Fourier Series for Even Function over Symmetric Range

Theorem

Let $\map f x$ be an even real function defined on the interval $\openint {-\lambda} \lambda$.

Then the Fourier series of $\map f x$ can be expressed as:

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

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

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

Proof

By definition of the Fourier series for $f$:

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

From Fourier Cosine Coefficients for Even Function over Symmetric Range:

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

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

From Fourier Sine Coefficients for Even Function over Symmetric Range

$b_n = 0$

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

$\blacksquare$

Also see

• Fourier Series for Odd Function over Symmetric Range

Sources

• 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 4$. Even and Odd Functions: $(6)$