Definition:Half-Range Fourier Cosine Series/Formulation 2

Definition

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:

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

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

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

Also known as

Some sources give the half-range Fourier series as Fourier's half-range series.

Some sources give them as just the half-range series.

Also see

• Fourier Series for Even Function over Symmetric Range, which justifies the definition

• Definition:Half-Range Fourier Cosine Series over Range Pi

• Definition:Half-Range Fourier Sine Series

• Definition:Half-Range Fourier Series
• Definition:Half-Range Fourier Series over Range Pi

Sources

• 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 7$. Fourier Series over a General Range $\tuple {a, b}$: $(6)$