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Question Number 145748 by mathmax by abdo last updated on 07/Jul/21

f(x)=arctan(2sinx)  developp f at fourier serie

f(x)=arctan(2sinx)developpfatfourierserie

Answered by mathmax by abdo last updated on 09/Jul/21

we have f^′ (x)=((2cosx)/(1+4sin^2 x))=((2cosx)/(1+4.((1−cos(2x))/2)))=((2cosx)/(1+2−2cos(2x)))  =((2cosx)/(3−2cos(2x))) =((2((e^(ix) +e^(−ix) )/2))/(3−2((e^(2ix) +e^(−2ix) )/2)))=((e^(ix)  +e^(−ix) )/(3−e^(2ix) −e^(−2ix) ))  =_(e^(ix)  =z)    ((z+z^(−1) )/(3−z^2 −z^(−2) ))=((z^2 (z+z^(−1) ))/(3z^2 −z^4  −1))=((z^3  +z)/(−z^4  +3z^2 −1))=−((z^3  +z)/(z^4 −3z^2  +1))  z^4 −3z^2  +1=0 ⇒u^2 −3u+1=0  (u=z^2 )  Δ=9−4=5 ⇒u_1 =((3+(√5))/2) and u_2 =((3−(√5))/2) ⇒  f^′ (x)=−((z^3  +z)/((z^2 −u_1 )(z^2 −u_2 )))=(z^3  +z)((1/(z^2 −u_1 ))−(1/(z^2 −u_2 )))×(1/( (√5)))  =(1/( (√5)))(((z(z^2 −u_1 )+u_1 z+z)/(z^2 −u_1 ))−((z(z^2 −u_2 )+u_2 z+z)/(z^2 −u_2 )))  =(1/( (√5))){(((1+u_1 )z)/(z^2 −u_1 ))−(((1+u_2 )z)/(z^2 −u_2 ))}  ∣(u_1 /z^2 )∣−1=((3+(√5))/2)−1>0 ⇒∣(u_1 /z_2 )∣>1  ∣(u_2 /z^2 )∣−1=((3−(√5))/2)−1<0 ⇒∣(u_2 /z^2 )∣<1 ⇒  f^′ (x)=(1/( (√5))){−((1+u_1 )/u_1 )×(z/(1−(z^2 /u_1 )))−((1+u_2 )/z)×(1/(1−(u_2 /z^2 )))}  =−(((1+u_1 )z)/( (√5)u_1 ))Σ_(n=0) ^∞  (z^(2n) /u_1 ^n )−(((1+u_2 ))/(z(√5)))Σ_(n=0) ^∞  (u_2 ^n /z^(2n) )  =−((1+u_1 )/(u_1 (√5)))Σ_(n=0) ^∞  (z^(2n+1) /u_1 ^n )−(((1+u_2 ))/( (√5)))Σ_(n=0) ^∞  (u_2 ^n /z^(2n+1) )  =−((1+u_1 )/(u_1 (√5)))Σ_(n=0) ^∞  u_1 ^(−n) e^(i(2n+1)x)  −((1+u_2 )/( (√5)))Σ_(n=0) ^∞  u_2 ^n  e^(−i(2n+1)x)  ⇒  f(x)=−((1+u_1 )/(u_1 (√(5(i(2n+1))))))Σ_(n=0) ^∞  u_1 ^(−n)  e^(i(2n+1)x)  +((1+u_2 )/( (√5)i(2n+1)))Σ_(n=0) ^∞  u_2 ^(−n)  e^(−i(2n+1)x)  +K  ...be continued....

wehavef(x)=2cosx1+4sin2x=2cosx1+4.1cos(2x)2=2cosx1+22cos(2x)=2cosx32cos(2x)=2eix+eix232e2ix+e2ix2=eix+eix3e2ixe2ix=eix=zz+z13z2z2=z2(z+z1)3z2z41=z3+zz4+3z21=z3+zz43z2+1z43z2+1=0u23u+1=0(u=z2)Δ=94=5u1=3+52andu2=352f(x)=z3+z(z2u1)(z2u2)=(z3+z)(1z2u11z2u2)×15=15(z(z2u1)+u1z+zz2u1z(z2u2)+u2z+zz2u2)=15{(1+u1)zz2u1(1+u2)zz2u2}u1z21=3+521>0⇒∣u1z2∣>1u2z21=3521<0⇒∣u2z2∣<1f(x)=15{1+u1u1×z1z2u11+u2z×11u2z2}=(1+u1)z5u1n=0z2nu1n(1+u2)z5n=0u2nz2n=1+u1u15n=0z2n+1u1n(1+u2)5n=0u2nz2n+1=1+u1u15n=0u1nei(2n+1)x1+u25n=0u2nei(2n+1)xf(x)=1+u1u15(i(2n+1))n=0u1nei(2n+1)x+1+u25i(2n+1)n=0u2nei(2n+1)x+K...becontinued....

Commented by mathmax by abdo last updated on 09/Jul/21

to simplify  calculus we can use this decomposition  f^′ (x)=((2cosx)/(3−2cos(2x)))=((2cosx)/(3−2(2cos^2 x−1)))=((2cosx)/(3−4cos^2 x+2))  =−((2cosx)/(4cos^2 x−5))=−((2cosx)/((2cosx−(√5))(2cosx +(√5))))  =(1/(2(√5)))((1/(2cosx−(√5)))−(1/(2cosx+(√5))))=(1/(2(√5)))(u(x)−v(x))  u(x)=(1/(2cosx−(√5)))=(1/(e^(ix)  +e^(−ix) −(√5)))=_(e^(ix)  =z)    (1/(z+z^(−1)  −(√5)))  =(z/(z^2  +1−(√5)z))=(z/(z^2 −(√5)z +1))  Δ=5−4=1 ⇒z_1 =(((√5)+1)/2) and z_2 =(((√5)−1)/2)  u(x)=(z/((z−z_1 )(z−z_2 )))=z((1/(z−z_1 ))−(1/(z−z_2 )))=(z/(z−z_1 ))−(z/(z−z_2 ))  ∣(z/z_1 )∣−1=(2/( (√5)+1))−1 =((2−(√5)−1)/((...)))<0 ⇒∣(z/z_1 )∣<1  ∣(z/z_2 )∣−1=(2/( (√5)−1))−1 =((2−(√5)+1)/( (√5)−1))=((3−(√5))/((..)))>0 ⇒∣(z/z_2 )∣>1 ⇒  u(x)=−(z/(z_1 (1−(z/z_1 ))))−(1/(1−(z_2 /z)))  =−(z/z_1 )Σ_(n=0) ^∞  (z^n /z_1 ^n )−Σ_(n=0) ^∞  (z_2 ^n /z^n )  z_1 z_2 =1 ⇒z_2 ^n  =(1/z_1 ^n ) ⇒u(x)=−(z/z_1 )Σ_(n=0) ^∞  (z^n /z_1 ^n )−Σ_(n=0) ^∞  (1/(z_1 ^n  z^n ))  =−Σ_(n=0) ^∞  ((((√5)+1)/2))^(−n−1)  e^(i(n+1)x)  −Σ_(n=0) ^∞  ((((√5)+1)/2))^(−n)  e^(−inx)   =−Σ_(n=0) ^∞  ((((√5)+1)/2))^(−n−1) (cos(n+1)x +isin(n+1)x)  −Σ_(n=0) ^∞  ((((√5)+1)/2))^(−n)  (cos(nx)−isin(nx))  u(x)real ⇒u(x)=−Σ_(n=0) ^∞ ((((√5)+1)/2))^(−(n+1))  cos(n+1)x  −Σ_(n=0) ^∞  ((((√5)+1)/2))^(−n)  cos(nx)  =−Σ_(n=1) ^∞  ((((√5)+1)/2))^(−n)  cos(nx)−Σ_(n=0) ^∞  ((((√5)+1)/2))^(−n) cos(nx)  =−1−2Σ_(n=1) ^∞  ((((√5)+1)/2))^(−n ) cos(nx)  ...be continued....

tosimplifycalculuswecanusethisdecompositionf(x)=2cosx32cos(2x)=2cosx32(2cos2x1)=2cosx34cos2x+2=2cosx4cos2x5=2cosx(2cosx5)(2cosx+5)=125(12cosx512cosx+5)=125(u(x)v(x))u(x)=12cosx5=1eix+eix5=eix=z1z+z15=zz2+15z=zz25z+1Δ=54=1z1=5+12andz2=512u(x)=z(zz1)(zz2)=z(1zz11zz2)=zzz1zzz2zz11=25+11=251(...)<0⇒∣zz1∣<1zz21=2511=25+151=35(..)>0⇒∣zz2∣>1u(x)=zz1(1zz1)11z2z=zz1n=0znz1nn=0z2nznz1z2=1z2n=1z1nu(x)=zz1n=0znz1nn=01z1nzn=n=0(5+12)n1ei(n+1)xn=0(5+12)neinx=n=0(5+12)n1(cos(n+1)x+isin(n+1)x)n=0(5+12)n(cos(nx)isin(nx))u(x)realu(x)=n=0(5+12)(n+1)cos(n+1)xn=0(5+12)ncos(nx)=n=1(5+12)ncos(nx)n=0(5+12)ncos(nx)=12n=1(5+12)ncos(nx)...becontinued....

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