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Question Number 144699 by mathmax by abdo last updated on 28/Jun/21
let f(x)=log(cht) developp f at fourier serie
$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{log}\left(\mathrm{cht}\right) \\ $$$$\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{fourier}\:\mathrm{serie} \\ $$
Answered by Olaf_Thorendsen last updated on 28/Jun/21
f is not periodic !
$${f}\:\mathrm{is}\:\mathrm{not}\:\mathrm{periodic}\:! \\ $$
Answered by mathmax by abdo last updated on 28/Jun/21
sorry developp f at integr serie
$$\mathrm{sorry}\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$
Answered by mathmax by abdo last updated on 29/Jun/21
f(x)=log(cht)⇒f^′ (x)=((sht)/(cht))=((e^t −e^(−t) )/(e^t +e^t )) =_(e^t =z) ((z−z^(−1) )/(z+z^(−1) ))=((z^2 −1)/(z^2 +1))=(z^2 −1)Σ_(n=0) ^∞ (−1)^n z^(2n) =Σ_(n=0) ^∞ (−1)^n z^(2n+2) −Σ_(n=0) ^∞ (−1)^n z^(2n) =Σ_(n=0) ^∞ (−1)^n e^((2n+2)t) −Σ_(n=0) ^∞ (−1)^n e^(2nt) =Σ_(n=0) ^∞ (−1)^n Σ_(p=0) ^∞ (({(2n+2)t}^p )/(p!))−Σ_(n=0) ^∞ (−1)^n Σ_(p=0) ^∞ (((2nt)^p )/(p!)) =Σ_(n=0) ^∞ (−1)^n Σ_(p=0) ^∞ (((2n+2)^p )/(p!))t^p −Σ_(n=0) ^∞ (−1)^n Σ_(p=0) ^∞ (((2n)^p )/(p!))t^p
$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{log}\left(\mathrm{cht}\right)\Rightarrow\mathrm{f}^{'} \left(\mathrm{x}\right)=\frac{\mathrm{sht}}{\mathrm{cht}}=\frac{\mathrm{e}^{\mathrm{t}} −\mathrm{e}^{−\mathrm{t}} }{\mathrm{e}^{\mathrm{t}} \:+\mathrm{e}^{\mathrm{t}} } \\ $$$$=_{\mathrm{e}^{\mathrm{t}} \:=\mathrm{z}} \:\:\frac{\mathrm{z}−\mathrm{z}^{−\mathrm{1}} }{\mathrm{z}+\mathrm{z}^{−\mathrm{1}} }=\frac{\mathrm{z}^{\mathrm{2}} −\mathrm{1}}{\mathrm{z}^{\mathrm{2}} \:+\mathrm{1}}=\left(\mathrm{z}^{\mathrm{2}} −\mathrm{1}\right)\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{z}^{\mathrm{2n}} \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{z}^{\mathrm{2n}+\mathrm{2}} −\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{z}^{\mathrm{2n}} \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{e}^{\left(\mathrm{2n}+\mathrm{2}\right)\mathrm{t}} \:−\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\left(−\mathrm{1}\right)^{\mathrm{n}} \:\mathrm{e}^{\mathrm{2nt}} \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \sum_{\mathrm{p}=\mathrm{0}} ^{\infty} \:\frac{\left\{\left(\mathrm{2n}+\mathrm{2}\right)\mathrm{t}\right\}^{\mathrm{p}} }{\mathrm{p}!}−\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\left(−\mathrm{1}\right)^{\mathrm{n}} \:\sum_{\mathrm{p}=\mathrm{0}} ^{\infty} \:\frac{\left(\mathrm{2nt}\right)^{\mathrm{p}} }{\mathrm{p}!} \\ $$$$=\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \:\sum_{\mathrm{p}=\mathrm{0}} ^{\infty} \:\frac{\left(\mathrm{2n}+\mathrm{2}\right)^{\mathrm{p}} }{\mathrm{p}!}\mathrm{t}^{\mathrm{p}} \:−\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{\mathrm{n}} \:\sum_{\mathrm{p}=\mathrm{0}} ^{\infty} \:\frac{\left(\mathrm{2n}\right)^{\mathrm{p}} }{\mathrm{p}!}\mathrm{t}^{\mathrm{p}} \\ $$

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