Half-Range Fourier Cosine Series over Negative Range

Theorem

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

Let $f$ be expressed using the half-range Fourier cosine series over $\openint 0 \lambda$:

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

where:

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

for all $n \in \Z_{\ge 0}$.

Then over the closed real interval $\openint {-\lambda} 0$, $\map C x$ takes the values:

$\map C x = \map f {-x}$

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That is, the real function expressed by the half-range Fourier cosine series over $\openint 0 \lambda$ is an even function over $\openint {-\lambda} \lambda$.

Proof

From Fourier Series for Even Function over Symmetric Range, $\map C x$ is the Fourier series of an even real function over the interval $\openint 0 \lambda$.

We have that $\map C x \sim \map f x$ over $\openint 0 \lambda$.

Thus over $\openint {-\lambda} 0$ it follows that $\map C x = \map f {-x}$.

$\blacksquare$

Sources

• 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 6$. Half-Range Cosine Series