Definite Integral of Fourier Series at Ends of Interval

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Theorem

Let $f: \R \to \R$ be a real function defined in the open interval $\openint {-\pi} \pi$.

Let $f$ fulfil the Dirichlet conditions in $\openint {-\pi} \pi$.

Let $a_0, a_1, \dotsc; b_1, \dotsc$ be the Fourier coefficients of $f$ in $\openint {-\pi} \pi$.

Consider the real function:

$\map F x = \ds \int_{-\pi}^x \map f t \rd t - \dfrac {a_0} 2 x$


Then:

$\map F \pi = \map F {-\pi} = \dfrac {a_0 \pi} 2$


Proof

From Definite Integral on Zero Interval:

\(\ds \map F {-\pi}\) \(=\) \(\ds \int_{-\pi}^{-\pi} \map f t \rd t - \dfrac {a_0} 2 \paren {-\pi}\)
\(\ds \) \(=\) \(\ds 0 - \dfrac {a_0} 2 \paren {-\pi}\)
\(\ds \) \(=\) \(\ds \dfrac {a_0 \pi} 2\)


Then:

\(\ds \map F \pi\) \(=\) \(\ds \int_{-\pi}^\pi \map f t \rd t - \dfrac {a_0} 2 \pi\)
\(\ds \) \(=\) \(\ds \int_{-\pi}^\pi \paren {\dfrac {\map {f^+} t + \map {f^-} t} 2} \rd t - \dfrac {a_0} 2 \pi\) Fourier's Theorem



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Sources

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