In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier.
In this article, f denotes a real-valued function on
which is periodic with period 2L.
If f is an odd function with period
, then the Fourier Half Range sine series of f is defined to be
sin
which is just a form of complete Fourier series with the only difference that
are zero, and the series is defined for half of the interval.
In the formula we have
f ( x ) sin
{\displaystyle b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {n\pi x}{L}}\right)\,dx,\quad n\in \mathbb {N} .}
If f is an even function with a period
, then the Fourier cosine series is defined to be
{\displaystyle a_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\cos \left({\frac {n\pi x}{L}}\right)\,dx,\quad n\in \mathbb {N} _{0}.}
This notion can be generalized to functions which are not even or odd, but then the above formulas will look different.