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Question Number 17625 by tawa tawa last updated on 08/Jul/17

$$\mathrm{Find}\mathrm{the}\mathrm{fourier}\mathrm{series}\mathrm{of}:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x},\mathrm{from}0<\mathrm{x}<\pi$$

Commented by tawa tawa last updated on 08/Jul/17

$$\mathrm{please}\mathrm{help}\mathrm{with}\mathrm{this}.$$

Answered by alex041103 last updated on 09/Jul/17

$$\frac{\pi }{2}-\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{sin\left(2nx\right)}{n}=f\left(x\right)=xwhenx\in (0,\pi )$$

Commented by tawa tawa last updated on 09/Jul/17

$$\mathrm{i}\mathrm{have}\mathrm{a}\mathrm{question}\mathrm{sir}?$$$$\mathrm{Is}\mathrm{the}\mathrm{period}\pi \mathrm{or}2\pi ?$$$$\mathrm{how}\mathrm{can}\mathrm{i}\mathrm{identify}\mathrm{if}\mathrm{it}\mathrm{is}\pi \mathrm{or}2\pi ?$$$$\mathrm{if}\mathrm{the}\mathrm{period}\mathrm{is}\pi ...\mathrm{how}\mathrm{can}\mathrm{i}\mathrm{reperesent}{\mathrm{a}}_{0,}{\mathrm{a}}_{\mathrm{n}},\mathrm{and}{\mathrm{b}}_{\mathrm{n}}$$$$\mathrm{if}\mathrm{it}\mathrm{the}\mathrm{period}\mathrm{is}2\pi ...\mathrm{how}\mathrm{can}\mathrm{i}\mathrm{reperesent}{\mathrm{a}}_{0,}{\mathrm{a}}_{\mathrm{n}},\mathrm{and}{\mathrm{b}}_{\mathrm{n}}$$

Commented by alex041103 last updated on 09/Jul/17

$$$$$${It}^{\prime }seasytoprovethatif$$$$f\left(x\right)=c+\underset{n=1}{\sum ^{\mathrm{\infty }}}[a_{n}cos\left(\frac{2\pi nx}{T}\right)+b_{n}sin\left(\frac{2\pi nx}{T}\right)]$$$$then$$$$c=\frac{1}{T}\underset{0}{\stackrel{T}{\int }}f\left(x\right)dx$$$$a_n=\frac{2}{T}\underset{0}{\stackrel{T}{\int }}f\left(x\right)cos\left(\frac{2\pi nx}{T}\right)dx$$$$b_b=\frac{2}{T}\underset{0}{\stackrel{T}{\int }}f\left(x\right)sin\left(\frac{2\pi nx}{T}\right)dx$$$$$$$$Inthecaseoff\left(x\right)=xwewillsettheperiod$$$$T=\pi$$$$\Rightarrow c=\frac{\pi }{2},a_n=0,b_n=-\frac{1}{n}$$$$\Rightarrow x\in (0,\pi ),f\left(x\right)=\frac{\pi }{2}-\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{sin\left(2nx\right)}{n}$$

Commented by tawa tawa last updated on 09/Jul/17

$$\mathrm{That}\mathrm{means}\mathrm{for}\mathrm{period}2\pi$$$${\mathrm{a}}_0=\frac{1}{\pi }{\int }_0^{2\pi }\mathrm{f}\left(\mathrm{x}\right)$$$${\mathrm{a}}_{\mathrm{n}}=\frac{1}{\pi }{\int }_0^{2\pi }\mathrm{f}\left(\mathrm{x}\right)\cos \left(\mathrm{nx}\right)$$$${\mathrm{b}}_{\mathrm{n}}=\frac{1}{\pi }{\int }_0^{2\pi }\mathrm{f}\left(\mathrm{x}\right)\sin \left(\mathrm{nx}\right)$$$$\mathrm{Again},$$$$\mathrm{That}\mathrm{means}\mathrm{for}\mathrm{period}\pi$$$${\mathrm{a}}_0=\frac{2}{\pi }{\int }_0^{\pi }\mathrm{f}\left(\mathrm{x}\right)$$$${\mathrm{a}}_{\mathrm{n}}=\frac{2}{\pi }{\int }_0^{\pi }\mathrm{f}\left(\mathrm{x}\right)\cos \left(\mathrm{nx}\right)$$$${\mathrm{b}}_{\mathrm{n}}=\frac{2}{\pi }{\int }_0^{\pi }\mathrm{f}\left(\mathrm{x}\right)\sin \left(\mathrm{nx}\right)$$$$??????????????$$

Commented by alex041103 last updated on 09/Jul/17

$$No.$$$$Forperiodof\pi wehave:$$$${\mathrm{b}}_{\mathrm{n}}=\frac{2}{\pi }{\int }_0^{\pi }\mathrm{f}\left(\mathrm{x}\right)\sin \left(\frac{2\pi }{\pi }\mathrm{nx}\right)dx=\frac{2}{\pi }\underset{0}{\stackrel{\pi }{\int }}f\left(x\right)sin\left(2nx\right)dx$$$${\mathrm{a}}_{\mathrm{n}}=\frac{2}{\pi }{\int }_0^{\pi }\mathrm{f}\left(\mathrm{x}\right)\cos \left(\frac{2\pi }{\pi }\mathrm{nx}\right)dx=\frac{2}{\pi }\underset{0}{\stackrel{\pi }{\int }}f\left(x\right)cos\left(2nx\right)dx$$$${\mathrm{a}}_0=\frac{2}{\pi }{\int }_0^{\pi }\mathrm{f}\left(\mathrm{x}\right)dx$$$$AndinfactifweuseT=2\pi \left(theperiodis2\pi \right)$$$$wearegoingtogetfourierseriesfor$$$$f\left(x\right)=xforx\in [0,2\pi ]$$

Commented by tawa tawa last updated on 09/Jul/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Commented by alex041103 last updated on 10/Jul/17

Commented by alex041103 last updated on 10/Jul/17

$$\mathrm{Blue}\to \mathrm{T}=2\pi ,\mathrm{i}.\mathrm{e}.\mathrm{for}0<\mathrm{x}<2\pi$$$$\mathrm{x}=\pi -2\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\frac{\sin \left(\mathrm{nx}\right)}{\mathrm{n}}$$$$\mathrm{Red}\to \mathrm{T}=\pi ,\mathrm{i}.\mathrm{e}.\mathrm{for}0<\mathrm{x}<\pi$$$$\mathrm{x}=\frac{\pi }{2}-\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\frac{\sin \left(2\mathrm{nx}\right)}{\mathrm{n}}$$

Commented by tawa tawa last updated on 10/Jul/17

$$\mathrm{i}\mathrm{really}\mathrm{appreciate}\mathrm{your}\mathrm{effort}\mathrm{sir}.\mathrm{God}\mathrm{bless}\mathrm{you}.$$

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