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FourierSeries

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FourierSeries[expr,t,n]

gives the n^(th)-order Fourier series expansion of expr in t.

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FourierSeries[expr,{t1,t2,…},{n1,n2,…}]

gives the multidimensional Fourier series.

Details and Options

  • The ^(th)-order Fourier series of is by default defined to be with .
  • The multidimensional Fourier series of is given by with .
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    FourierParameters {1,1}parameters to define Fourier series
    GenerateConditionsFalsewhether to generate results that involve conditions on parameters
  • Common settings for FourierParameters include:
  • {1,1}
    {1,–2Pi}
    {a,b}

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Find the 3^(rd)-order Fourier series of :

In[1]:=1
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https://wolfram.com/xid/0rsxv7kte-jdooow
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0rsxv7kte-re7s1u
Out[2]=2

Compute an order {2,2} Fourier series:

In[1]:=1
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https://wolfram.com/xid/0rsxv7kte-d0xjh6
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0rsxv7kte-dkce36
Out[2]=2

Scope  (4)Survey of the scope of standard use cases

Find the 3^(rd)-order Fourier series of an exponential function:

In[1]:=1
✖
https://wolfram.com/xid/0rsxv7kte-qg38w
Out[1]=1

Fourier series for a Gaussian function:

In[1]:=1
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https://wolfram.com/xid/0rsxv7kte-ww41v
Out[1]=1

Fourier series for Abs:

In[1]:=1
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https://wolfram.com/xid/0rsxv7kte-bnxxkz
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0rsxv7kte-dknzck
Out[2]=2

Fourier series for a basis function has only one term:

In[1]:=1
✖
https://wolfram.com/xid/0rsxv7kte-c3urt2
Out[1]=1

Options  (1)Common values & functionality for each option

FourierParameters  (1)

Use a nondefault setting for FourierParameters:

In[1]:=1
✖
https://wolfram.com/xid/0rsxv7kte-h4tsuu
Out[1]=1
Wolfram Research (2008), FourierSeries, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSeries.html.
✖
Wolfram Research (2008), FourierSeries, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSeries.html.

Text

Wolfram Research (2008), FourierSeries, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSeries.html.

✖
Wolfram Research (2008), FourierSeries, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSeries.html.

CMS

Wolfram Language. 2008. "FourierSeries." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FourierSeries.html.

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Wolfram Language. 2008. "FourierSeries." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FourierSeries.html.

APA

Wolfram Language. (2008). FourierSeries. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierSeries.html

✖
Wolfram Language. (2008). FourierSeries. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierSeries.html

BibTeX

@misc{reference.wolfram_2025_fourierseries, author="Wolfram Research", title="{FourierSeries}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FourierSeries.html}", note=[Accessed: 25-May-2025 ]}

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@misc{reference.wolfram_2025_fourierseries, author="Wolfram Research", title="{FourierSeries}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FourierSeries.html}", note=[Accessed: 25-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_fourierseries, organization={Wolfram Research}, title={FourierSeries}, year={2008}, url={https://reference.wolfram.com/language/ref/FourierSeries.html}, note=[Accessed: 25-May-2025 ]}

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@online{reference.wolfram_2025_fourierseries, organization={Wolfram Research}, title={FourierSeries}, year={2008}, url={https://reference.wolfram.com/language/ref/FourierSeries.html}, note=[Accessed: 25-May-2025 ]}