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:


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

where for all $n \in \Z_{\ge 0}$:
 * $a_n = \dfrac 2 \lambda \displaystyle \int_0^\lambda \map f x \cos \frac {n \pi x} \lambda \rd x$

Proof
By definition of the Fourier series for $f$:


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

From Fourier Cosine Coefficients for Even Function over Symmetric Range:


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

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

From Fourier Sine Coefficients for Even Function over Symmetric Range


 * $b_n = 0$

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

Also see

 * Fourier Series for Odd Function over Symmetric Range