--- title: "FourierCosSeries" language: "en" type: "Symbol" summary: "FourierCosSeries[expr, t, n] gives the n\\[Null]^th-order Fourier cosine series expansion of expr in t. FourierCosSeries[expr, {t1, t2, ...}, {n1, n2, ...}] gives the multidimensional Fourier cosine series of expr." keywords: - Fourier cosine series - Fourier cos series - Fourier cosine expansion - Fourier cos expansion - orthogonal function series - orthogonal function expansion - Joseph Fourier - boundary value problem - Sturm-Liouville problem canonical_url: "https://reference.wolfram.com/language/ref/FourierCosSeries.html" source: "Wolfram Language Documentation" related_guides: - title: "Fourier Analysis" link: "https://reference.wolfram.com/language/guide/FourierAnalysis.en.md" - title: "Integral Transforms" link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md" related_functions: - title: "FourierCosCoefficient" link: "https://reference.wolfram.com/language/ref/FourierCosCoefficient.en.md" - title: "FourierDCT" link: "https://reference.wolfram.com/language/ref/FourierDCT.en.md" - title: "FourierTrigSeries" link: "https://reference.wolfram.com/language/ref/FourierTrigSeries.en.md" - title: "FourierSinSeries" link: "https://reference.wolfram.com/language/ref/FourierSinSeries.en.md" - title: "FourierSeries" link: "https://reference.wolfram.com/language/ref/FourierSeries.en.md" - title: "Fourier" link: "https://reference.wolfram.com/language/ref/Fourier.en.md" - title: "Integrate" link: "https://reference.wolfram.com/language/ref/Integrate.en.md" --- # FourierCosSeries FourierCosSeries[expr, t, n] gives the n$$^{\text{th}}$$-order Fourier cosine series expansion of expr in t. FourierCosSeries[expr, {t1, t2, …}, {n1, n2, …}] gives the multidimensional Fourier cosine series of expr. ## Details and Options * The $n$$$^{\text{th}}$$-order Fourier cosine series of $f(t)$ is by default defined to be $\sum _{k=0}^n a_k \cos (k t)$ with $a_k=\frac{2}{\pi }\int _0^{\pi }f(t) \cos (k t)dt$ and $a_0=\frac{1}{\pi }\int_0^{\pi } f(t) \, dt$. * The $m$-dimensional Fourier cosine series of $f\left(t_1,t_2,\ldots \right)$ is given by $\sum _{k_1=0}^{n_1} \sum _{k_2=0}^{n_2} \cdots a_{k_1,k_2,\ldots }\cos \left(k_1 t_1\right)\cos \left(k_2 t_2\right)\cdots$ with $a_{k_1,k_2,\ldots }=\left(\frac{2}{\pi }\right)^m\int _0^{\pi }\int _0^{\pi }\cdots f\left(t_1,t_2,\ldots \right) \cos \left(k_1 t_1\right)\cos \left(k_2 t_2\right)\cdots dt_1dt_2\cdots$. * The following options can be given: | | | | | ------------------ | ------------- | ----------------------------------------------------------------- | | Assumptions | \$Assumptions | assumptions on parameters | | FourierParameters | {1, 1} | parameters to define Fourier cosine series | | GenerateConditions | False | whether to generate results that involve conditions on parameters | * Common settings for ``FourierParameters`` include: | | | | | --- | --- | --- | | {1, 1} | $a_k=\frac{2}{\pi }\int _0^{\pi }f(t) \cos (k t)dt$ | $\sum _{k=0}^n a_k \cos (k t)$ | | {1, 2Pi} | $a_k=4\int _0^{1/2}f(t) \cos (2 \pi k t)dt$ | $\sum _{k=0}^n a_k \cos (2 \pi k t)$ | | {a, b} | $a_k=\left\| \frac{2 b}{\pi }\right\| ^{\frac{a+1}{2}} \int _0^{\pi /\| b\| }f(t) \cos (b k t)dt$ | $\left\| \frac{2b}{ \pi }\right\| ^{\frac{1-a}{2}}\sum _{k=0}^n a_k \cos (b k t)$ | * The Fourier cosine series of $f(t)$ is equivalent to the Fourier series of $\begin{cases} f(t) & t\geq 0 \\ f(-t) & t<0 \end{cases}$. ## Examples (6) ### Basic Examples (2) Find the 5$$^{\text{th}}$$-order Fourier cosine series of $t^2$ : ```wl In[1]:= FourierCosSeries[t ^ 2, t, 5] Out[1]= (π^2/3) + 4 (-Cos[t] + (1/4) Cos[2 t] - (1/9) Cos[3 t] + (1/16) Cos[4 t] - (1/25) Cos[5 t]) In[2]:= Plot[%, {t, -3Pi, 3Pi}] Out[2]= [image] ``` --- Find the ``{2, 2}`` - order Fourier cosine series: ```wl In[1]:= FourierCosSeries[x Exp[-y], {x, y}, {2, 2}] Out[1]= (1 + E^-π) Cos[y] - ((4 + 4 E^-π) Cos[x] Cos[y]/π^2) + ((2/5) - (2 E^-π/5)) Cos[2 y] + (8 (-1 + E^-π) Cos[x] Cos[2 y]/5 π^2) + (8 Cos[x] (-1 + Cosh[π] - Sinh[π])/π^2) + 2 (1 - Cosh[π] + Sinh[π]) In[2]:= Plot3D[%, {x, -Pi, Pi}, {y, -Pi, Pi}] Out[2]= [image] ``` ### Scope (3) Find the $$4^{\text{th}}$$-order Fourier cosine series of a quadratic polynomial: ```wl In[1]:= FourierCosSeries[t ^ 2 + 3t + 7, t, 4] Out[1]= (1/2) (14 + 3 π + (2 π^2/3)) - (4 (3 + π) Cos[t]/π) + Cos[2 t] - (4 (3 + π) Cos[3 t]/9 π) + (1/4) Cos[4 t] ``` --- Fourier cosine series for a piecewise function: ```wl In[1]:= FourierCosSeries[UnitStep[t(Pi / 2 - t)], t, 10] Out[1]= (1/2) + (2 Cos[t]/π) - (2 Cos[3 t]/3 π) + (2 Cos[5 t]/5 π) - (2 Cos[7 t]/7 π) + (2 Cos[9 t]/9 π) In[2]:= Plot[%, {t, 0, Pi}] Out[2]= [image] ``` --- The Fourier cosine series for a basis function has only one term: ```wl In[1]:= FourierCosSeries[Cos[3 t], t, 5] Out[1]= Cos[3 t] ``` ### Options (1) #### FourierParameters (1) Use a nondefault setting for ``FourierParameters`` : ```wl In[1]:= FourierCosSeries[UnitStep[t(1 / 4 - t)], t, 10, FourierParameters -> {1, 2Pi}] Out[1]= (1/2) + (2 Cos[2 π t]/π) - (2 Cos[6 π t]/3 π) + (2 Cos[10 π t]/5 π) - (2 Cos[14 π t]/7 π) + (2 Cos[18 π t]/9 π) In[2]:= Plot[%, {t, 0, 1 / 2}] Out[2]= [image] ``` ## See Also * [`FourierCosCoefficient`](https://reference.wolfram.com/language/ref/FourierCosCoefficient.en.md) * [`FourierDCT`](https://reference.wolfram.com/language/ref/FourierDCT.en.md) * [`FourierTrigSeries`](https://reference.wolfram.com/language/ref/FourierTrigSeries.en.md) * [`FourierSinSeries`](https://reference.wolfram.com/language/ref/FourierSinSeries.en.md) * [`FourierSeries`](https://reference.wolfram.com/language/ref/FourierSeries.en.md) * [`Fourier`](https://reference.wolfram.com/language/ref/Fourier.en.md) * [`Integrate`](https://reference.wolfram.com/language/ref/Integrate.en.md) ## Related Guides * [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.en.md) * [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md) ## History * [Introduced in 2008 (7.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn70.en.md)