.com/xid/0yv4747dycuq-ej7og1
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Apply the Fourier sine transform to the ODE:
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https://wolfram.com/xid/0yv4747dycuq-wkf3px
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https://wolfram.com/xid/0yv4747dycuq-ilph5p
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Find the inverse Fourier sine transform with and
:

https://wolfram.com/xid/0yv4747dycuq-g6xb71
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Compare with DSolveValue:
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https://wolfram.com/xid/0yv4747dycuq-6h29lw
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Partial Differential Equations (1)
Solve the infinite diffusion problem for ,
:
with initial condition
for
and boundary condition
for
:
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https://wolfram.com/xid/0yv4747dycuq-jscm4x
Fourier sine transform with respect to :
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https://wolfram.com/xid/0yv4747dycuq-9py0g3
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https://wolfram.com/xid/0yv4747dycuq-4bllfe
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Compute the inverse sine transform:
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https://wolfram.com/xid/0yv4747dycuq-1r2yv3
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Compare with DSolveValue:
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https://wolfram.com/xid/0yv4747dycuq-dkim4w
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Consider the special case with and
:
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https://wolfram.com/xid/0yv4747dycuq-bvc9pm
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Evaluation of Integrals (2)
Calculate the following definite integral for :
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https://wolfram.com/xid/0yv4747dycuq-8tmc8r
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Compute the Fourier sine transform of an exponential function:
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https://wolfram.com/xid/0yv4747dycuq-ee38yx
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Apply the Fourier sine inversion formula:
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https://wolfram.com/xid/0yv4747dycuq-l8n5u3
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Solve for the definite integral:

https://wolfram.com/xid/0yv4747dycuq-1yiih3
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Compare with Integrate:
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https://wolfram.com/xid/0yv4747dycuq-jdypq8
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Calculate the following definite integral for :
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https://wolfram.com/xid/0yv4747dycuq-r687jy
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Compute the Fourier sine transform of an exponential function:
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https://wolfram.com/xid/0yv4747dycuq-drholg
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https://wolfram.com/xid/0yv4747dycuq-gkfkxr
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https://wolfram.com/xid/0yv4747dycuq-tbzmtb
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Solve for the definite integral:
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https://wolfram.com/xid/0yv4747dycuq-6ioxzg
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Compare with Integrate:
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https://wolfram.com/xid/0yv4747dycuq-5jgzyd
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Properties & Relations (4)Properties of the function, and connections to other functions
By default, the Fourier sine transform of is:

https://wolfram.com/xid/0yv4747dycuq-iyfnht
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For , the definite integral becomes:
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https://wolfram.com/xid/0yv4747dycuq-p3u34y
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Compare with FourierSinTransform:

https://wolfram.com/xid/0yv4747dycuq-d9s0gi
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Use Asymptotic to compute an asymptotic approximation:
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https://wolfram.com/xid/0yv4747dycuq-lwbzos
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FourierSinTransform and InverseFourierSinTransform are mutual inverses:

https://wolfram.com/xid/0yv4747dycuq-bp78fy
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https://wolfram.com/xid/0yv4747dycuq-bhj9ka
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https://wolfram.com/xid/0yv4747dycuq-it7bfn
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
https://wolfram.com/xid/0yv4747dycuq-byd6ra
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Results from FourierSinTransform and FourierTransform differ by a factor of for odd functions:
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https://wolfram.com/xid/0yv4747dycuq-hrqxcb
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https://wolfram.com/xid/0yv4747dycuq-ef1iz8
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The results differ by a factor of for ω>0:

https://wolfram.com/xid/0yv4747dycuq-b2ebmr
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Possible Issues (1)Common pitfalls and unexpected behavior
The result from an inverse Fourier sine transform may not have the same form as the original:

https://wolfram.com/xid/0yv4747dycuq-5jbdmw
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https://wolfram.com/xid/0yv4747dycuq-csfig0
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The Fourier sine transform may be given in terms of generalized functions such as DiracDelta:
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https://wolfram.com/xid/0yv4747dycuq-mjrqa7
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https://wolfram.com/xid/0yv4747dycuq-j88uu6
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Neat Examples (2)Surprising or curious use cases
The Fourier sine transform represented in terms of MeijerG:

https://wolfram.com/xid/0yv4747dycuq-3m6m5
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Create a table of basic Fourier sine transforms:

https://wolfram.com/xid/0yv4747dycuq-wwndk9
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https://wolfram.com/xid/0yv4747dycuq-zsny25
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Wolfram Research (1999), FourierSinTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSinTransform.html (updated 2025).
Text
Wolfram Research (1999), FourierSinTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSinTransform.html (updated 2025).
Wolfram Research (1999), FourierSinTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSinTransform.html (updated 2025).
CMS
Wolfram Language. 1999. "FourierSinTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/FourierSinTransform.html.
Wolfram Language. 1999. "FourierSinTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/FourierSinTransform.html.
APA
Wolfram Language. (1999). FourierSinTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierSinTransform.html
Wolfram Language. (1999). FourierSinTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierSinTransform.html
BibTeX
@misc{reference.wolfram_2025_fouriersintransform, author="Wolfram Research", title="{FourierSinTransform}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/FourierSinTransform.html}", note=[Accessed: 23-February-2025
]}
BibLaTeX
@online{reference.wolfram_2025_fouriersintransform, organization={Wolfram Research}, title={FourierSinTransform}, year={2025}, url={https://reference.wolfram.com/language/ref/FourierSinTransform.html}, note=[Accessed: 23-February-2025
]}