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FourierSinTransform
✖
FourierSinTransform

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✖
FourierSinTransform[expr,t,ω]

gives the symbolic Fourier sine transform of expr.

✖
FourierSinTransform[expr,{t1,t2,…},{ω1,ω2,…}]

gives the multidimensional Fourier sine transform of expr.

Details and Options

  • 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, (2/pi)^(n/2)int_(t in TemplateBox[{}, PositiveReals]^n )f(t) sin(omega t)dt.
  • 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:
  • FourierTransforminfinite continuous-time functions (FT)
    FourierSequenceTransforminfinite discrete-time functions (DTFT)
    FourierCoefficientfinite continuous-time functions (FS)
    Fourierfinite 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 Automaticdigits of absolute accuracy sought
    Assumptions $Assumptionsassumptions to make about parameters
    FourierParameters {0,1}parameters to define the Fourier sine transform
    GenerateConditions Falsewhether to generate answers that involve conditions on parameters
    PerformanceGoal$PerformanceGoalaspects of performance to optimize
    PrecisionGoal Automaticdigits of precision sought
    WorkingPrecision Automaticthe precision used in internal computations
  • Common settings for FourierParameters include:
  • {0,1}
    {1,1}
    {-1,1}
    {0,2Pi}
    {a,b}

Examples

open allclose all

Basic Examples  (6)Summary of the most common use cases

Compute the Fourier sine transform of a function:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-23b8wo
Out[1]=1

Plot the function and its Fourier sine transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-7v3k9i
Out[2]=2

Fourier sine transform of an exponential function:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-6zrucp
Out[1]=1

For a different convention, change the parameters:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-y0sapp
Out[1]=1

Fourier sine transform of the reciprocal of a square root:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-mg0u2w
Out[1]=1

Compute the Fourier sine transform of a multivariate function:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-iun92t
Out[1]=1

Plot the function and its transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-r7glj0
Out[2]=2

Compute the transform at a single point:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-imjbib
Out[1]=1

Scope  (37)Survey of the scope of standard use cases

Basic Uses  (3)

Fourier sine transform of a function for a symbolic parameter :

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-4ynz91
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-21w99q
Out[2]=2

Fourier sine transforms involving trigonometric functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-37i4a9
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-ylueav
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-n6cguf
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-9m2bjy
Out[4]=4

Evaluate the Fourier sine transform for a numerical value of the parameter :

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-7s4bdl
Out[1]=1

Algebraic Functions  (3)

Fourier sine transform of power functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-34yl88
Out[1]=1

Sine transform of rational functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-e0m0lo
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-fpdfnk
Out[2]=2

Plot the transform for :

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-4mwsj1
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-3p3hty
Out[4]=4

Plot the transform for :

In[5]:=5
✖
https://wolfram.com/xid/0yv4747dycuq-x4savd
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0yv4747dycuq-ci0tju
Out[6]=6

Transform when :

In[7]:=7
✖
https://wolfram.com/xid/0yv4747dycuq-c0ysf5
Out[7]=7

Plot the transform:

In[8]:=8
✖
https://wolfram.com/xid/0yv4747dycuq-d9kc2j
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/0yv4747dycuq-k3l3y7
Out[9]=9

Plot the transform for :

In[10]:=10
✖
https://wolfram.com/xid/0yv4747dycuq-vamovl
Out[10]=10

Fourier sine transform of a quotient of two polynomials:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-y5uzc3
Out[1]=1

Plot the transform for:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-thzach
Out[2]=2

Exponential and Logarithmic Functions  (3)

Fourier sine transforms for exponential functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-u1kxbo
Out[1]=1

Plot the transform for :

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-da2xmy
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-el4sge
Out[3]=3

Plot the transform for :

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-guvbpy
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0yv4747dycuq-c33cpq
Out[5]=5

Plot the transform:

In[6]:=6
✖
https://wolfram.com/xid/0yv4747dycuq-nxe6en
Out[6]=6

Fourier sine transform of a Gaussian:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-g6o6mg
Out[1]=1

Transform when :

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-519vo9
Out[2]=2

Plot the transform:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-0jv9wn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-hymiq7
Out[4]=4

Plot the transform for :

In[5]:=5
✖
https://wolfram.com/xid/0yv4747dycuq-up5x46
Out[5]=5

Sine transforms of logarithmic functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-evmn0t
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-4exxuo
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-nf71us
Out[3]=3

Plot the transform for :

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-h4wyh6
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0yv4747dycuq-l0ukce
Out[5]=5

Plot the transform for :

In[6]:=6
✖
https://wolfram.com/xid/0yv4747dycuq-kh8443
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0yv4747dycuq-6y7tru
Out[7]=7

Plot the transform:

In[8]:=8
✖
https://wolfram.com/xid/0yv4747dycuq-6rfwo
Out[8]=8

Trigonometric Functions  (3)

Composition of elementary functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-jttj4i
Out[1]=1

Plot the transform for :

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-xf6870
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-gfk34x
Out[3]=3

Plot the transform for :

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-k0xhvg
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0yv4747dycuq-by62s2
Out[5]=5

Plot the transform for :

In[6]:=6
✖
https://wolfram.com/xid/0yv4747dycuq-o44udk
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0yv4747dycuq-1raban
Out[7]=7

Plot the transform for :

In[8]:=8
✖
https://wolfram.com/xid/0yv4747dycuq-dzakc7
Out[8]=8

Fourier sine transform of the product of exponential and trigonometric functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-mpau2u
Out[1]=1

Plot the transform for , :

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-nlr0o8
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-6dnrw0
Out[3]=3

Plot the transform for , :

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-229x0h
Out[4]=4

Fourier sine transforms of arctangent functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-1lpb4
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-zaxnsl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-tigy9
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-f59a5a
Out[4]=4

Special Functions  (8)

Fourier sine transforms of expressions involving the Sinc function:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-geedsn
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-sun6r0
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-xq2xg3
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-ksuktw
Out[4]=4

Fourier sine transform of ExpIntegralEi:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-paksr2
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-qhk5hj
Out[2]=2

Transform of Erf:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-e4ovt2
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-i3eyn
Out[2]=2

Transform of Erfc:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-83z1yo
Out[1]=1

Plot the transform for :

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-bl2a36
Out[2]=2

Expression involving the SinIntegral:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-ifruz9
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-ojs53e
Out[2]=2

CosIntegral:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-lbgyzq
Out[1]=1

Plot the transform for :

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-tq0qk4
Out[2]=2

Sine transforms for BesselJ functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-sheqd6
Out[1]=1

Plot the transform for :

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-v57fbv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-0po19w
Out[3]=3

Plot the transform for and :

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-i8mbty
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0yv4747dycuq-htktsp
Out[5]=5

Plot the transform for and :

In[6]:=6
✖
https://wolfram.com/xid/0yv4747dycuq-hsl6h0
Out[6]=6

Sine transforms for BesselY functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-p811xa
Out[1]=1

Plot the transform for :

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-s0a2t0
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-ykk7rt
Out[3]=3

Plot the transform for and :

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-ev7j53
Out[4]=4

Piecewise Functions and Distributions  (4)

Fourier sine transform of a piecewise function:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-8sy1xc
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-zl93cr
Out[3]=3

Restriction of a sine function to a half-period:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-ubddcp
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-bfynq8
Out[2]=2

Triangular function:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-oann5g
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-2l2h82
Out[2]=2

Transforms in terms of FresnelS:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-5lhjyp
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-2bb6ob
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-sb1j2z
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-29gotf
Out[4]=4

Periodic Functions  (2)

Fourier sine transform of sine:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-u2x9am
Out[1]=1

Fourier sine transform of SquareWave:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-ffviam
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-rrfprq
Out[2]=2

Generalized Functions  (4)

Fourier sine transforms of expressions involving HeavisideTheta:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-60mbhv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-qs90mg
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-jbyyj6
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-v1jvk5
Out[4]=4

Fourier sine transforms involving DiracDelta:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-uvvh37
Out[1]=1

Plot the transform:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-b16dou
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-xxxbkb
Out[3]=3

Plot the transform:

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-e0aopw
Out[4]=4

Fourier sine transform involving HeavisideLambda:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-6mp5hw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-ffj755
Out[2]=2

Fourier sine transform involving HeavisidePi:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-8ghc31
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-xju3an
Out[2]=2

Multivariate Functions  (2)

Fourier sine transform of an exponential functions in two variables:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-9cug3a
Out[1]=1

Plot of both:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-8yb0o2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-75byoz
Out[3]=3

Plot of both:

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-s8c5jc
Out[4]=4

Fourier sine transform of product of exponential and SquareWave:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-cczu71
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-24pq4p
Out[2]=2

Formal Properties  (3)

Fourier sine transform of a first-order derivative:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-cgv9xz
Out[1]=1

Fourier sine transform of a second-order derivative:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-7cahab
Out[1]=1

Fourier sine transform threads itself over equations:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-u3a9a5
Out[1]=1

Numerical Evaluation  (2)

Calculate the Fourier sine transform at a single point:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-saipi6
Out[1]=1

Alternatively, calculate the Fourier sine transform symbolically:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-vt3ppm
Out[1]=1

Then evaluate it for specific value of :

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-czsagq
Out[2]=2

Options  (8)Common values & functionality for each option

AccuracyGoal  (1)

The option AccuracyGoal sets the number of digits of accuracy:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-5ymjb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-f6u3ua
Out[2]=2

With default settings:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-fhbr
Out[3]=3

Assumptions  (1)

Fourier sine transform of BesselJ is a piecewise function:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-nagwk1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-fk9tc3
Out[2]=2

FourierParameters  (3)

Fourier sine transform for the unit box function with different parameters:

Use a nondefault setting for a different definition of the transform:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-ejscab
Out[1]=1

To get the inverse, use the same FourierParameters setting:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-j5l0np
Out[2]=2

Set up your particular global choice of parameters once per session:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-tcyt2a
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-kb7shy

Restore defaults:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-uc9bv5
Out[3]=3

GenerateConditions  (1)

Use GenerateConditions True to get the parameter conditions necessary for the result to be valid:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-dosv41
Out[1]=1

PrecisionGoal  (1)

The option PrecisionGoal sets the relative tolerance in the integration:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-n3x5d0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-9119x
Out[2]=2

With default settings:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-b3dg8e
Out[3]=3

WorkingPrecision  (1)

If a WorkingPrecision is specified, the computation is done at that working precision:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-d1i4yb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-duvdjv
Out[2]=2

With default settings:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-cx1m7u
Out[3]=3

Applications  (4)Sample problems that can be solved with this function

Ordinary Differential Equations  (1)

Consider the following ODE with initial condition :

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-ej7og1
Out[1]=1

Apply the Fourier sine transform to the ODE:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-wkf3px
Out[2]=2

Solve for the transform:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-ilph5p
Out[3]=3

Find the inverse Fourier sine transform with and :

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-g6xb71
Out[4]=4

Compare with DSolveValue:

In[5]:=5
✖
https://wolfram.com/xid/0yv4747dycuq-6h29lw
Out[5]=5

Partial Differential Equations  (1)

Solve the infinite diffusion problem for , : with initial condition for and boundary condition for :

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-jscm4x

Fourier sine transform with respect to :

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-9py0g3
Out[2]=2

With and , solve this ODE:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-4bllfe
Out[3]=3

Compute the inverse sine transform:

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-1r2yv3
Out[4]=4

Compare with DSolveValue:

In[5]:=5
✖
https://wolfram.com/xid/0yv4747dycuq-dkim4w
Out[5]=5

Consider the special case with and :

In[6]:=6
✖
https://wolfram.com/xid/0yv4747dycuq-bvc9pm
Out[6]=6

Evaluation of Integrals  (2)

Calculate the following definite integral for :

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-8tmc8r
Out[1]=1

Compute the Fourier sine transform of an exponential function:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-ee38yx
Out[2]=2

Apply the Fourier sine inversion formula:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-l8n5u3
Out[3]=3

Solve for the definite integral:

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-1yiih3
Out[4]=4

Compare with Integrate:

In[5]:=5
✖
https://wolfram.com/xid/0yv4747dycuq-jdypq8
Out[5]=5

Calculate the following definite integral for :

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-r687jy
Out[1]=1

Compute the Fourier sine transform of an exponential function:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-drholg
Out[2]=2

Apply Parseval's identity:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-gkfkxr
Out[3]=3

Or equivalently:

In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-tbzmtb
Out[4]=4

Solve for the definite integral:

In[5]:=5
✖
https://wolfram.com/xid/0yv4747dycuq-6ioxzg
Out[5]=5

Compare with Integrate:

In[6]:=6
✖
https://wolfram.com/xid/0yv4747dycuq-5jgzyd
Out[6]=6

Properties & Relations  (4)Properties of the function, and connections to other functions

By default, the Fourier sine transform of is:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-iyfnht
Out[1]=1

For , the definite integral becomes:

In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-p3u34y
Out[2]=2

Compare with FourierSinTransform:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-d9s0gi
Out[3]=3

Use Asymptotic to compute an asymptotic approximation:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-lwbzos
Out[1]=1

FourierSinTransform and InverseFourierSinTransform are mutual inverses:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-bp78fy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-bhj9ka
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-it7bfn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-byd6ra
Out[4]=4

Results from FourierSinTransform and FourierTransform differ by a factor of for odd functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-hrqxcb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-ef1iz8
Out[2]=2

The results differ by a factor of for ω>0:

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-b2ebmr
Out[3]=3

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:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-5jbdmw
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-csfig0
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0yv4747dycuq-mjrqa7
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0yv4747dycuq-j88uu6
Out[4]=4

Neat Examples  (2)Surprising or curious use cases

The Fourier sine transform represented in terms of MeijerG:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-3m6m5
Out[1]=1

Create a table of basic Fourier sine transforms:

In[1]:=1
✖
https://wolfram.com/xid/0yv4747dycuq-wwndk9
In[2]:=2
✖
https://wolfram.com/xid/0yv4747dycuq-zsny25
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).

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: 11-July-2025 ]}

✖
@misc{reference.wolfram_2025_fouriersintransform, author="Wolfram Research", title="{FourierSinTransform}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/FourierSinTransform.html}", note=[Accessed: 11-July-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: 11-July-2025 ]}

✖
@online{reference.wolfram_2025_fouriersintransform, organization={Wolfram Research}, title={FourierSinTransform}, year={2025}, url={https://reference.wolfram.com/language/ref/FourierSinTransform.html}, note=[Accessed: 11-July-2025 ]}