Fourier Sine Coefficients for Even Function over Symmetric Range
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Theorem
Let $\map f x$ be an even real function defined on the interval $\openint {-\lambda} \lambda$.
Let the Fourier series of $\map f x$ be expressed as:
- $\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
Then for all $n \in \Z_{> 0}$:
- $b_n = 0$
Proof
As suggested, let the Fourier series of $\map f x$ be expressed as:
- $\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
By definition of Fourier series:
\(\ds b_n\) | \(=\) | \(\ds \frac 1 \lambda \int_{-\lambda}^{-\lambda + 2 \lambda} \map f x \sin \frac {n \pi x} \lambda \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 \lambda \int_{-\lambda}^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x\) |
From Sine Function is Odd:
- $\sin a = -\map \sin {-a}$
for all $a$.
By Odd Function Times Even Function is Odd, $\map f x \sin \dfrac {n \pi x} l$ is odd.
The result follows from the corollary to Definite Integral of Odd Function:
- $\ds \frac 1 \lambda \int_{-\lambda}^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x = 0$
$\blacksquare$
Also see
- Fourier Sine Coefficients for Odd Function over Symmetric Range
- Fourier Cosine Coefficients for Odd Function over Symmetric Range
Sources
- 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter One: $\S 4$. Even and Odd Functions