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Question Number 100966 by mhmd last updated on 29/Jun/20

find the fourier series of the function f(x)= { ((x −2≤x≤0)),((x+2 0≤x≤2)) :} help me sir ?

$${find}\:{the}\:{fourier}\:{series}\:{of}\:{the}\:{function} \\ $$$${f}\left({x}\right)=\begin{cases}{{x}\:\:\:\:\:\:\:\:\:−\mathrm{2}\leqslant{x}\leqslant\mathrm{0}}\\{{x}+\mathrm{2}\:\:\:\:\:\:\:\:\mathrm{0}\leqslant{x}\leqslant\mathrm{2}}\end{cases}\:\:\:\:\:\:{help}\:{me}\:{sir}\:? \\ $$

Commented by bobhans last updated on 29/Jun/20

f(x) = (a_o /2) + Σ_(n=1) ^∞ b_n cos (((nπx)/L)) b_n =(2/4) [ ∫_(−2) ^0 x sin (((nπx)/4)) dx + ∫_0 ^2 (x+2)sin (((nπx)/4)) dx ] b_n = (1/2)[ ∫_(−2) ^2 x sin (((nπx)/4)) dx + ∫_0 ^2 2 sin (((nπx)/4)) dx ] b_n = (1/2)[ −((4x)/(nπ)) cos (((nπx)/4))+((16)/((nπ)^2 )) sin (((nπx)/4)) ]_(−2) ^2 − (1/2) [ (8/(nπ)) cos (((nπx)/4)) ]_0 ^2

$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\frac{\mathrm{a}_{\mathrm{o}} }{\mathrm{2}}\:+\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\mathrm{b}_{\mathrm{n}} \mathrm{cos}\:\left(\frac{\mathrm{n}\pi\mathrm{x}}{\mathrm{L}}\right) \\ $$$$\mathrm{b}_{\mathrm{n}} =\frac{\mathrm{2}}{\mathrm{4}}\:\left[\:\underset{−\mathrm{2}} {\overset{\mathrm{0}} {\int}}\mathrm{x}\:\mathrm{sin}\:\left(\frac{\mathrm{n}\pi\mathrm{x}}{\mathrm{4}}\right)\:\mathrm{dx}\:+\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\left(\mathrm{x}+\mathrm{2}\right)\mathrm{sin}\:\left(\frac{\mathrm{n}\pi\mathrm{x}}{\mathrm{4}}\right)\:\mathrm{dx}\:\right] \\ $$$$\mathrm{b}_{\mathrm{n}} =\:\frac{\mathrm{1}}{\mathrm{2}}\left[\:\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\mathrm{x}\:\mathrm{sin}\:\left(\frac{\mathrm{n}\pi\mathrm{x}}{\mathrm{4}}\right)\:\mathrm{dx}\:+\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\mathrm{2}\:\mathrm{sin}\:\left(\frac{\mathrm{n}\pi\mathrm{x}}{\mathrm{4}}\right)\:\mathrm{dx}\:\right] \\ $$$$\mathrm{b}_{\mathrm{n}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}\left[\:−\frac{\mathrm{4x}}{\mathrm{n}\pi}\:\mathrm{cos}\:\left(\frac{\mathrm{n}\pi\mathrm{x}}{\mathrm{4}}\right)+\frac{\mathrm{16}}{\left(\mathrm{n}\pi\right)^{\mathrm{2}} }\:\mathrm{sin}\:\left(\frac{\mathrm{n}\pi\mathrm{x}}{\mathrm{4}}\right)\:\right]_{−\mathrm{2}} ^{\mathrm{2}} \\ $$$$−\:\frac{\mathrm{1}}{\mathrm{2}}\:\left[\:\frac{\mathrm{8}}{\mathrm{n}\pi}\:\mathrm{cos}\:\left(\frac{\mathrm{n}\pi\mathrm{x}}{\mathrm{4}}\right)\:\right]_{\mathrm{0}} ^{\mathrm{2}} \: \\ $$$$ \\ $$

Commented by mhmd last updated on 29/Jun/20

how are you sir ? am dont anderstand how [bn cos(((nπx)/L))] because the fourier series is given by f(x)=((ao)/2)+some (n≥1)[an cos(((nπx)/L))+bn sin(((nπx)/l))] how bn cos(((nπx)/L)) and where an cos(((nπx)/L)) blease sir can you exact this blease im very want this equestion ?

$${how}\:{are}\:{you}\:{sir}\:?\:{am}\:{dont}\:{anderstand}\:{how}\:\left[{bn}\:{cos}\left(\frac{{n}\pi{x}}{{L}}\right)\right]\: \\ $$$${because}\:{the}\:{fourier}\:{series}\:{is}\:{given}\:{by}\: \\ $$$${f}\left({x}\right)=\frac{{ao}}{\mathrm{2}}+{some}\:\left({n}\geqslant\mathrm{1}\right)\left[{an}\:{cos}\left(\frac{{n}\pi{x}}{{L}}\right)+{bn}\:{sin}\left(\frac{{n}\pi{x}}{{l}}\right)\right] \\ $$$${how}\:{bn}\:{cos}\left(\frac{{n}\pi{x}}{{L}}\right)\:{and}\:{where}\:{an}\:{cos}\left(\frac{{n}\pi{x}}{{L}}\right)\:{blease}\:{sir}\:{can}\:{you}\:{exact}\:{this}\:{blease} \\ $$$${im}\:{very}\:{want}\:{this}\:{equestion}\:? \\ $$

Commented by mathmax by abdo last updated on 29/Jun/20

sir bobhans you have supposed that f is even and this is not given in the question you must use f(x) =(a_o /2) +Σ_(n=1) ^∞ (a_n cos(nwx) +b_n sin(nwx)) a_n =(1/T)∫_(−(T/2)) ^(T/2) f(x)cos(nx)dx and b_n =(1/T)∫_(−(T/2)) ^(T/2) f(x)sin(nx)dx (w =((2π)/T))

$$\mathrm{sir}\:\mathrm{bobhans}\:\mathrm{you}\:\mathrm{have}\:\mathrm{supposed}\:\mathrm{that}\:\mathrm{f}\:\mathrm{is}\:\mathrm{even}\:\mathrm{and}\:\mathrm{this}\:\mathrm{is}\:\mathrm{not}\:\mathrm{given}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{question}\:\mathrm{you}\:\mathrm{must}\:\mathrm{use}\:\mathrm{f}\left(\mathrm{x}\right)\:=\frac{\mathrm{a}_{\mathrm{o}} }{\mathrm{2}}\:+\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\left(\mathrm{a}_{\mathrm{n}} \mathrm{cos}\left(\mathrm{nwx}\right)\:+\mathrm{b}_{\mathrm{n}} \mathrm{sin}\left(\mathrm{nwx}\right)\right) \\ $$$$\mathrm{a}_{\mathrm{n}} =\frac{\mathrm{1}}{\mathrm{T}}\int_{−\frac{\mathrm{T}}{\mathrm{2}}} ^{\frac{\mathrm{T}}{\mathrm{2}}} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{cos}\left(\mathrm{nx}\right)\mathrm{dx}\:\mathrm{and}\:\mathrm{b}_{\mathrm{n}} =\frac{\mathrm{1}}{\mathrm{T}}\int_{−\frac{\mathrm{T}}{\mathrm{2}}} ^{\frac{\mathrm{T}}{\mathrm{2}}} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{sin}\left(\mathrm{nx}\right)\mathrm{dx}\:\:\:\left(\mathrm{w}\:=\frac{\mathrm{2}\pi}{\mathrm{T}}\right) \\ $$

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