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Fourier Series - Math

Fourier Series Fourier Sine Series Fourier Cosine Series Fourier Series Convergence of Fourier Series for2T-Periodic Functions Convergence of Half-Range Expansions: Cosine Series Convergence of Half-Range Expansions: Sine Series Sawtooth Wave Triangular Wave Parseval s Identity and Bessel s Inequality Complex Fourier Series Dirichlet Kernel and ConvergenceFourier Sine SeriesDefinition. Consider the orthogonal system{sin(n xT)} n=1on[ T,T]. A Fouriersine Series with coefficients{bn} n=1is the expressionF(x) = n=1bnsin(n xT)Theorem. A Fourier sine seriesF(x)is an odd2T-periodic The coefficients{bn} n=1in a Fourier sine seriesF(x)are determined by theformulas (inner product on[ T,T])bn= F,sin(n xT) sin(n xT),sin(n xT) =2T T0F(x) sin(n xT) Cosine SeriesDefinition. Consider the orthogonal system{cos(m xT)} m=0on[ T,T]. A Fouriercosine Series with coefficients{am} m=0is the expressionF(x) = m=0amcos(m xT)Theorem. A Fourier cosine seriesF(x)is an even2T-periodic The coefficients{am} m=0in a Fourier cosine seriesF(x)are determined bythe formulas (inner product on[ T,T])am= F,cos(m xT) cos(m xT),cos(m xT) = 2T T0F(x) cos(m xT)dx m >0,1T T0F(x)dxm= SeriesDefinition.

Fourier Sine Series Definition. Consider the orthogonal system fsin nˇx T g1 n=1 on [ T;T].A Fourier sine series with coefficients fb ng1 n=1 is the expression F(x) = X1 n=1 b nsin nˇx T Theorem. A Fourier sine series F(x) is an odd 2T-periodic function. Theorem.

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