Derivative's Fourier Coefficients
📂Fourier AnalysisDerivative's Fourier Coefficients
Given that a function f defined on the interval [−L, L) is continuous and piecewise smooth, then the Fourier coefficients of f′ are as follows.
an′=Lnπbn
bn′=−Lnπan
cn′=Linπcn
Here, an, bn are the Fourier coefficients of f, and cn are the complex Fourier coefficients of f.
Proof
cn′=2L1∫−LLf′(t)e−iLnπtdt=2L1[f(t)e−iLnπt]−LL+Linπ2L1∫−LLf(t)e−iLnπtdt=2L1f(t)[e−inπ−einπ]+Linπcn=2L1f(t)[(−1)−n−(−1)n]+Linπcn=2L1f(t)(−1)n[(−1)−2n−1]+Linπcn=Linπcn
The second equality is due to the integration by parts.
an′=L1∫−LLf′(t)cosLnπtdt=L1[f(t)cosLnπt]−LL+LnπL1∫−LLf(t)sinLnπtdt=L1f(t)(cosnπ−cosnπ)+Lnπbn=Lnπbn
The second equality is due to the integration by parts.
bn′=L1∫−LLf′(t)sinLnπtdt=L1[f(t)sinLnπt]−LL−LnπL1∫−LLf(t)cosLnπtdt=L1f(t)(sinnπ+sinnπ)−Lnπan=−Lnπan
The second equality is due to the integration by parts.
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