--- title: "FourierCosTransform" language: "en" type: "Symbol" summary: "FourierCosTransform[f[t], t, \\[Omega]] gives the symbolic Fourier cosine transform of f[t] in the variable t as F[\\[Omega]] in the variable \\[Omega]. FourierCosTransform[f[t], t, OverscriptBox[StyleBox[\\[Omega], TI, FontSlant->Plain], ^]] gives the numeric Fourier cosine transform at the numerical value OverscriptBox[StyleBox[\\[Omega], TI, FontSlant->Plain], ^]. FourierCosTransform[f[t1, ..., tn], {t1, ..., tn}, {\\[Omega] 1, ..., \\[Omega] n}] gives the multidimensional Fourier cosine transform of f[t1, ..., tn]." keywords: - even function Fourier transform - real Fourier Transform - cosine transform - distributions - Dirac delta function - function transform - integral operator - integral transform canonical_url: "https://reference.wolfram.com/language/ref/FourierCosTransform.html" source: "Wolfram Language Documentation" related_guides: - title: "Integral Transforms" link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md" - title: "Fourier Analysis" link: "https://reference.wolfram.com/language/guide/FourierAnalysis.en.md" related_functions: - title: "FourierSinTransform" link: "https://reference.wolfram.com/language/ref/FourierSinTransform.en.md" - title: "FourierTransform" link: "https://reference.wolfram.com/language/ref/FourierTransform.en.md" - title: "FourierDCT" link: "https://reference.wolfram.com/language/ref/FourierDCT.en.md" - title: "FourierCosSeries" link: "https://reference.wolfram.com/language/ref/FourierCosSeries.en.md" - title: "FourierCosCoefficient" link: "https://reference.wolfram.com/language/ref/FourierCosCoefficient.en.md" - title: "InverseFourierCosTransform" link: "https://reference.wolfram.com/language/ref/InverseFourierCosTransform.en.md" - title: "Convolve" link: "https://reference.wolfram.com/language/ref/Convolve.en.md" - title: "Asymptotic" link: "https://reference.wolfram.com/language/ref/Asymptotic.en.md" related_tutorials: - title: "Integral Transforms and Related Operations" link: "https://reference.wolfram.com/language/tutorial/Calculus.en.md#26017" --- # FourierCosTransform FourierCosTransform[f[t], t, ω] gives the symbolic Fourier cosine transform of f[t] in the variable t as F[ω] in the variable ω. FourierCosTransform[f[t], t, Overscript[ω, ^ ]] gives the numeric Fourier cosine transform at the numerical value Overscript[ω, ^ ]. FourierCosTransform[f[t1, …, tn], {t1, …, tn}, {ω1, …, ωn}] gives the multidimensional Fourier cosine transform of f[t1, …, tn]. ## Details and Options * The Fourier cosine 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 sine transform, and they are still used in applications like signal processing, statistics and image and video compression. * The Fourier cosine transform of the time domain function $f(t)$ is the frequency domain function $F(\omega )$ for $\omega \text{$>$=}0$ : [image] * The Fourier cosine transform of a function $f(t)$ is by default defined to be $\sqrt{\frac{2}{\pi }}\int _0^{\infty }f(t) \cos (\omega \text{ }t)dt$. * The multidimensional Fourier cosine transform of a function $f\left(t_1,t_2,\ldots ,t_n\right)$ is by default defined to be $\left(\frac{2}{\pi }\right)^{n/2}\int _0^{\infty }\int _0^{\infty }\cdots f\left(t_1,t_2,\ldots ,t_n\right) \cos \left(\omega _1 t_1\right)\cos \left(\omega _2t_2\right)\ldots \cos \left(\omega _n t_n\right)dt_1dt_2\ldots dt_n$ or when using vector notation, $\left(\frac{2}{\pi }\right)^{n/2}\int _{t\in \mathbb{R}_{>\, 0}{}^n} f(t) \cos (\omega t)dt$. * Different choices of definitions can be specified using the option ``FourierParameters``. * The integral is computed using numerical methods if the third argument, $\omega$, is given a numerical value. * The asymptotic Fourier cosine 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 cosine 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, ``FourierCosTransform`` not only works with absolutely integrable functions on $[0,\infty )$, 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 lower limit of the integral is effectively taken to be $0^-$, so that the Fourier cosine transform of the Dirac delta function $\delta (t)$ is equal to $\frac{1}{\sqrt{2 \pi }}$. » * 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 cosine 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} | $ $$\sqrt{\frac{2}{\pi }}\int _0^{\infty }f(t) \cos (\omega \text{ }t)dt$ | | {1, 1} | $2\int _0^{\infty }f(t)\cos (\omega \text{ }t)dt$ | | {-1, 1} | $\frac{1}{ \pi }\int _0^{\infty }f(t)e^{i\text{ }\omega \text{ }t}dt$ | | {0, 2Pi} | $2\int _0^{\infty }f(t)\cos (2\pi \omega \text{ }t)dt$ | | {a, b} | $2\sqrt{\frac{\|b\|}{(2\pi )^{1-a}}}\int _0^{\infty }f(t)\cos (b \omega t)dt$ | --- ## Examples (64) ### Basic Examples (6) Compute the Fourier cosine transform of a function: ```wl In[1]:= FourierCosTransform[UnitBox[t - 1 / 2], t, ω] Out[1]= Sqrt[(2/π)] Sinc[ω] ``` Plot the function and its Fourier cosine transform: ```wl In[2]:= {Plot[UnitBox[t - 1 / 2], {t, 0, 2}, Exclusions -> None], Plot[%, {ω, 0, 10}]} Out[2]= {[image], [image]} ``` --- Fourier cosine transform of reciprocal square root: ```wl In[1]:= FourierCosTransform[1 / Sqrt[t], t, ω] Out[1]= (1/Sqrt[ω]) ``` --- For a different convention, change the parameters: ```wl In[1]:= FourierCosTransform[1 / Sqrt[t], t, ω, FourierParameters -> {1, 2π}] Out[1]= (Sqrt[2 π]/Sqrt[ω]) ``` --- Fourier cosine transform of a Gaussian is another Gaussian: ```wl In[1]:= FourierCosTransform[E ^ (-t ^ 2), t, ω] Out[1]= (E^-(ω^2/4)/Sqrt[2]) ``` Plot both Gaussians: ```wl In[2]:= {Plot[E ^ (-t ^ 2), {t, 0, 5}, PlotRange -> Full], Plot[%, {ω, 0, 5}, PlotRange -> Full]} Out[2]= {[image], [image]} ``` --- Compute the Fourier cosine transform of a multivariate function: ```wl In[1]:= FourierCosTransform[1 / Sqrt[x ^ 2 + y ^ 2], {x, y}, {u, v}] Out[1]= (1/Sqrt[u^2 + v^2]) ``` Plot the result: ```wl In[2]:= Plot3D[%, {u, 0, 2}, {v, 0, 2}, PlotRange -> {0, 10}, Mesh -> None] Out[2]= [image] ``` --- Compute the transform at a single point: ```wl In[1]:= FourierCosTransform[Exp[-t ^ 2] Sin[t], t, 0.3] Out[1]= 0.326023 ``` ### Scope (39) #### Basic Uses (3) Fourier cosine transform of a function for a symbolic parameter $\omega$ : ```wl In[1]:= FourierCosTransform[HeavisideTheta[t], t, ω] Out[1]= Sqrt[2 π] DiracDelta[ω] ``` --- Fourier cosine transforms involving trigonometric functions: ```wl In[1]:= FourierCosTransform[Sin[t] / (t E ^ t), t, ω] Out[1]= (ArcTan[1 - ω] + ArcTan[1 + ω]/Sqrt[2 π]) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 4}] Out[2]= [image] In[3]:= FourierCosTransform[Cos[α t ^ 2], t, ω, Assumptions -> α > 0] Out[3]= (Cos[(ω^2/4 α)] + Sin[(ω^2/4 α)]/2 Sqrt[α]) ``` Plot the transform for $\alpha =1/4$ : ```wl In[4]:= Plot[% /. α -> 1 / 4, {ω, 0, 4}] Out[4]= [image] ``` --- Evaluate the Fourier cosine transform for a numerical value of the parameter $\omega$ : ```wl In[1]:= FourierCosTransform[Exp[-t] / Sqrt[t], t, 1.3] Out[1]= 0.990686 ``` #### Algebraic Functions (4) Fourier cosine transform of power functions: ```wl In[1]:= FourierCosTransform[t ^ (n - 1), t, ω, Assumptions -> 0 < n < 1] Out[1]= Sqrt[(2/π)] ω^-n Cos[(n π/2)] Gamma[n] ``` --- Cosine transform for rational functions: ```wl In[1]:= FourierCosTransform[1 / (1 + t ^ 2), t, ω] Out[1]= E^-ω Sqrt[(π/2)] ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 4}] Out[2]= [image] In[3]:= FourierCosTransform[t / (1 + t ^ 2), t, ω] Out[3]= (MeijerG[{{0}, {}}, {{0, 0}, {(1/2)}}, (ω^2/4)]/Sqrt[2]) ``` Plot the transform: ```wl In[4]:= Plot[%, {ω, 0, 4}] Out[4]= [image] ``` --- Fourier cosine transform of a quotient of two nonlinear polynomials: ```wl In[1]:= FourierCosTransform[(1 - t ^ 2) / (t ^ 2 + 1) ^ 2, t, ω]//TrigToExp Out[1]= E^-ω Sqrt[(π/2)] ω ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 4}] Out[2]= [image] ``` --- Fourier cosine transform of a quotient of quadratic and quartic polynomials: ```wl In[1]:= FourierCosTransform[t ^ 2 / (t ^ 4 + 1), t, ω] Out[1]= (1/2) E^-(ω/Sqrt[2]) Sqrt[π] (Cos[(ω/Sqrt[2])] - Sin[(ω/Sqrt[2])]) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 10}] Out[2]= [image] ``` #### Exponential and Logarithmic Functions (3) Fourier cosine transform of an exponential function: ```wl In[1]:= FourierCosTransform[Exp[-α t], t, ω, Assumptions -> Re[α] > 0] Out[1]= (Sqrt[(2/π)] α/α^2 + ω^2) ``` Transform for $\alpha =3$ : ```wl In[2]:= FourierCosTransform[Exp[-3t], t, ω] Out[2]= (3 Sqrt[(2/π)]/9 + ω^2) ``` Plot the transform: ```wl In[3]:= Plot[%, {ω, 0, 4}] Out[3]= [image] ``` --- Fourier cosine transforms of products of exponential and trigonometric functions: ```wl In[1]:= FourierCosTransform[Exp[α t] Sin[ β t], t, ω, Assumptions -> {α < 0, β > 0}] Out[1]= (Sqrt[(2/π)] β (α^2 + β^2 - ω^2)/(α^2 + (β - ω)^2) (α^2 + (β + ω)^2)) ``` Plot the transform for $\alpha =3/4$ and $\beta =1/4$ : ```wl In[2]:= Plot[% /. {α -> 3 / 4, β -> 1 / 4}, {ω, 0, 4}] Out[2]= [image] In[3]:= FourierCosTransform[Exp[-t ^ 2] Sin[t], t, ω] Out[3]= ((DawsonF[(1/2) - (ω/2)] + DawsonF[(1 + ω/2)]) Sign[-1 + ω^2]/Sqrt[2 π] Sign[-1 + ω]) ``` Plot the transform: ```wl In[4]:= Plot[%, {ω, 0, 4}] Out[4]= [image] ``` --- Cosine transforms of logarithmic functions: ```wl In[1]:= FourierCosTransform[Log[t]UnitBox[t - 1 / 2], t, ω] Out[1]= -(Sqrt[(2/π)] SinIntegral[ω]/ω) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 10}] Out[2]= [image] In[3]:= FourierCosTransform[Log[t] / Sqrt[t], t, ω] Out[3]= -(2 EulerGamma + π + Log[16] + 2 Log[ω]/2 Sqrt[ω]) ``` Plot the transform: ```wl In[4]:= Plot[%, {ω, 0, 4}] Out[4]= [image] In[5]:= FourierCosTransform[Log[1 + t] / t, t, ω] Out[5]= (MeijerG[{{0}, {1}}, {{0, 0, 0, (1/2)}, {}}, (ω^2/4)]/2 Sqrt[2] π) ``` Plot the transform: ```wl In[6]:= Plot[%, {ω, 0, 4}] Out[6]= [image] In[7]:= FourierCosTransform[Log[1 + 9 / t ^ 2], t, ω] Out[7]= (Sqrt[(2/π)] (π - E^-3 ω π)/ω) ``` Plot the transform: ```wl In[8]:= Plot[%, {ω, 0, 4}] Out[8]= [image] ``` #### Trigonometric Functions (5) Expressions involving trigonometric functions: ```wl In[1]:= FourierCosTransform[Sin[2t] ^ 2 / t ^ 2, t, ω] Out[1]= (1/4) Sqrt[(π/2)] (4 - ω + Abs[-4 + ω]) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 4}] Out[2]= [image] ``` --- Composition of elementary functions: ```wl In[1]:= FourierCosTransform[Cos[α t ^ 2], t, ω, Assumptions -> α > 0] Out[1]= (Cos[(ω^2/4 α)] + Sin[(ω^2/4 α)]/2 Sqrt[α]) ``` Plot the transform for $\alpha =3/4$ : ```wl In[2]:= Plot[% /. α -> 3 / 4, {ω, 0, 10}] Out[2]= [image] ``` --- Ratio of sine and product of exponential and linear functions: ```wl In[1]:= FourierCosTransform[(Sin[t]/t Exp[t]), t, ω] Out[1]= (ArcTan[1 - ω] + ArcTan[1 + ω]/Sqrt[2 π]) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 10}] Out[2]= [image] ``` --- Fourier cosine transform of arctangent functions: ```wl In[1]:= FourierCosTransform[ArcTan[2 / t], t, ω] Out[1]= (MeijerG[{{0}, {}}, {{0, 0}, {-(1/2)}}, ω^2]/Sqrt[2]) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 10}] Out[2]= [image] In[3]:= FourierCosTransform[ArcTan[2 / t ^ 2], t, ω] Out[3]= E^-ω Sqrt[2 π] Sinc[ω] ``` Plot the transform: ```wl In[4]:= Plot[%, {ω, 0, 4}] Out[4]= [image] ``` --- Fourier cosine transform of ``Sech`` is another ``Sech`` : ```wl In[1]:= FourierCosTransform[Sech[t], t, ω] Out[1]= Sqrt[(π/2)] Sech[(π ω/2)] ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 4}] Out[2]= [image] ``` #### Special Functions (7) ``Sinc`` function: ```wl In[1]:= FourierCosTransform[Sinc[t], t, ω] Out[1]= (1/2) Sqrt[(π/2)] (1 + Sign[1 - ω]) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 4}, Exclusions -> None] Out[2]= [image] ``` --- Fourier cosine transforms of expressions involving ``ExpIntegralEi`` : ```wl In[1]:= FourierCosTransform[ExpIntegralEi[-t], t, ω] Out[1]= -(Sqrt[(2/π)] ArcTan[ω]/ω) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 4}] Out[2]= [image] ``` --- Expression involving ``Erfc`` : ```wl In[1]:= FourierCosTransform[t Erfc[α t], t, ω, Assumptions -> α > 0] Out[1]= (E^-(ω^2/4 α^2) ((1/α^2) + (2/ω^2)) - (2/ω^2)/Sqrt[2 π]) ``` Plot the transform for $\alpha =2$ : ```wl In[2]:= Plot[% /. α -> 2, {ω, 0, 10}] Out[2]= [image] ``` --- Expression involving ``SinIntegral`` : ```wl In[1]:= FourierCosTransform[SinIntegral[t] - π / 2, t, ω] Out[1]= (1/ω)Sqrt[(2/π)] (-ArcCoth[ω] - ArcTanh[ω] + ArcCoth[ω] UnitStep[1 - ω] + ArcTanh[ω] (UnitStep[-1 + ω] + UnitStep[-ω])) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 10}] Out[2]= [image] ``` --- ``CosIntegral`` : ```wl In[1]:= FourierCosTransform[CosIntegral[α t], t, ω, Assumptions -> {α > 0, ω > α}] Out[1]= -(Sqrt[(π/2)]/ω) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 4}] Out[2]= [image] ``` --- Cosine transforms for ``BesselJ`` functions: ```wl In[1]:= FourierCosTransform[BesselJ[0, α t], t, ω, Assumptions -> {α > 0, 0 < ω < α}] Out[1]= (Sqrt[(2/π)]/Sqrt[α^2 - ω^2]) ``` Plot the transform for $\alpha =2$ : ```wl In[2]:= Plot[% /. α -> 2, {ω, 0, 2.5}] Out[2]= [image] In[3]:= FourierCosTransform[BesselJ[2 n, α t], t, ω, Assumptions -> {α > 0, 0 < ω < α}] Out[3]= (Sqrt[(2/π)] Cos[2 n ArcSin[(ω/α)]]/Sqrt[α^2 - ω^2]) ``` Plot the transform for $\alpha =2$ and $n=2$ : ```wl In[4]:= Plot[% /. {α -> 2, n -> 2}, {ω, 0, 2.5}] Out[4]= [image] In[5]:= FourierCosTransform[BesselJ[n, α t] / t ^ n, t, ω, Assumptions -> {α > 0, 0 < ω < α, n∈PositiveIntegers}] Out[5]= (2^(1/2) - n α^-n (α^2 - ω^2)^-(1/2) + n/Gamma[(1/2) + n]) ``` Plot the transform for $\alpha =2$ and $n=2$ : ```wl In[6]:= Plot[% /. {α -> 2, n -> 2}, {ω, 0, 2.5}] Out[6]= [image] In[7]:= FourierCosTransform[BesselJ[0, Sqrt[t]], t, ω] Out[7]= (Sqrt[(2/π)] Sin[(1/4 ω)]/ω) ``` Plot the transform: ```wl In[8]:= Plot[%, {ω, 0, 4}] Out[8]= [image] ``` --- Cosine transforms for ``BesselY`` functions: ```wl In[1]:= FourierCosTransform[BesselY[0, α t], t, ω, Assumptions -> {α > 0, ω > α}] Out[1]= -(Sqrt[(2/π)]/Sqrt[-α^2 + ω^2]) ``` Plot the transform for $\alpha =2$ : ```wl In[2]:= Plot[% /. α -> 2, {ω, 0, 4}] Out[2]= [image] In[3]:= FourierCosTransform[BesselY[n, α t] t ^ n, t, ω, Assumptions -> {α > 0, ω > α, Abs[Re[n]] < 1 / 2}] Out[3]= -(2^(1/2) + n α^n (-α^2 + ω^2)^-(1/2) - n/Gamma[(1/2) - n]) ``` Plot the transform for $\alpha =2$ and $n=1/3$ : ```wl In[4]:= Plot[% /. {α -> 2, n -> 1 / 3}, {ω, 0, 4}] Out[4]= [image] ``` #### Piecewise Functions and Distributions (4) Fourier cosine transform of a piecewise function: ```wl In[1]:= f[t_] = Piecewise[{{t, 0 ≤ t ≤ 1}, {0, t > 1}}]; Plot[f[t], {t, 0, 1.5}, Exclusions -> None] Out[1]= [image] In[2]:= FourierCosTransform[f[t], t, ω] Out[2]= (Sqrt[(2/π)] (-1 + Cos[ω] + ω^2 Sinc[ω])/ω^2) ``` --- Restriction of a sine function to a half-period: ```wl In[1]:= Plot[Sin[α t]UnitBox[α t / π - 1 / 2] /. α -> 3π / 2, {t, 0, 1}] Out[1]= [image] In[2]:= Simplify[FourierCosTransform[Sin[α t]UnitBox[α t / π - 1 / 2], t, ω], α > π] Out[2]= (Sqrt[(2/π)] α (1 + Cos[(π ω/α)])/α^2 - ω^2) ``` --- Triangular function: ```wl In[1]:= f[t_] = Piecewise[{{t, 0 ≤ t ≤ 1}, {2 - t, 1 < t ≤ 2}, {0, 2 < t }}]; Plot[f[t], {t, 0, 3}] Out[1]= [image] In[2]:= FourierCosTransform[f[t], t, ω] Out[2]= (4 Sqrt[(2/π)] Cos[ω] Sin[(ω/2)]^2/ω^2) ``` --- Transforms in terms of ``FresnelC`` : ```wl In[1]:= FourierCosTransform[UnitStep[t - 1] / Sqrt[t], t, ω] Out[1]= (1 - 2 FresnelC[Sqrt[(2/π)] Sqrt[ω]]/Sqrt[ω]) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 4}] Out[2]= [image] In[3]:= FourierCosTransform[(1 - UnitStep[t - 1]) / Sqrt[t], t, ω] Out[3]= (2 FresnelC[Sqrt[(2/π)] Sqrt[ω]]/Sqrt[ω]) ``` Plot the transform: ```wl In[4]:= Plot[%, {ω, 0, 4}] Out[4]= [image] ``` #### Periodic Functions (2) Fourier cosine transform of cosine: ```wl In[1]:= FourierCosTransform[Cos[t], t, ω] Out[1]= Sqrt[(π/2)] (DiracDelta[-1 + ω] + DiracDelta[1 + ω]) ``` --- Fourier cosine transform of ``SquareWave`` : ```wl In[1]:= Plot[SquareWave[t + 1 / 4], {t, 0, 4}, Exclusions -> None] Out[1]= [image] In[2]:= FourierCosTransform[SquareWave[t + 1 / 4], t, ω] Out[2]= Underoverscript[∑, K[1] = 1, ∞](1/K[1])2 Sqrt[(2/π)] (DiracDelta[ω - π K[1]] + DiracDelta[ω + π K[1]]) (Sin[(1/4) π K[1]] - Sin[(3/4) π K[1]]) ``` #### Generalized Functions (4) Fourier cosine transforms of expressions involving ``HeavisideTheta`` : ```wl In[1]:= Plot[HeavisideTheta[t - 1], {t, 0, 2}, Exclusions -> None] Out[1]= [image] In[2]:= FourierCosTransform[HeavisideTheta[t - 1], t, ω] Out[2]= Sqrt[2 π] DiracDelta[ω] - (Sqrt[(2/π)] Sin[ω]/ω) In[3]:= Plot[t HeavisideTheta[t] HeavisideTheta[1 - t], {t, 0, 1.5}, Exclusions -> None] Out[3]= [image] In[4]:= FourierCosTransform[t HeavisideTheta[t] HeavisideTheta[1 - t], t, ω] Out[4]= (Sqrt[(2/π)] (-1 + Cos[ω] + ω^2 Sinc[ω])/ω^2) ``` --- Fourier cosine transforms involving ``DiracDelta`` : ```wl In[1]:= FourierCosTransform[DiracDelta[t], t, ω] Out[1]= (1/Sqrt[2 π]) ``` Plot the transform: ```wl In[2]:= Plot[%, {ω, 0, 4}] Out[2]= [image] In[3]:= FourierCosTransform[DiracDelta[t - 1], t, ω] Out[3]= Sqrt[(2/π)] Cos[ω] ``` Plot the transform: ```wl In[4]:= Plot[%, {ω, 0, 4}] Out[4]= [image] In[5]:= FourierCosTransform[DiracDelta'[t - 1], t, ω] Out[5]= Sqrt[(2/π)] ω Sin[ω] ``` Plot the transform: ```wl In[6]:= Plot[%, {ω, 0, 4}] Out[6]= [image] ``` --- Fourier cosine transform involving ``HeavisideLambda`` : ```wl In[1]:= Plot[HeavisideLambda[t - 1], {t, 0, 4}] Out[1]= [image] In[2]:= FourierCosTransform[HeavisideLambda[t - 1], t, ω] Out[2]= (4 Sqrt[(2/π)] Cos[ω] Sin[(ω/2)]^2/ω^2) ``` --- Fourier cosine transform involving ``HeavisidePi`` : ```wl In[1]:= Plot[HeavisidePi[t - (3/2)], {t, 0, 3}, Exclusions -> None] Out[1]= [image] In[2]:= FourierCosTransform[HeavisidePi[t - (3/2)], t, ω] Out[2]= (Sqrt[(2/π)] (Sin[2 ω] - ω Sinc[ω])/ω) ``` #### Multivariate Functions (2) Fourier cosine transforms of exponentials in two variables: ```wl In[1]:= FourierCosTransform[Exp[-x - y], {x, y}, {u, v}] Out[1]= (2/π (1 + u^2) (1 + v^2)) ``` Plot of both: ```wl In[2]:= {Plot3D[Exp[-x - y], {x, 0, 5}, {y, 0, 5}, Mesh -> None], Plot3D[%, {u, 0, 5}, {v, 0, 5}, Mesh -> None]} Out[2]= {[image], [image]} In[3]:= FourierCosTransform[Exp[-x ^ 2 - y ^ 2], {x, y}, {u, v}] Out[3]= (1/2) E^-(u^2/4) - (v^2/4) ``` Plot of both: ```wl In[4]:= {Plot3D[Exp[-x ^ 2 - y ^ 2], {x, 0, 5}, {y, 0, 5}, Mesh -> None], Plot3D[%, {u, 0, 5}, {v, 0, 5}, Mesh -> None]} Out[4]= {[image], [image]} ``` --- Fourier cosine transform of product of exponential and ``SquareWave`` : ```wl In[1]:= Plot3D[E ^ (-x)SquareWave[y + 1 / 4], {x, 0, 5}, {y, 0, 5}, Mesh -> None] Out[1]= [image] In[2]:= FourierCosTransform[E ^ (-x)SquareWave[y + 1 / 4], {x, y}, {u, v}] Out[2]= (Sqrt[(2/π)] Underoverscript[∑, K[1] = 1, ∞](2 Sqrt[(2/π)] (DiracDelta[v - π K[1]] + DiracDelta[v + π K[1]]) (Sin[(1/4) π K[1]] - Sin[(3/4) π K[1]])/K[1])/1 + u^2) ``` #### Formal Properties (3) Fourier cosine transform of a first-order derivative: ```wl In[1]:= FourierCosTransform[f'[t], t, ω] Out[1]= -Sqrt[(2/π)] f[0] + ω FourierSinTransform[f[t], t, ω] ``` --- Fourier cosine transform of a second-order derivative: ```wl In[1]:= FourierCosTransform[f''[t], t, ω] Out[1]= -ω^2 FourierCosTransform[f[t], t, ω] - Sqrt[(2/π)] Derivative[1][f][0] ``` --- Fourier cosine transform threads itself over equations: ```wl In[1]:= FourierCosTransform[f'[t] == 1 / (t ^ 2 + 1), t, ω] Out[1]= -Sqrt[(2/π)] f[0] + ω FourierSinTransform[f[t], t, ω] == E^-ω Sqrt[(π/2)] ``` #### Numerical Evaluation (2) Calculate the Fourier cosine transform at a single point: ```wl In[1]:= FourierCosTransform[(1/Sqrt[t]), t, .9] Out[1]= 1.05409 ``` --- Alternatively, calculate the Fourier cosine transform symbolically: ```wl In[1]:= FourierCosTransform[(1/Sqrt[t]), t, ω] Out[1]= (1/Sqrt[ω]) ``` Then evaluate it for specific value of $\omega$ : ```wl In[2]:= N[% /. ω -> .9] Out[2]= 1.05409 ``` ### Options (8) #### AccuracyGoal (1) The option ``AccuracyGoal`` sets the number of digits of accuracy: ```wl In[1]:= exact = FourierTransform[HeavisideLambda[t - 1], t, 9 / 10] Out[1]= -(50/81) (-1 + E^9 I / 10)^2 Sqrt[(2/π)] In[2]:= FourierTransform[HeavisideLambda[t - 1], t, .9, AccuracyGoal -> 5] - exact Out[2]= 5.546899278785489`*^-8 - 5.2648017112577605`*^-8 I ``` With default settings: ```wl In[3]:= FourierTransform[HeavisideLambda[t - 1], t, .9] - exact Out[3]= 3.1922080812041287`*^-9 - 1.3353430194928961`*^-8 I ``` #### Assumptions (1) Fourier cosine transform of ``BesselJ`` is a piecewise function: ```wl In[1]:= FourierCosTransform[BesselJ[0, t], t, ω, Assumptions -> ω > 1] Out[1]= 0 In[2]:= FourierCosTransform[BesselJ[0, t], t, ω, Assumptions -> ω < 1] Out[2]= (Sqrt[(2/π)]/Sqrt[1 - ω^2]) ``` #### FourierParameters (3) Fourier cosine transform for the unit box function with different parameters: ```wl In[1]:= params = {{0, 1}, {1, 1}, {-1, 1}, {0, 2 π}}; funs = funs = Table[FourierCosTransform[UnitBox[t - (1/2)], t, ω, FourierParameters -> p], {p, params}] Out[1]= {Sqrt[(2/π)] Sinc[ω], 2 Sinc[ω], (Sinc[ω]/π), (Sin[2 π ω]/π ω)} ``` Create a nicely formatted table of the results: ```wl In[2]:= header = { "Parameters", HoldForm@FourierCosTransform[UnitBox[t - (1/2)], t, ω]}; Grid[Prepend[Transpose[{params, funs}], header], IconizedObject[«Grid options»]]//TraditionalForm Out[2]//TraditionalForm= $$\begin{array}{cc} \text{Parameters} & \text{FourierCosTransform}\left[\Pi \left(t-\frac{1}{2}\right),t,\omega \right] \\ \{0,1\} & \sqrt{\frac{2}{\pi }} \text{sinc}(\omega ) \\ \{1,1\} & 2 \text{sinc}(\omega ) \\ \{-1,1\} & \frac{\text{sinc}(\omega )}{\pi } \\ \{0,2 \pi \} & \frac{\sin (2 \pi \omega )}{\pi \omega } \\ \end{array}$$ ``` --- Use a nondefault setting for a different definition of transform: ```wl In[1]:= FourierCosTransform[Exp[-t], t, ω, FourierParameters -> {1, 1}] Out[1]= (2/1 + ω^2) ``` To get the inverse, use the same ``FourierParameters`` setting: ```wl In[2]:= InverseFourierCosTransform[%, ω, t, FourierParameters -> {1, 1}] Out[2]= E^-t ``` --- Set up your particular global choice of parameters once per session: ```wl In[1]:= SetOptions[FourierCosTransform, FourierParameters -> {0, 2π}] Out[1]= {AccuracyGoal -> ∞, Assumptions :> $Assumptions, FourierParameters -> {0, 2 π}, GenerateConditions -> False, Method -> Automatic, PerformanceGoal :> $PerformanceGoal, PrecisionGoal -> Automatic, WorkingPrecision -> Automatic} In[2]:= FourierCosTransform[1, t, ω]//TraditionalForm Out[2]//TraditionalForm= $$\delta (\omega )$$ ``` Restore defaults: ```wl In[3]:= SetOptions[FourierCosTransform, FourierParameters -> {0, 1}] Out[3]= {AccuracyGoal -> ∞, Assumptions :> $Assumptions, FourierParameters -> {0, 1}, GenerateConditions -> False, Method -> Automatic, PerformanceGoal :> $PerformanceGoal, PrecisionGoal -> Automatic, WorkingPrecision -> Automatic} ``` #### GenerateConditions (1) Use ``GenerateConditions -> True`` to get parameter conditions for when a result is valid: ```wl In[1]:= FourierCosTransform[Exp[α t], t, ω, GenerateConditions -> True] Out[1]= ConditionalExpression[-((Sqrt[2/Pi]*α)/(α^2 + ω^2)), Re[α] < 0] ``` #### PrecisionGoal (1) The option ``PrecisionGoal`` sets the relative tolerance in the integration: ```wl In[1]:= exact = FourierCosTransform[8 / (t ^ 2 + 1), t, 1 / 2] Out[1]= 4 Sqrt[(2 π/E)] In[2]:= FourierCosTransform[8 / (t ^ 2 + 1), t, .5, PrecisionGoal -> 2] - exact Out[2]= -0.0000226833 ``` With default settings: ```wl In[3]:= FourierCosTransform[8 / (t ^ 2 + 1), t, .5] - exact Out[3]= -7.219056641361021`*^-8 ``` #### WorkingPrecision (1) If a ``WorkingPrecision`` is specified, the computation is done at that working precision: ```wl In[1]:= exact = FourierCosTransform[8 / (t ^ 2 + 1), t, 1 / 2] Out[1]= 4 Sqrt[(2 π/E)] In[2]:= FourierCosTransform[8 / (t ^ 2 + 1), t, .5, WorkingPrecision -> 18] - exact Out[2]= 1.4845507295621740990329542386`7.387592358182837*^-10 ``` With default settings: ```wl In[3]:= FourierCosTransform[8 / (t ^ 2 + 1), t, .5] - exact Out[3]= -7.219056641361021`*^-8 ``` ### Applications (4) #### Ordinary Differential Equations (1) Consider the following ODE with initial condition $y'[0]=0$ : ```wl In[1]:= OdEqn = y''[t] + Ω^2y[t] == Cos[t] Out[1]= Ω^2 y[t] + Derivative[2][y][t] == Cos[t] ``` Apply the Fourier cosine transform to the ODE: ```wl In[2]:= FourierCosTransform[OdEqn, t, ω] Out[2]= -ω^2 FourierCosTransform[y[t], t, ω] + Ω^2 FourierCosTransform[y[t], t, ω] - Sqrt[(2/π)] Derivative[1][y][0] == Sqrt[(π/2)] (DiracDelta[-1 + ω] + DiracDelta[1 + ω]) ``` Solve for the Fourier cosine transform of $y[t]$ : ```wl In[3]:= SolveValues[%, FourierCosTransform[y[t], t, ω]][[1]] /. y'[0] -> 1 Out[3]= (2 Sqrt[(2/π)] + Sqrt[2 π] DiracDelta[-1 + ω] + Sqrt[2 π] DiracDelta[1 + ω]/2 (-ω^2 + Ω^2)) ``` Find the inverse Fourier cosine transform with $\kappa =1$ and $\Omega =1/2$ : ```wl In[4]:= icf[t_] = InverseFourierCosTransform[%, ω, t] /. {κ -> 1, Ω -> 1 / 2}//TrigToExp Out[4]= -2 I E^(I t/2) - (2 E^-I t/3) - (2 E^I t/3) ``` Compare with ``DSolveValue`` : ```wl In[5]:= DSolveValue[{OdEqn /. {κ -> 1, Ω -> 1 / 2}, y'[0] == 1, y[0] == icf[0]}, y[t], t]//TrigToExp Out[5]= -2 I E^(I t/2) - (2 E^-I t/3) - (2 E^I t/3) ``` #### Partial Differential Equations (1) Solve the heat equation for $x>0$, $t>0$: $u_t=\alpha ^2u_{\text{xx}}$ with initial condition $u(x,0)=f(x)$ for $x>0$ and Neumann boundary condition $u_x(0,t)=g(t)$ for $t>0$. ```wl In[1]:= eqn = D[u[x, t], t] == α ^ 2 D[u[x, t], x, x]; ``` Apply the Fourier cosine transform to the ODE on $x$ : ```wl In[2]:= FourierCosTransform[eqn, x, ω] Out[2]= FourierCosTransform[u^(0, 1)[x, t], x, ω] == α^2 (-ω^2 FourierCosTransform[u[x, t], x, ω] - Sqrt[(2/π)] u^(1, 0)[0, t]) ``` With $\text{u1}(\omega ,t)=\mathcal{F}_x[u(x,t)](\omega )$ and $\text{u1}(\omega ,0)=\omega _0$, solve this ODE: ```wl In[3]:= DSolveValue[{D[u1[ω, t], t] == α^2 (-ω^2u1[ω, t] - Sqrt[(2/π)] u^(1, 0)[0, t]), u1[ω, 0] == FourierCosTransform[f[x], x, ω][ω]}, u1[ω, t], t] /. {u^(1, 0)[0, K[1]] -> g[K[1]]}//Expand Out[3]= E^-t α^2 ω^2 FourierCosTransform[f[x], x, ω][ω] + E^-t α^2 ω^2 Inactive[Integrate][(-E^(α^2*ω^2*K[1]))*Sqrt[2/Pi]*α^2*g[K[1]], {K[1], 0, t}] ``` Compute the inverse cosine transform of the exponential functions: ```wl In[4]:= h[x_] = InverseFourierCosTransform[Sqrt[(2/π)]E^-t α^2 ω^2, ω, x]; l[x] = InverseFourierCosTransform[-Sqrt[(2/π)]α^2E^(K[1] - t) α^2 ω^2 , ω, x]; ``` The convolution property gives the inverse cosine transform of the first summand to get the solution: ```wl In[5]:= 1 / 2Integrate[f[τ](h[x + τ] + h[x - τ]), {τ, 0, ∞}] + Integrate[l[x]g[K[1]], {K[1], 0, t}] Out[5]= (1/2) Subsuperscript[∫, 0, ∞]((E^-((x - τ)^2/4 t α^2)/Sqrt[π] Sqrt[t α^2]) + (E^-((x + τ)^2/4 t α^2)/Sqrt[π] Sqrt[t α^2])) f[τ]\[DifferentialD]τ + Subsuperscript[∫, 0, t]-(E^(x^2/4 α^2 (-t + K[1])) α^2 g[K[1]]/Sqrt[π] Sqrt[α^2 (t - K[1])])\[DifferentialD]K[1] ``` Consider the special case with $f(x)=\text{UnitBox}\left[x-\frac{1}{2}\right]$, $g(t)=0$ and $\alpha =1$ : ```wl In[6]:= % /. {f -> Function[u, UnitBox[u - 1 / 2]], g -> Function[u, 0], α -> 1} Out[6]= (1/2) (-Erf[(-1 + x/2 Sqrt[t])] + Erf[(1 + x/2 Sqrt[t])]) ``` Compare with ``DSolveValue`` : ```wl In[7]:= DSolveValue[{eqn /. α -> 1, {u[x, 0] == UnitBox[x - 1 / 2], Derivative[1, 0][u][0, t] == 0}}, u[x, t], {x, t}, Assumptions -> {t > 0 && x > 0}] Out[7]= (1/2) (-Erf[(-1 + x/2 Sqrt[t])] + Erf[(1 + x/2 Sqrt[t])]) ``` Plot the initial conditions and solutions for different values of $t$. ```wl In[8]:= Plot[{UnitBox[x - 1 / 2], Evaluate@Table[%, {t, {.005, .1, .3, .8}}]}, {x, 0, 4}, PlotLegends -> {UnitBox[x - 1 / 2], .005, .1, .3, .8}, Exclusions -> None] Out[8]= [image] ``` #### Evaluation of Integrals (2) Calculate the following definite integral: ```wl In[1]:= Inactive[Integrate][(1/(ω ^ 2 + 9)(ω ^ 2 + 4)), {ω, 0, ∞}] Out[1]= Inactive[Integrate][1/((4 + ω^2)*(9 + ω^2)), {ω, 0, Infinity}] ``` Fourier cosine transform preserves integration of products over $[0,\infty )$ : ```wl In[2]:= Inactive[Integrate][FourierCosTransform[(1/(ω ^ 2 + 9)), ω, t]FourierCosTransform[1 / (ω ^ 2 + 4), ω, t], {t, 0, ∞}] Out[2]= Inactive[Integrate][((1/12)*Pi)/E^(5*t), {t, 0, Infinity}] ``` Solve the definite integral: ```wl In[3]:= Activate[%] Out[3]= (π/60) ``` Compare with ``Integrate`` : ```wl In[4]:= Integrate[(1/(ω ^ 2 + 9)(ω ^ 2 + 4)), {ω, 0, ∞}] Out[4]= (π/60) ``` --- Calculate the following definite integral for $\alpha >0$ : ```wl In[1]:= Inactive[Integrate][(1/(α ^ 2 + ω ^ 2) ^ 2), {ω, 0, ∞}] Out[1]= Inactive[Integrate][1/(α^2 + ω^2)^2, {ω, 0, Infinity}] ``` Compute Fourier cosine transform of an exponential function: ```wl In[2]:= FourierCosTransform[E^-α x, x, ω, Assumptions -> α > 0] Out[2]= (Sqrt[(2/π)] α/α^2 + ω^2) ``` Apply Parseval's identity: ```wl In[3]:= Inactive[Integrate][Abs[E^-α x] ^ 2, {x, -∞, ∞}] == Inactive[Integrate][Abs[(Sqrt[(2/π)] α/α^2 + ω^2)] ^ 2, {ω, -∞, ∞}] Out[3]= Inactive[Integrate][E^(-2*Re[x*α]), {x, -Infinity, Infinity}] == Inactive[Integrate][(2*Abs[α/(α^2 + ω^2)]^2)/Pi, {ω, -Infinity, Infinity}] ``` Or equivalently: ```wl In[4]:= Integrate[E^-2α x, {x, 0, ∞}, Assumptions -> α > 0] == HoldForm[(2α ^ 2/π)]Inactive[Integrate][( 1/(α^2 + ω^2) ^ 2), {ω, 0, ∞}] Out[4]= (1/2 α) == (2 α^2/π) Inactive[Integrate][1/(α^2 + ω^2)^2, {ω, 0, Infinity}] ``` Solve for the definite integral: ```wl In[5]:= SolveValues[ReleaseHold[%], Inactive[Integrate][( 1/(α^2 + ω^2) ^ 2), {ω, 0, ∞}]][[1]] Out[5]= (π/4 α^3) ``` Compare with ``Integrate`` : ```wl In[6]:= Integrate[( 1/(α^2 + ω^2) ^ 2), {ω, 0, ∞}, Assumptions -> α > 0] Out[6]= (π/4 α^3) ``` ### Properties & Relations (4) By default, the Fourier cosine transform of $f(t)$ is: ```wl In[1]:= HoldForm[FourierCosTransform[f[t], t, ω] = HoldForm[Sqrt[2 / π]] * Integrate[f[t]Cos[ω t], {t, 0, ∞}]]//TraditionalForm Out[1]//TraditionalForm= $$\text{FourierCosTransform}[f(t),t,\omega ]=\sqrt{\frac{2}{\pi }} \int_0^{\infty } f(t) \cos (\omega t) \, dt$$ ``` For $f(t)=e^{-t^2}\cos (t)$, the definite integral becomes: ```wl In[2]:= Sqrt[2 / π]Integrate[Exp[-t ^ 2]Cos[t]Cos[ω t], {t, 0, ∞}] Out[2]= (E^-(1/4) (-1 + ω)^2 + E^-(1/4) (1 + ω)^2/2 Sqrt[2]) ``` Compare with ``FourierCosTransform`` : ```wl In[3]:= FourierCosTransform[Exp[-t ^ 2]Cos[t], t, ω]//TrigToExp//Simplify Out[3]= (E^-(1/4) (-1 + ω)^2 + E^-(1/4) (1 + ω)^2/2 Sqrt[2]) ``` --- Use ``Asymptotic`` to compute an asymptotic approximation: ```wl In[1]:= Asymptotic[Inactive[FourierCosTransform][E ^ (-t ^ 3), t, ω], ω -> 0] Out[1]= Sqrt[(2/π)] Gamma[(4/3)] ``` --- ``FourierCosTransform`` and ``InverseFourierCosTransform`` are mutual inverses: ```wl In[1]:= InverseFourierCosTransform[FourierCosTransform[f[t], t, ω], ω, t] Out[1]= f[t] In[2]:= FourierCosTransform[InverseFourierCosTransform[G[ω], ω, t], t, ω] Out[2]= G[ω] In[3]:= FourierCosTransform[1 / (t ^ 2 + 1), t, ω] Out[3]= E^-ω Sqrt[(π/2)] In[4]:= InverseFourierCosTransform[%, ω, t] Out[4]= (1/1 + t^2) ``` --- Results from ``FourierCosTransform`` and ``FourierTransform`` agree for even functions: ```wl In[1]:= FourierCosTransform[Sin[t] / t, t, ω] Out[1]= (1/2) Sqrt[(π/2)] (1 + Sign[1 - ω]) In[2]:= FourierTransform[Sin[t] / t, t, ω ] Out[2]= (1/2) Sqrt[(π/2)] (Sign[1 - ω] + Sign[1 + ω]) ``` The results agree for $\omega >0$ : ```wl In[3]:= Simplify[% - %%, ω > 0] Out[3]= 0 ``` ### Possible Issues (1) The result from an inverse Fourier cosine transform may not have the same form as the original: ```wl In[1]:= FourierCosTransform[UnitStep[1 + t] UnitStep[1 - t], t, ω] Out[1]= Sqrt[(2/π)] Sinc[ω] In[2]:= InverseFourierCosTransform[%, ω, t] Out[2]= (1/2) (1 + Sign[1 - t]) ``` Fourier cosine transform may be given in terms of generalized functions such as ``DiracDelta`` : ```wl In[3]:= FourierCosTransform[1, t, ω] Out[3]= Sqrt[2 π] DiracDelta[ω] ``` ### Neat Examples (2) Fourier cosine transform as a Meijer function: ```wl In[1]:= FourierCosTransform[t / (t ^ 3 + 1), t, ω] Out[1]= (MeijerG[{{(2/3)}, {}}, {{0, (1/6), (1/3), (2/3), (2/3)}, {(1/2), (5/6)}}, (ω^6/46656)]/Sqrt[6] π) ``` --- Create a table of basic Fourier cosine transforms: ```wl In[1]:= flist = {t ^ n, E ^ (-a t), Exp[-t ^ 2], Sinc[t], DiracDelta[t - a], Log[t], UnitStep[t], UnitBox[t], t Erfc[a t], ConditionalExpression[BesselJ[2, t], 0 < ω < 1], ConditionalExpression[BesselY[0, a t], ω > a]}; In[2]:= Grid[Prepend[{#, Assuming[{a > 0}, Simplify[FourierCosTransform[#1, t, ω]]]}& /@ flist, {f[t], FourierCosTransform[f[t], t, ω]}], IconizedObject[«Grid options»]]//TraditionalForm Out[2]//TraditionalForm= $$\begin{array}{cc} f(t) & \text{FourierCosTransform}[f(t),t,\omega ] \\ t^n & -\sqrt{\frac{2}{\pi }} \omega ^{-n-1} \sin \left(\frac{\pi n}{2}\right) \Gamma (n+1) \\ e^{-a t} & \frac{\sqrt{\frac{2}{\pi }} a}{a^2+\omega ^2} \\ e^{-t^2} & \frac ... if }0<\omega <1$} & \fbox{$\frac{2-4 \omega ^2}{\sqrt{2 \pi } \sqrt{1-\omega ^2}}\text{ if }0<\omega <1$} \\ \fbox{$Y_0(a t)\text{ if }\omega >a$} & \fbox{$-\frac{\sqrt{\frac{2}{\pi }}}{\sqrt{\omega ^2-a^2}}\text{ if }\omega >a$} \\ \end{array}$$ ``` ## See Also * [`FourierSinTransform`](https://reference.wolfram.com/language/ref/FourierSinTransform.en.md) * [`FourierTransform`](https://reference.wolfram.com/language/ref/FourierTransform.en.md) * [`FourierDCT`](https://reference.wolfram.com/language/ref/FourierDCT.en.md) * [`FourierCosSeries`](https://reference.wolfram.com/language/ref/FourierCosSeries.en.md) * [`FourierCosCoefficient`](https://reference.wolfram.com/language/ref/FourierCosCoefficient.en.md) * [`InverseFourierCosTransform`](https://reference.wolfram.com/language/ref/InverseFourierCosTransform.en.md) * [`Convolve`](https://reference.wolfram.com/language/ref/Convolve.en.md) * [`Asymptotic`](https://reference.wolfram.com/language/ref/Asymptotic.en.md) ## Tech Notes * [Integral Transforms and Related Operations](https://reference.wolfram.com/language/tutorial/Calculus.en.md#26017) ## Related Guides * [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md) * [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.en.md) ## History * Introduced in 1999 (4.0) \| [Updated in 2025 (14.2)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn142.en.md)