Half-range series

If $f(x)$ is piecewise continuous on an interval $[-L, L]$, we can represent the function at all points in $x$ on the interval by its Fourier series expansion, except for possibly finitely many points (i.e. at jump discontinuities). The series may contain only sin terms or only cos terms or both. This depends on the function itself and there is no control over it. Now, if the function is defined on a half interval, say $[0, L]$, it is possible to write a half-range Fourier series. This can be a cosine series (with only cos terms) or a sine series (with only sin terms) for $f(x)$ on $[0, L]$.

Suppose we have a function $f(x)$ defined on $[0, L]$ and we reflect it along the vertical axis. The result is an even function in $[-L, L]$. Since the function is even, all odd coefficients $b_{n}$ are automatically zero.

Definition $\mathbf{1 . 5}$ The Fourier cosine series of $f(x)$ on $[0, L]$ is

with $a_{n}$ as:

where $a_{n}$ are the Fourier cosine coefficients of $f(x)$ on $[0, L]$.

Similarly, we can define a Fourier series representation of a function $f(x)$ on $[0, L]$ containing only sin terms. If we consider a function $f(x)$ defined on $[0, L]$ and reflect it over the origin, we obtain the odd extension of $f(x)$ in $[-L, L]$. Now, since the function is odd, all even coefficients $a_{n}$ are automatically zero.

Definition 1.6 The Fourier sine series of $f(x)$ on $[0, L]$ is

with $b_{n}$ as:

where $b_{n}$ are the Fourier sine coefficients of $f(x)$ on $[0, L]$. Note that the theorem of convergence as applied to the Fourier series may be directly applied to the half-range Fourier sine or cosine series.