Definition:Half-Range Fourier Cosine Series/Formulation 1

Definition

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:

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

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

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

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: $\S 6$. Half-Range Cosine Series: $(2)$
• 1968: Peter D. Robinson: Fourier and Laplace Transforms ... (previous) ... (next): $\S 1.3$. Fourier Series and Finite Fourier Transforms