FourierCosCoefficient
FourierCosCoefficient[expr,t,n]
gives the n coefficient in the Fourier cosine series expansion of expr, where expr is a periodic function of t with period 1.
Details and Options
- To use FourierCosCoefficient, you first need to load the Fourier Series Package using Needs["FourierSeries`"].
- The n
coefficient in the Fourier cosine series expansion of expr is by default defined to be 2Integrate[expr Cos[2π n t],{t,-
,
}] for n>0 and Integrate[expr,{t,-
,
}] for n==0.
- If n is numeric, it should be an explicit integer.
- Different choices for the definition of the Fourier cosine series expansion can be specified using the option FourierParameters.
- With the setting FourierParameters->{a,b}, expr is assumed to have a period of
, and the n
coefficient computed by FourierCosCoefficient is 2
Integrate[expr Cos[2π b n t],{t,-
,
}] for n>0 and
Integrate[expr,{t,-
,
}] for n==0.
- In addition to the option FourierParameters, FourierCosCoefficient can also accept the options available to Integrate. These options are passed directly to Integrate.
Examples
Basic Examples (1)Summary of the most common use cases

https://wolfram.com/xid/0bzj3rpxbx11p9bwapai6-kihn4a
Use different definitions for calculating a coefficient in a Fourier cosine series:

https://wolfram.com/xid/0bzj3rpxbx11p9bwapai6-f4lpax


https://wolfram.com/xid/0bzj3rpxbx11p9bwapai6-c1de4y

Compare with the answer from a numerical approximation:

https://wolfram.com/xid/0bzj3rpxbx11p9bwapai6-m9obl


https://wolfram.com/xid/0bzj3rpxbx11p9bwapai6-ww3t6


https://wolfram.com/xid/0bzj3rpxbx11p9bwapai6-v3yqm

Wolfram Research (2008), FourierCosCoefficient, Wolfram Language function, https://reference.wolfram.com/language/FourierSeries/ref/FourierCosCoefficient.html.
Text
Wolfram Research (2008), FourierCosCoefficient, Wolfram Language function, https://reference.wolfram.com/language/FourierSeries/ref/FourierCosCoefficient.html.
Wolfram Research (2008), FourierCosCoefficient, Wolfram Language function, https://reference.wolfram.com/language/FourierSeries/ref/FourierCosCoefficient.html.
CMS
Wolfram Language. 2008. "FourierCosCoefficient." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/FourierSeries/ref/FourierCosCoefficient.html.
Wolfram Language. 2008. "FourierCosCoefficient." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/FourierSeries/ref/FourierCosCoefficient.html.
APA
Wolfram Language. (2008). FourierCosCoefficient. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/FourierSeries/ref/FourierCosCoefficient.html
Wolfram Language. (2008). FourierCosCoefficient. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/FourierSeries/ref/FourierCosCoefficient.html
BibTeX
@misc{reference.wolfram_2025_fouriercoscoefficient, author="Wolfram Research", title="{FourierCosCoefficient}", year="2008", howpublished="\url{https://reference.wolfram.com/language/FourierSeries/ref/FourierCosCoefficient.html}", note=[Accessed: 11-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_fouriercoscoefficient, organization={Wolfram Research}, title={FourierCosCoefficient}, year={2008}, url={https://reference.wolfram.com/language/FourierSeries/ref/FourierCosCoefficient.html}, note=[Accessed: 11-July-2025
]}