# Definition:Half-Range Fourier Cosine Series

## Definition

### Formulation 1

Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$.

Then the half-range Fourier cosine series of $\map f x$ over $\openint 0 \lambda$ is the series:

$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$

where for all $n \in \Z_{\ge 0}$:

$a_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \cos \frac {n \pi x} \lambda \rd x$

### Formulation 2

Let $\map f x$ be a real function defined on the interval $\openint a b$.

Then the half-range Fourier cosine series of $\map f x$ over $\openint a b$ is the series:

$\displaystyle \map f x \sim \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty A_m \cos \frac {m \pi \paren {x - a} } {b - a}$

where for all $m \in \Z_{\ge 0}$:

$A_m = \displaystyle \frac 2 {b - a} \int_a^b \map f x \cos \frac {m \pi \paren {x - a} } {b - a} \rd x$

## Half-Range Fourier Cosine Series on Range of $\pi$

Let $\map f x$ be a real function defined on the interval $\openint 0 \pi$.

Then the half-range Fourier cosine series of $\map f x$ over $\openint 0 \pi$ is the series:

$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty a_n \cos n x$

where for all $n \in \Z_{\ge 0}$:

$a_n = \displaystyle \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$