Chpter 5 5. [; ] f(x) 3 f(x) = X + ( n cos nx + b n nx ) (5:) n= (5.) ; ;...;b ;b ;... n; m > 3 Z Z Z mx cos mx mx nx nx cos ( : n 6= m dx = : n = m ( : n 6= m dx = : n = m nx cos dx = (5.) (5.) (5.) 3 n ;b n n = b n = Z Z f(x) cos nx dx ; n =; ; ;... (5.3) f(x) nx dx ; n =; ; 3...: (5.4) (5.) 3 73
5. <x< ( : <x f(x) = x : <x< (5:5). (5.3) (5.4) Z = Z xdx = x cos nxdx = n [x nx] n n = = n [()n ] ; n=;... b n = Z = ()n+ n x nxdx = n [x cos nx] + n Z Z nxdx cos nxdx ; n=;... (5.6) f(x) = X 4 + [ ()n n n= cos nx + ()n+ n nx] (5:7) (5.7) P m= (5.7) x = (m+) 3 = 4 + X X m= n= () n n (m +) = 8 u() = u() = u(x) <x< 3 nx u(x) = X n= c n nx (5:8) (5:9) (5.9) [;] <x< u(x) =u(x); <x< 74
u(x) (5.9) (5.) f(x) <x< u(x) = X n= c n cos nx (5:) (5.9) (5.) 5. ( <x<) 5. f(x) = ( : <x : <x< (5:). x n = Z [ cos nxdx b n = Z Z cos nxdx] =; n =; ;... nxdx = () n ; n =; ;... (5.) n f(x) = 4 X n= (n +)x n + (5:3) x = (5.) f() = (5.3) ff( +)+f( )g x = c x = c ff(c +)+f(c )g (5:4) ( (5.) (5.) f(x) X + ( n cos nx + b n nx ) (5:5) n= 5. (5.3) S N (x) = 4 n= (n +)x n + (5:6) 75
x = x = 6 // 5.//////////////////////////// ; cos nx cos nx (5.5) = i fexp(inx ) exp(inx )g = fexp(inx ) + exp(inx )g (5.7) f(x) X + f ( n ib n )e i nx = X n= n= + ( n + ib n )e nx i c n exp(i nx ) (5.8) c n ( n ) c n = Z f(x)exp(i nx )dx; (n =; 6; 6...) (5:9) g Z exp(i nx ) exp(imx )dx = ( : n 6= m : n = m (5:) (5.8) n ;b n c n c = ; c m = ( m ib m ); m > c m = (c m ) 3 ;m< (5.) 76
5. ( ) m(x) ( : x = m(x) = : x 6= m(x) x = m(x)dx =: m(x) P.A.M. (distribution) q n f g enx n (x) r n enx (5:) x p p n <x< p p n dxr n enx = (5:3) n n! (x) = lim n! r n enx (5:4), n (x) 5. 77
// 5.//////////////////////////// f(x) jxj! N jxj N ( ) jxj exp(x ) f(x)(x)dx n! lim f(x) n (x)dx = f() (5:5) n n (x) x = f(x) x ' f(x) n (x)dx ' f() n (x)dx = f() (5:6) : r n j f(x)dx f()j enx r n = j ff(x) f()gdxj enx r n Mxjf () (x)j jxjdx enx = p n Mxjf () (x)j!: (n!): (5.7) (5.4) (5.5) f(x) f() f(x) (x)dx n! lim f(x) n(x)dx (5:8) = f(x) d Z n(x) dx dx = f(x) d r n dx ( )dx enx r n f (x) dx!f () : (n!) enx 78
f(x)! : jxj! (x) f(x) (x)dx = f (x)(x)dx = f () (5:9) 5.3 (x) () (x) =(x) () (x) = jj (x) (5.3) (). f(x)(x)dx = n! lim f(x) n (x)dx = n! lim f(x) n (x)dx = (x) =(x) () f(x)(x)dx = n! lim = lim n! r n = lim n! r n f(x) n (x)dx f(x)(x)dx f(x)exp(n x )dx f( y jj ) exp(ny ) dy jj = jj f() = jj f(x)(x)dx (x) = jj(x) ri(i =; ;...) m i (r) = X i m i (3) (r ri): (5:3) (3) (r) 3 r =(x; y; x) 79
(3) (r) =(x)(y)(z) (5:3) (x) (5.4)(5.5) x (x) = lim! x ; (5:33) " (x) =lim "! x + " : (5:34) 5.4 (5.33)(5.34). (5.33) > dz f(z)eiz z (5:35) f(z) jzj! 5.3 z = f(z) z = f(z)e iz =z z = ( 5.3) z f(z) fz n g Res(f : z n ) C C I C dx f(z)eiz z = Z R Z + f(x)e ix dx + x Z R f(x)e ix dx x ide i f(ei )e iei e i + Z idre i f(rei )e irei Re i (5.36) 3 z = e i, 4 z = Re i R // 5.3//////////////////////////// 8
f(z) I C dx f(z)eiz z =i X n Res(f(z) eiz z : z n ) (5:37)!,R! (5.36) 3 4 Z lim! idf(e i )e iei = if(); lim R! Z idf(re i )e irei = (5:38) x=x x = f(x) x x (5.36)(5.39) dx = Z i lim f! R R! f(x)e ix dx + x Z R f(x)e ix dxg (5:39) x f(x) x X x dx f() = n Res(f(z) eiz z : z n ) (5:4) f(z)! z n I m z n > e izn! (!) jzj! f(z) z = z n m! e Im(z n) lim! (5.33) (5.34) x f(x) dx = f() (5:4) x " x + " = i [ x i" x + i" ] (5:4) " dz f(z) (5:43) z + " 8
f(z) z =+i" z = i" " X dz f(z) =i f(i") + z + " i n " ires(f(z) z + " : z n) z n f(z) "! " lim "! " dz f(z) =f() (5:44) z + " ; " (5.33)(5.34) 5.4 x = (x) = dk exp(ikx) (5:45) // 5.4//////////////////////////// 5.3 (5.) (5.5) f(x) () [; ] () (3) [; ] jf(x)j Z jf(x)jdx 8
[; ] (x = ; x n = ) (x ;x ); (x ;x );...(x n ;x n ) nx jf(x k ) f(x k )j (5:46) f(x) V (f :[; ]) V (f :[; ]) f(x) V (f :[; ]) = jf() f()j x x ( 5.5) x x x = (n +) ; (n = 6; 6...) [;] // 5.5//////////////////////////// V (x x :[;]) > 4 [ 3 + 5 + 7 +...] () (3) f(x) (5.) S N (x) = + ( k cos kx + b k kx ) (5:47) 83
k = b k = Z Z (5.48)(5.49) (5.47) S N (x) = Re = k= Z Z f(t)dt + dtf(t)[ + f(x) cos kx dx k =; ; ;... (5:48) f(x) kx dx k =; ; 3;... (5:49) [cos kx cos Z f(t) cos kt kt Z dt + f(t) kt dt] k(x t) ] (5.5) e ikt +)t ei(n = Re = Re ei(n + )t e it= = (N + )t +(t) e it e it= e it= ( t ) (5.5) Re e ikt = (5.5) cos kt = (N + )t ( t ) S N (x) = Z dtf(t) (N +)(xt) (xt) = Z (N+)t dtf(x t) t (5.5) t = S N (x) = Z (N+) df(x ) + Z (N+) df(x +) (5.53) = Z Z (N+) df(x +) df(x +)( Z ) (N +) df(x +) (N+) (5.54) (N +) N! (5.33) () = [f(x +)( ( ) )]! = 84
(5.54) N! (5.53) S N (x)! N! Z (N +)= df(x ) = Z (N +)= df(x +) = (N+)! () lim S N(x) = N! = Z Z df(x )( )+ df(x )()+ (5.3) Z Z Z g(x)(x)dx = g() Z (x)dx =;(x) =(x) Z (5.55) g(x)(x)dx = g( +) df(x +)( ) df(x +)() (5.55) lim S N(x) = ff(x ) + f(x +)g (5:56) N! x f(x) x f(x) x (Dirichlet) () f(x 6 ) x = c f(x) f(x) n n = Z f(x)cos nx dx = [ nx n f(x)] n = nc [f nf() n n Z Z f (x) nx dx f(c +)g + f nc f(c )+nf()g] f (x) nx dx (5.57) 85
f (x) [; ] =n b n n = nc n ff(c +) f(c )g + O( n ) (5:58) b n = nc cos n ff(c +) f(c )g + O( n ) (5:59) n ;b n =n f(x) m (C m, f(x) m f (m) (x) ) f (m+) (x) n ; b n n ;b n O( ) (5:6) m+ n 5. f(x) = ( : <x :<x< (5:6) f(x) = 4 n N S N (x) S N (x) = 4 = n= Z x X n= (n +)x n + (n +)x n + = 4 Z x n= cos(n +)d (5:6) (N +) d (5.63) x! + S N (x) " j (N +) (N +) j <" N (N >N ) (N +) N S N (x) = Z (N +)x 86 d (5:64)
Z d = Si =:7897975... ( ) (5.64) (N +)x = x! + ;N! N! x! + S N (x) N! x! + lim x! + lim N! S N(x) =f( + )= lim lim S N (x) = N! x! + lim x! + S N (x) =:7897975... (5:65) N! (N +)x= x = S N (x) S N (x) 5. 5.4 f(x) f(x) 3 f(x) ' N (x) = + ( k cos kx + k kx ) (5:66) n ; n ' N (x) Mx jf(x) ' N (x)j x[;] [; ] ( ) N = Z jf(x) ' N (x)j dx : (5:67) 87
N ;... N ;... N N N = = + Z Z ff(x) f(x)'(x)+'(x) gdx dxff(x) f(x) f(x) 4 + ( k cos kx + k kx X + ( k cos kx k6=l + k kx )( l cos lx ( k cos kx + k k kx + k kx ) 3 (5.) (5.68) N = + = N Z 4 + Z f(x) dx ( k + k) f(x) dx 4 + 4 ( ) + N Z f(x) dx 4 ( ) kx cos ) + l lx )g (5.68) ( k k + k b k ) ( k + b k ) f( k k ) +( k b k ) g (5.69) ( k + k) (5:7) = ; n = n ; n = b n (n =; ;...;N) (5:7) 3 ( ) (5.7) (5.7) Z f(x) dx + ( k + b k) N! Z f(x) dx 88 + X ( k + b k) (5:7)
Z X f(x) dx = + ( k + b k ) (5:73) f(x) S N (x) lim N! Z jf(x) S N (x)j dx = (5:74) f(x) [; ] (5.56) (5.74) 3 ( ) 5.5 3 (5.5)(5.8) 3 ( ) [; b] f' n (x)g hf;gi = Z b dx(x)f(x)g(x) (5:75) (x) [; b] g(x) g(x) (x) f' n (x)g h' n ;' m i = nm ( : n = m nm = : n 6= m Z b (x)' n (x)' m (x)dx = nm (5:76) (5.75) hf; gi (i) hf;fi hf;fi =*) f(x) (ii) (iii) hf; gi = hg;fi hf + f ;gi = hf ;gi + hf ;gi (iv) hf; gi = hf;gi (5.77) 89
jjfjj = q hf;fi (5:78) ( ) (i) jjfjj jjfjj =*) f(x) = (ii) (iii) jjf + gjj jjfjj + jjgjj (3 ) jjfjj = jj jjfjj (5.79) 5.5 jhf;gij jjfjj jjgjj (5:8).f(x);g(x) jjf + gjj = jj jjfjj + hf; gi + hf;gi + jj jjgjj = jjgjj ; = hf;gi jjgjj fjjgjj jjfjj jhf;gij g jjgjj = 5.6 3. jjf + gjj (jjfjj + jjgjj) hf;gi + hg;fi jjfjj jjgjj hf; gi = hg; fi frehf; gig jjfjj jjgjj (5:8) (5.88) z x; y z = x + iy 9
x x + y = jzj jrehf;gij jrehf; gij + jimhf;gij = jhf; gij jhf;gij jjfjj jjgjj jrehf; gij jjfjj jjgjj (5.8) 3 3 3 3 ( ) ( ) [; b] f(x) f' n (x)g f(x) X c k ' k (x) (5:8) ' m (x) hf;' m i = = Z b X dx(x)f(x)' m (x) c k h' k ;' m i = c m (5.83) c m hf;' m i N (x) = k ' k (x) (5:84) (5.75) k N N = jjf N jj =(b ) (5:85) N = b [jjfjj hf; N ih N ;fi + h N ; N i] = = b [jjfjj b [jjfjj (c k k +c k k )+ jc k j + 9 j k j ] jc k k j ] (5.86)
k = c k = hf;' k i (5:87) N jjfjj jjfjj X jc k j ; jc k j (5:88) (5.88) f' n (x)g f(x) (5.8) (Weierstrss) : [; b] f(x) "> x jf(x) P (x)j <" x P (x) [; b] f(x) (5.5) 3 x = x P (x) 5.7 x x (x) =. x f;x;x ;...;x n ;...g ff n (x)g (x) = f (x) jjf jj ; (5.89) g (x) = f (x) hf ; i (x); 9
(x) = g (x) jjg jj ; (5.9) g (x) = f (x) hf ; i (x) hf ; i (x); (x) = g (x) jjg jj ; (5.9). g n (x) = f n (x) n X k= hf n ; k i k (x); n (x) = g n(x) jjg n jj ; (5.9). q jjg n jj g n (x) hg n ;g n i (5.9) f n (x) (x), (x)..., n (x) n (5.9) (5.9) h k ;g n i =; k =; ; ;...;n h k ; n i =; k =; ; ;...;n (5:93) n hg;fi = R g(x) f(x)dx ff n (x) =x n jjf jj = g (x) = x hx; dx = ; (n =; ;...)g (x) = p (5:94) p ip = x; jjg jj = (x) = x dx = s 3 ; 3 x (5.95) 93
g (x) = x hx ; jjg jj = 8 45 ; s s p ip 3 3 hx ; xi x = x 3 ; (x) = p 4 (3x ) (5.96) n x n ( ) n ( ) n (x) s n + n (x) = P n (x) (5:97) P n (x) n P n (x) ( x ) d dx P n(x) x d dx P n(x)+n(n +)P n (x) = (5:98) 3 P n (x) P nm (x) m = ( ) 4 (5.8) 5.6 : z = x + iy f(z) z = z f(z) = n (z z ) n + n+ (z z ) n +...+ + (z z )+ (z z ) +... (5:99) z = z f(z) n z = z C( A-) I C f(z)dz (5:) 94
z C f(z) I C f(z)dz =i (5:) f(z) z = z Res(f(z) :z ) I C f(z)dz =ires(f(z) :z ) (5:) z f(z) // A-//////////////////////////// f(z) z = f(x)e ikx dx (5:3) k > A- z C z = Re i R! e ikz = e ik(rei ) = e ikr cos kr e kr e ( R lim R! Z dre kr jf(re i )j = dxf(x)e ikx = I C <<) dzf(z)e ikz (5:4) C dxf(x)e ikx =i X <Arg(z n )< 95 Res(f(z)e ikz : z n ): (5.5)
k> <Arg(z) < ( z- ) k < < Arg(z) < ( z- ) (z ) X dxf(x)e ikx = i Res(f(z)e ikz : zn) (5.6) <Arg(zn )< 96
97