? E Bc'PVSJFS ) ?  5f ?￥ 'PVSJFS) ?Z 7 (a) (b)  m  ? !? @è￥M?? p1 | ?M1° @è ￥? @V?μ ? ?? Et A t() sin=ω ò ?o? @ m 16.1.5(a) ft A 1 2 () (sin= ft A t 2 () |sin | tt|sin |)+ωω ó ?o? @ m 16.1.5(b) = ω ) x)  C |  k| ? ω=1 fx 1 ( f 2 ( ],[ ππ? Z 71 Fourier) ?b 3   0 a = 1 1 ()f xdx π π π ? ∫ 2A π =  a n = 1 1 ()cosf xnxdx 2 2 (1 A nπ =? )?  2, 4,6,n = null  π π π ? ∫ n a = 1 1 ()cos 0f x nxdx π π π ? = ∫  1, 3, 5,n = null   1 b = 1 1 ()sin 2 A fx xdx π π π ? = ∫   b n = 1 1 ()sin 0f x nxdx π π π ? = ∫  2,3, 4,n = null b 1 ()f x ～ 2 1 2cos2 sin 24 k 1 A AA x kππ ∞ = +? kx ? ∑ b   0 a = 2 1 ()f xdx π π π ? ∫ 4A π =  a n = 2 1 ()cosf xnxdx 2 4 (1 A nπ =? )?  2, 4,6,n = …  π π π ? ∫ n a = 2 1 ()cos 0fx nxdx π π π ? = ∫  1, 3, 5,n = …   b n = 2 1 ()sin 0f x nxdx π π π ? = ∫  1, 2, 3,n = null b 2 ()f x ～ ∑ ∞ = ? ? 1 2 14 2cos42 k k kxAA ππ b ? |/ f ? ],[ ππ? Z 7? Fourier) ? ò xxf sgn)( = ; ó fx x() |cos |= ; 1 ? 2 2 2 )( π?= x xf ; ? fx() ? ? ? ∈ ?∈ = );,0[,0 ),0,[, π π x xx ? fx() ? ? ? ∈ ?∈ = ).,0[, ),0,[, π π xbx xax 3   1 f ? ?[()fx 0 n a =  0,1, 2,n = …   b n = 1 ()sinf xnxdx π π π ? ∫ 2(1 cos( ))n n π π ? =  1, 2, 3,n = null b ()fx～ ∑ ∞ = ? ? 1 12 )12sin(4 k k xk π b   1 }f ? ?[()fx 0 n b =  1, 2, 3,n = null  0 a = 1 ()f xdx π π π ? ∫ 4 π =  n a = 1 ()cosf xnxdx 2 2 4( 1) (1 n nπ ? =? )?  2, 4,6,n = null  π π π ? ∫ n a = 1 ()cos 0fx nxdx π π π ? = ∫   1, 3, 5,n = null b ()fx～ ∑ ∞ = ? ? ? 1 2 2cos 14 )1(42 k k kx k ππ b   1 }f ? ?[()fx 0 n b =  1, 2, 3,n = null  0 a = 1 ()f xdx π π π ? ∫ 2 5 3 π=?  n a = 1 ()cosf xnxdx 2 2( 1) n n ? =  1, 2, 3,n = null b π π π ? ∫ ()fx～ nx nn n cos )1(2 6 5 1 2 2 ∑ ∞ = ? +? π b   0 a = 1 ()f xdx π π π ? ∫ 2 π =?  n a =  1 ()cosf xnxdx 2 1(1) n nπ ?? =  1, 2, 3,n = null  π π π ? ∫ b n = 1 ()sinf xnxdx π π π ? ∫ cos( )n n π =?  1, 2, 3,n = null b ()fx～ ∑ ∞ = + + +? 0 2 )12( )12cos(2 4 k k xk π π nx n n n sin )1( 1 1 ∑ ∞ = + ? + b   0 a = 1 ()f xdx π π π ? ∫ () 2 baπ ? =  2 n a = 1 ()cosf xnxdx π π π ? ∫ 2 ()(1(1) n ab nπ ??? = )  1, 2, 3,n = null  b n = 1 ()sinf xnxdx ()cos(ab n n )π+ =?  1, 2, 3,n = null b π π π ? ∫ ()fx～ ∑ ∞ = + +? + ? ? 0 2 )12( )12cos()(2 4 )( k k xkbaba π π nx n ba n n sin )1( )( 1 1 ∑ ∞ = + ? ++ b à |/ f ?Z 7???) ? ò xxf +=π)( ],0[ π∈x ; ó fx x () e= ?2 ],0[ π∈x ; ? fx() ? ? ? ∈ ∈ = ];,[, ),,0[,2 2 2 ππ π π x xx ? fx() ? ? ? ? ? ∈ ∈ = ].2,1[,0 ),1,0[, 2 cos x x xπ 3   b n = 0 2 ()sinf xnxdx π π ∫ 12(1) 2 n n ? ? =?  1, 2, 3,n = null b ()fx～ 1 12(1) 2s n n nx n ∞ = ?? ∑ ib   b n = 0 2 ()sinf xnxdx π π ∫ 2 2 21(1) (4 ) n ne n π π ? ? ??? ? ? = +  1, 2, 3,n = null b ()fx～ [ ] nx n en n n sin 4 )1(12 1 2 2 ∑ ∞ = ? + ?? π π b   b n = 0 2 ()sinf xnxdx π π ∫ 2 2(1)2sin 2 n n n n π π π ? ? ??? ? ? ? ? =  1, 2, 3,n = null b ()fx～ nx n n n n n sin 2 sin 4 )1( 2 1 2 1 ∑ ∞ = + ? ? ? ? ? ? +? π π b   1 b = 2 0 21 ()sin 2 fx xdx π = ∫   b n = 2 0 2 ()sin 2 f xnxdx ∫ 2 2( sin ) 2 (1) n n n π π ? = ?  2,3, 4,n = null b ()fx～ x n n n n x n 2 sin 1 2 sin 2 2 sin 1 2 2 π π π π π ∑ ∞ = ? ? + b á |/ f ?Z 7???) ? ò fx x x() ( )=?π ],0[ π∈x ; ó fx x () e= ],0[ π∈x ; 3 ? fx() ? ? ? ∈ ∈ = ];,[,1 ),,0[,2sin 24 4 ππ π x xx  ? 22 )( ππ ?+?= xxxf ],0[ π∈x . 3   0 a = 0 2 ()f xdx π π ∫ 2 3 π =  n a = 0 2 ()cosf xnxdx π π ∫ 2 2(1 ( 1) ) n n +? =?  1, 2, 3,n = null b ()fx～ ∑ ∞ = ? 1 2 2 2cos 6 k k kxπ b   0 a = 0 2 ()f xdx π π ∫ 2 (1e π π =?)  n a = 0 2 ()cosf xnxdx π π ∫ 2 2(1)1 (1 ) n e n π π ??? ? ?? = +  1, 2, 3,n = null b ()fx～ )1( 1 ? π π e [ ] nx n e n n cos 1 1)1(2 1 2 ∑ ∞ = + ?? + π π b   0 a = 2 0 4 ()f xdx π π ∫ 2 π π + =  1 a = 2 0 4 ()cos2f xxdx π π ∫ 1 π =?  n a = 2 0 4 ()cos2f xnxd π π ∫ 2 2 sin (1) 2 n n nn π π ?? =? ?? ? ??  2,3,4,n = null b ()fx～ 11 1 ()cos2 2 2 211 sin 1 cos 2 12 n n nx nn π π ∞ = ?? ?? ?? ? ?? ∑ b 2 x π π +?   0 a = 0 2 ()f xdx π π ∫ 2 π =  n a = 0 2 ()cosf xnxdx π π ∫ 2 4(1) cos 2 n n n π π ?? ?? ?? ?? =  1, 2, 3,n = null b ()fx～ nx n n n n cos 2 cos)1( 4 4 1 2 ∑ ∞ = ? ? ? ? ? ? ?? + π π π b ? p?l ?iB?é1 π2 ￥ uW ]2,[ π+aa ￥f ? ￥ Fourier) ?# " ?￥9 ? Tb fx() 4 3 ! fx()~ a anxbn nn n 0 1 2 ++ = ∞ ∑ (cos sinx) 5 22 0 1 ()cos ( cos sin )cos 2 aa nn n a f x mxdx a nx b nx mxdx ππ ∞ ++ = ?? =+ + ?? ?? ∑ ∫∫  2 0 1 cos ( cos cos sin cos ) 2 a aa n a mxdx a nx mxdx b nx mxdx π ∞ + = =+ ∑ ∫ m a π=  0,1, 2,m = … 22 0 1 ( )sin ( cos sin ) sin 2 aa nn n a f xmxdx anxbnx mxd ππ ∞ ++ = ?? =+ + ?? ?? ∑ ∫∫ x  2 0 1 sin ( cos sin sin sin ) 2 a aa n a mxdx a nx mxdx b nx mxdx π ∞ + = =+ ∑ ∫ m b π=  1, 2,m = … ?[ a n = ∫ + π π 2 cos)( 1 a a nxdxxf  null,2,1,0=n  b n = ∫ + π π 2 sin)( 1 a a nxdxxf  null,2,1=n b ? |/ f ?·? uWZ 7? Fourier) ? ò 2 )( x xf ? = π ]2,0[ π∈x ; ó fx x()= 2 ]2,0[ π∈x ; ? xxf =)(  ; x ∈[,]01 ? fx() ? ? ? ∈ ?∈ = );1,0[,0 ),0,1[,e 3 x x x ? fx() ? ? ? ∈ ?∈ = ),0[,0 ),0,[, Tx TxC C ^è ? ). 3   n a = 2 0 1 ()cos 0f x nxdx π π = ∫  0,1, 2,n = null  b n = 2 0 1 ()sinf xnxdx π π ∫ 1 n =  1, 2, 3,n = null b ()fx～ nx n n sin 1 1 ∑ ∞ = b   0 a = 2 2 0 18 () 3 fxdx π π π = ∫  n a = 2 0 1 ()cosf xnxdx π π ∫ 2 4 n =  1, 2, 3,n = null  b n = 2 0 1 ()sinf xnxdx π π ∫ 4 n π =?  1, 2, 3,n = null b ()fx～ ∑ ∞ = ? ? ? ? ? ? ?+ 1 2 2 sincos 1 4 3 4 n nx n nx n π π b 5   0 a = 1 0 2() 1f xdx= ∫  n a = 1 0 2()cos2f xnxd0=π ∫  1, 2, 3,n = null  b n = 1 0 2()sin2f xnxdπ ∫ 1 nπ =?  1, 2, 3,n = null b ()fx～ nx n n π π 2sin 11 2 1 1 ∑ ∞ = ? b   0 a = 1 3 1 1 () (1 ) 3 f xdx e ? ? =? ∫  n a = 1 1 ()cosf xnxdπ ? ∫ 3 22 3 1(1) 9 n e n π ? ? ?=? ? ? +  1, 2, 3,n = null  b n = 1 1 ()sinf xnxdπ ? ∫ 3 22 1(1) 9 n n e n π π ? ? ?=?+? ? ? +  1, 2, 3,n = null b ()fx～ )1( 6 1 3? ?e () ( ) ∑ ∞ = ?? ? ? ? ? ? ? + ?? ? + ?? + 1 22 3 22 3 sin 9 )1(1 cos 9 )1(13 n nn xn n en xn n e π π π π π b   0 a = 1 () T T f xdx C T ? = ∫  n a = 1 ()cos T T nx f xdx TT π ? ∫ 0=  1, 2, 3,n = null  b n = 1 ()sin T T nx f xdx TT π ? ∫ 1(1) n C nπ ? ?=?+? ? ?  1, 2, 3,n = null b ()fx～ x T n n CC n π π )12( sin 12 12 2 1 ? ? ? ∑ ∞ = b ?  V eA e?è ^?￥μè @1   m  <≤ <≤ ,, ,0 0 0 TtT Tt ? ? ? = sin5 ,0 )( t tI ω ? 1? ? q ? ùω ω π2 =T bC ! S?Y HW T T 0 8 =  nm 16.1.6 p  [, ￥ Fourier) ?b It() ]0 T 3 0 a = 0 25( () 2 T fxdx T π ? = ∫ )  1 a = 0 22 ()cos T x f xd T 5 4π =?  π ∫ n a = 0 22 ()cos T nx f xdx TT π ∫ 2 51 (1) 1 (1) 2 cos cos 21 4 1 4 nn nn ππ π +? 1n ? ? =?+ ? ? + ?? ? ?  6 2,3, 4,n = null 1 b = 0 22 ()sin T x f xd TT π ∫ 5(7 2) 8 π π + =  b n = 0 22 ()sin T nx f xdx TT π ∫ 51 (1) 1 (1) sin sin 21 4 1 4 nn nn π π π +?? ? =? ? ? +? ? ?  b2,3, 4,n = null ()fx～ tt ω π ω ππ sin 8 35 4 5 cos 4 5 )22( 4 5 ? ? ? ? ? ? ++???  tn n n n n n n ω ππ π cos 1 2 4 )1( cos 1 1 4 )1( cos 1 1 2 5 2 2 ∑ ∞ = ? ? ? ? ? ? ? + ? ? ? + + +  tn n n n n n ω ππ π sin 4 )1( sin 1 1 4 )1( sin 1 1 2 5 2 ∑ ∞ = ? ? ? ? ? ? ? ? ? + + + b ? ! fx() ],[ ππ?  V ' V￡ ü ò ?? ?i ],[ ππ?∈x ? ? )()( π+= xfxf 5  ab nn21 21 0 ?? == ó ?? ?i ],[ ππ?∈x ? ? )()( π+?= xfxf 5 ab  nn22 0== ￡    21n a ? = 1 ()cos(2 1)f xnx π π π ? ? ∫ d  0 0 11 ()cos(2 1) ()cos(2 1)f xnxd fxnx π π ππ ? =?+ ∫∫ d?  00 ( )cos[(2 1) (2 1) ] ( )cos(2 1) ( )ft n t n dt fx n xdx t xπ π?+?=+  0=  1, 2, 3,n = … 21n b ? = 1 ()sin(2 1)f xnx π π π ? ? ∫ d  0 0 11 ()sin(2 1) ()sin(2 1)f xnxd fxnx π π d π π ? =?+ ∫∫ ?  00 ( )sin[(2 1) (2 1) ] ( )sin(2 1) ( )ft n t n dt fx n xdx t x ππ π π?+?=+  0=  b1, 2, 3,n = …  2n a = 1 ()cos(2 )f xnx π π π ? ∫ dx  0 0 11 ()cos(2 ) ()cos(2 )f xnxdx fxnx π π ππ ? + ∫∫ dx  00 ()cos(2 2 ) ( )cos(2 ) ( )ft nt n dt fx nxdx t xπ π=? ? + =+  0=  1, 2, 3,n = … 7 2n b = 1 ()sin(2 )f xnx π π π ? ∫ dx  0 0 11 ()sin(2 ) ()sin(2 )f xnxdx fxnx π π ππ ? + ∫∫ dx  00 ()sin(2 2 ) ( )sin(2 ) ( )ft nt n dt fx nxdx t xπ π=? ? + =+  0=  b1, 2, 3,n = … ? ! fx() (2/,0 )π  V ' V ?sY ?é?8 1"￥ ü ?? ? P ? [, ￥ Fourier) ?￥? T1]?π π ò fx a n x n n ()~ cos( )21 1 ? = ∞ ∑ ; ó fx b nx n n ()~ sin2 1= ∞ ∑  3  A ? 1 }f ?7 Ofx() 2n a = 0 2 ()cos(2 )f xnx π π ∫ dx  2 0 2 ( )cos(2 ) ( )cos(2 )fx nxdx fx nxdx π π π ππ + ∫  7 txπ= ?   22 00 ( )cos(2 ) ( )cos(2 )f xnxdx ftntπ=+? ∫ dt  [] 2 0 2 () ( )cos(2 )f xf x nxdx0 π π π =+? ∫ =  ?[ () ( ) 0fx f xπ+?=  ? ^ V[?/ ?Z Té?ü?fx() ? ? ? ? ? ? ? ? ? ? ? ∈?? ∈ ?∈? ??∈+? = ), 2 ()( ) 2 ,0()( )0, 2 ()( ) 2 ,()( )( ~ π π π π π π ππ xxf xxf xxf xxf xf b  A ? 1 f ?7 Ofx() 21n b ? = [] 0 2 ()sin(2 1)f xnx π dx π ? ∫  [] [] 2 0 2 ()sin(2 1) ()sin(2 1)fx n xdx fx n xdx π π π ππ =?+ ∫ ? 7 txπ=?  8 [] [] 22 00 ()sin(2 1) ( )sin(2 1)f xnxdxftnt ππ π=?+? ∫∫ dt?  [][] 2 0 2 () ( )sin(2 1)f xf x nx π π π =+?? ∫ dx0=  ?[ () ( ) 0fx f xπ+?= ? ^ V[?/ ?Z Té?ü?fx() ? ? ? ? ? ? ? ? ? ? ? ∈?? ∈ ?∈?? ??∈+ = ), 2 ()( ) 2 ,0()( )0, 2 ()( ) 2 ,()( )( ~ π π π π π π ππ xxf xxf xxf xxf xf b ? !? ù1 π2 ￥f ?  [,fx() ]?π π ￥ Fourier" ?1 a ?  p/ f ?￥ Fourier" ? ? n b n ~ a n ~ b n  ò gx f x() ( )= ? ; ó hx f x C() ( )= +  C ^è ? ) ? ∫ ? ?= π π π dttxftfxF )()( 1 )(  L?s ¨? V[?D b 3   n a =null 1 ()cos ( )cosg x nxdx f x nxdx π ππ?? =? ∫  7 tx=?  1 ()cosf tntd π π π ? = ∫ x ?[ nn aa = ~  ),2,1,0( null=n  n b = null 11 ()sin ( )sing x nxdx f x nxdx ππ ππ?? =? ∫∫  7 tx=?  1 ()sinf t td π π π ? =? ∫ x ?[ nn bb ?= ~  ),2,1( null=n b  y1 [,xC ]π π+∈?  ?[ [,]x CCπ π∈ ?? ? b 9 11 ()cos ( )cos CC n CC a h x nxdx f x C nxdx ππ ππ π π ?? ?? ?? ==+ ∫∫ null  7 txC= +  1 ()cos ( )f tntCd π π x π ? =? ∫  11 ()cos cos ()sin sinf tntnCdx ft tCd ππ x π π ?? =+ ∫∫ cos sin n anCb=+C ),2,1,0( null=n  n b null 11 ()sin ( )sin C CC h x nxdx f x C nxdx ππ ππ π π ?? ?? ?? ==+ ∫∫  7 txC= +  1 ()sin ( )f tntCd π π x π ? =? ∫  11 ()sin cos ()cos sinf t nt nCdx f t nt nCdx ππ π π ?? ∫∫ cos sin n bnCa=?C ),2,1( null=n b   n a =null 11 ()cos () ( ) cosF x nxdx f t f x t dx nxdx πππ ??? ? =? ? ?? ∫∫∫ ? ?  ?DQ?  11 ()cos ()f x t nxdx f t dt ππ ππ ππ ?? ?? =? ?? ?? ∫∫ b ? H0n= 2 00 11 1 ( ) () ()a f x t dx f t dt a f t dt a ππ π ππ π ππ π ?? ? ?? =?= ?? ?? ∫∫ ∫ null 0 = ? H0n> 11 ()[cos()cossin()sin] () n a f x t n x t nt n x t nt dx f t dt ππ ππ ππ ?? =???? ∫∫ null  1 (cos sin)() nn a nt b nt f t dt π π π ? =? ∫ 2 n ab 2 n = ?  ),2,1( null=n b n b = null 111 ()sin () ( ) cosF x nxdx f t f x t dt nxdx πππ ??? ? =? ? ?? ∫∫∫ ? ?  ?DQ?  11 ()cos ()f x t nxdx f t dt ππ ππ ππ ?? ?? =? ?? ?? ∫∫  11 ()[sin()cos cos()sin] ()f x t n x t nt n x t nt dx f t dt ππ ππ ππ ?? =??+? ∫∫  10 1 (cos sin)() nn b nt a nt f t dt π π π ? =+ ∫ 2 nn ab=  ),2,1( null=n b 11