1 1.1 [] f(x) f(x + T ) = f(x) (1.1), f(x), T f(x) x T 1 ) f(x) = sin x, T = 2 sin (x + 2) = sin x, sin x 2 [] n f(x + nt ) = f(x) (1.2) T [] 2 f(x) g(x) T, h 1 (x) = af(x)+ bg(x) 2 h 2 (x) = f(x)g(x) T, 2 f(x) g(x), f(x) T 1, g(x) T 2, p 1 (x) = af(x) + bg(x) T 2 T 1 T 2 2 f(x) g(x) p 2 (x) = f(x)g(x),?
[] x < T f(x), x < T f(x), < x < f(x) f(x) f(x) f(x + nt ) = f(x) x < T, n =, 1,, 1, (1.3) f(x) T x 2 f(x) T 2T x 3 f(x), f() = f(t ), f(x), f() f(t ), f(x) f(x) ()
1.2 [] x T f(x) g(x) (f, g) = T f(x)g(x) dx (1.4) (f, g) =, f(x) g(x) x T a b, a, b θ( θ ) a b = a b cos θ a, b a = (a 1, a 2,, a n ), b = (b 1, b 2,, b n ), n a b = a i b i i=1, a, b,, a b = [] {1, cos x, sin x, cos 2x, sin 2x,, (1) 1 cos mx dx = (m = 1, 2, ) (1.5) 1 sin mx dx = (m = 1, 2, ) (1.6) cos mx sin nx dx = (m = 1, 2, n = 1, 2, ) (1.7) cos mx cos nx dx = (m n m = 1, 2, n = 1, 2, ) (1.8) sin mx sin nx dx = (m n m = 1, 2, n = 1, 2, ) (1.9) (2) 1 2 dx = 2 (1.1) cos 2 mx dx = (m = 1, 2, ) (1.11) sin 2 mx dx = (m = 1, 2, ) (1.12)
(1.5) (1.12), [ sin mx 1 cos mx dx = m 1 sin mx dx = ] 2 = 1 m (sin 2m sin ) = 1 ( ) = (1.5) m [ ] cos mx 2 = 1 m m (cos 2m cos ) = 1 (1 1) = (1.6) m cos mx sin nx dx = 1 {sin (m + n)x sin (m n)x dx 2 = 1 { sin (m + n)x dx sin (m n)x dx 2 = 1 [ ] 2 [ ] 2 cos (m + n)x cos (m n)x 2 m + n m n = (m n) (1.7) cos mx sin nx dx = = 1 2 cos mx sin mx dx {sin (m + m)x sin (m m)x dx = 1 sin 2mx dx 2 = 1 [ ] cos 2mx 2 2 2m = (m = n) (1.7) cos mx cos nx dx = 1 {cos (m + n)x + cos (m n)x dx 2 = 1 { cos (m + n)x dx + cos (m n)x dx 2 = 1 [ ] 2 [ ] 2 sin (m + n)x sin (m n)x + 2 m + n m n = (m n) (1.8) sin mx sin nx dx = 1 {cos (m n)x cos (m + n)x dx 2 = 1 { cos (m n)x dx cos (m + n)x dx 2 = 1 [ ] 2 [ ] 2 sin (m n)x sin (m + n)x 2 m n m + n = (m n) (1.9)
1 1 dx = [x] 2 = 2 (1.1) cos mx cos mx dx = = 1 2 = 1 2 { = 1 2 cos 2 mx dx (1 + cos 2mx) dx 1 dx + cos 2mx dx [ ] sin 2mx 2 [x] 2 + 2m { = 1 {2 + 2 = (1.11) sin mx sin mx dx = (1.12)
1.3 [] f(x) 2 f(x), f(x) a 2 + a 1 cos x + b 1 sin x + a 2 cos 2x + b 2 sin 2x + a 3 cos 3x + b 3 sin 3x + = a (a n cos nx + b n sin nx) (1.13), f(x), cos nx, sin nx(n = 1, 2, ) 2, 2 n,, 2 { (2L) f(x) = a (a n cos nx + b n sin nx) {{ () e x = 1 + x 1! + x2 2! + + xn n! + sin x = x x3 3! + x5 5! + ( 1)n x 2n+1 (2n + 1)! + cos x = 1 x2 2! + x4 x2n + ( 1)n 4! (2n)! + f(x) = f() + f () 1! f (n) () = x n n! x + f () 2! x 2 + f () x 3 + 3! { x, x, ()
[] 2 f(x), a n b n f(x) = a (a n cos nx + b n sin nx) (1.14) (1) a n (1.14) cos mx(m = 1, 2, ), 2 f(x) cos mx dx = { 2 a (a n cos nx + b n sin nx) cos mx dx (1.15) = a cos mx dx + 2 a n cos nx cos mx dx + b n sin nx cos mx dx {{ = a [ ] sin mx 2 { n m + a n 2 m n = m = + a m a = a m (m = 1, 2, ) (1.16) f(x) dx = { 2 a = a 2 (a n cos nx + b n sin nx) dx dx + a n cos nx dx +b n sin nx dx {{{{ = a 2 [x]2 + = a (1.17), (1.16) (1.17) a n = 1 f(x) cos mx dx = a m (m =, 1, 2, ) (1.18) f(x) cos nx dx (n =, 1, 2, ) (1.19) (2) b n (1.14) sin mx(m = 1, 2, ), 2 f(x) sin mx dx = { 2 a (a n cos nx + b n sin nx) sin mx dx (1.2)
= a sin mx dx + 2 a n cos nx sin mx dx +b n sin nx sin mx dx {{ = a [ ] { cos mx 2 n m + b n 2 m n = m = + b m = b m (m = 1, 2, ) (1.21) b n = 1 f(x) sin mx dx = b m (m = 1, 2, ) (1.22) f(x) sin nx dx (n = 1, 2, ) (1.23)
1.4 ( 2) [] : f( x) = f(x), y x x cos x x f( x) = x = x = f(x) f( x) = cos ( x) = cos x = f(x) : f( x) = f(x), x sin x x x f( x) = x = f(x) f( x) = sin ( x) = sin x = f(x)
[] (P1) f(x) f(x) = f e (x) + f o (x) f e = f(x) + f( x), f o = 2 f(x) f( x) 2 f e (x) f o (x) f(x) (P2),, h(x) = f e (x)g e (x) : h( x) = f e ( x)g e ( x) = f e (x)g e (x) = h(x) h(x) = f o (x)g o (x) : h( x) = f o ( x)g o ( x) = { f o (x){ g o (x) = f o (x)g o (x) = h(x) h(x) = f e (x)g o (x) : h( x) = f e ( x)g o ( x) = f e (x){ g o (x) = f e (x)g o (x) = h(x) (P3) f e (x), f o (x), f e (x) dx = 2 f o (x) dx = f e (x) dx
[] ( 1) f e (x), b n = b n = 1 {{ f e (x) sin {{{{ nx dx = a n = 1 {{ f e (x) cos {{{{ nx dx = 2 f e (x) cos nx dx ( 2) f o (x), a n = a n = 1 {{ f o (x) cos {{{{ nx dx = b n = 1 {{ f o (x) sin {{{{ nx dx = 2 f o (x) sin nx dx ( 3) f(x) f e (x), f o (x), f(x) = f e (x)+ f o (x) a n = 1 = 1 = 1 = 2 b n = 1 = 1 = 1 = 2 f(x) cos nx dx {f e (x) + f o (x) cos nx dx f e (x) cos nx dx f e (x) cos nx dx f(x) sin nx dx {f e (x) + f o (x) sin nx dx f o (x) sin nx dx f o (x) sin nx dx ( 4) f(x) g(x), a n (f), b n (f) a n (g), b n (g), c d cf(x)+dg(x), ca n (f)+da n (g), cb n (f) + db n (g)
2 f(x) ( P.16[ 2](1)) { 1 ( x < ) T = 2 f(x) = ( x < 2) (1) a n a n = 1 f(x) cos nx dx = 1 { f(x) cos nx dx + = 1 { 1 cos nx dx + = 1 cos nx dx = 1 [ ] sin nx (n ) n = 1 (sin n sin ) = n n = a = 1 f(x) dx = 1 { 1 dx + (2) b n f(x) cos nx dx cos nx dx dx = 1 dx = 1 [x] = 1 [ ] = 1 b n = 1 f(x) sin nx dx = 1 { f(x) sin nx dx + = 1 { 1 sin nx dx + = 1 sin nx dx = 1 [ ] cos nx n = 1 (cos n cos ) n = 1 n {( 1)n 1 = { 2 n f(x) sin nx dx sin nx dx (n ) (n ) ( b 2n 1 = ) 2 (2n 1) f(x) = a (a n cos nx + b n sin nx) = 1 [ ] 1 sin nx {1 ( 1)n n = 1 2 + 2 (sin x + 1 3 sin 3x + 1 sin 5x + ) 5 [ = 1 2 + 2 ] 1 sin (2n 1)x 2n 1
2 f(x) ( P.18[ 2](2)) T = 2 f(x) = x ( x < ) f(x), ( 1), b n = (n = 1, 2, 3, ), a n (n =, 1, 2, ) a n = 1 = 2 f(x) cos nx dx f(x) cos nx dx = 2 x cos nx dx = 2 ( ) sin nx x dx (n ) n = 2 { [ ] sin nx x 1 sin nx dx n n = 2 [ ] cos nx n n = 2 n 2 {( 1)n 1 = { (n ) (n ) 4 n 2 ( ) 4 a 2n 1 = (2n 1) 2 n = a = 1 f(x) dx = 2 x dx = 2 [ ] x 2 2 = 2 ( 2 2 ) = f(x) = a (a n cos nx + b n sin nx) = [ ] 2 cos nx {( 1)n 1 [ = 2 4 = 2 4 ( cos x + n 2 ) cos 3x cos 5x + + 3 2 5 ] 2 cos (2n 1)x (2n 1) 2
2 f(x) ( P.18[ 2](3)) T = 2 f(x) = x( x < ) f(x), ( 2), a n = (n =, 1, 2, ), b n (n = 1, 2, 3, ) b n = 1 = 2 f(x) sin nx dx f(x) sin nx dx = 2 x sin nx dx = 2 ( ) cos nx x dx n = 2 {[ ( )] cos nx x + 1 n n = 2 { [ ] sin nx ( 1) n + n n = 2 { 2 (n ) n ( 1)n+1 = n (n ) 2 n cos nx dx f(x) = a (a n cos nx + b n sin nx) = = 2 n+1 sin nx 2( 1) n ( sin x sin 2x 2 + sin 3x 3 )
1 [ 2 ] f(x) = a (a n cos nx + b n sin nx) a n = 1 f(x) cos nx dx (n =, 1, ) a n = 1 f(x) cos nx dx (n =, 1, ) b n = 1 f(x) sin nx dx (n = 1, 2, ) b n = 1 f(x) sin nx dx (n = 1, 2, ) f(x) = x ( x < 2) f(x) 2 2 x, f(x) = x ( x < 2) b n = 1 f(x) sin nx dx = 1 x sin nx dx b n = 1 f(x) sin nx dx = 1 x sin nx dx, f(x) = x ( x < ), f(x) = x ( x < 2) f(x) = x ( x < ), b n = 1 f(x) sin nx dx = 1 + 2) sin nx dx + (x 1 x sin nx dx f(x) 2 x
1.5 [] f(x) x, x { f( x) ( x ) f e (x) = (1.24) f(x) ( x ), x 2 f e (x + 2n) = f e (x) x <, n =, 1,, 1, (1.25) f(x) = a 2 + a n cos nx, a n = 2 f(x) cos nx dx (n =, 1, 2, ) (1.26) [] f(x) x, x { f( x) ( x ) f o (x) = (1.27) f(x) ( x ), x 2 f o (x + 2n) = f o (x) x <, n =, 1,, 1, (1.28) f(x) = b n sin nx, b n = 2 f(x) sin nx dx (n = 1, 2, ) (1.29) f(x) x < () f e (x) < x < < x < f e (x) f o (x) < x < < x < f o (x) ( 1) ( 2) ()
1.6 ( 2L) [ ( 2L) ] x = L t h(t) = a (a n cos nt + b n sin nt) t : 2 (1.3) t x 2 2L, 2 t 2L x ( ), f(x) 2L(L > ), t = L x f(x) = h( L x) = a (a n cos n L x + b n sin n L x) x : 2L (1.31) a n dt = (dt = dx) dx L L a n = 1 = 1 = 1 L L L L L h(t) cos nt dt h( L x) cos n L x L dx f(x) cos n x dx (1.32) L b n b n = 1 L L L f(x) sin n L x dx (1.33) [ ( 2L) ] f(x) x L f(x) = a 2 + a n = 2 L L a n cos n L x (b n = ) (1.34) f(x) cos n x dx (1.35) L f(x) = b n sin n L x (a n = ) (1.36) b n = 2 L f(x) sin n x dx (1.37) L L
2L f(x) { 1 ( x < L) T = 2L f(x) = (L x < 2L) (1) a n a n = 1 2L f(x) cos n L L x dx = 1 { L f(x) cos n 2L L L x dx + f(x) cos n L L x dx = 1 { L 1 cos n 2L L L x dx + cos n L L x dx = 1 L cos n L L x dx = 1 [ L L n sin n ] L L x (n ) = 1 (sin n sin ) = n n = a = 1 2L f(x) dx = 1 { L 2L 1 dx + dx = 1 L dx = 1 L L L L L [x]l = 1 [L ] = 1 L (2) b n 2L b n = 1 f(x) sin n L L x dx = 1 { L f(x) sin n 2L L L x dx + f(x) sin n L L x dx = 1 { L 1 sin n 2L L L x dx + sin n L L x dx L = 1 sin n L L x dx = 1 [ L L n cos n ] L L x = 1 (cos n cos ) n = 1 n {( 1)n 1 = { 2 n (n ) (n ) f(x) = a (a n cos n L x + b n sin n L x) = 1 [ ] 1 n {1 ( 1)n sin n L x (sin L x + 1 3 sin 3 L x + 1 5 sin 5 ) L x + [ = 1 2 + 2 = 1 2 + 2 1 2n 1 (2n 1) sin x L ]
L = ( 2L = 2) f(x) = 1 2 + 2 (sin x + 1 3 sin 3x + 1 sin 5x + ) 5, 2 P.17[ 2](1) f(x) L = 1( 2L = 2), T = 2 f(x) = f(x) = 1 2 + 2 (sin x + 1 3 sin 3x + 1 sin 5x + ) 5 { 1 ( x < 1) (1 x < 2)
2 [ 2 f(x) ] f(x) = a (a n cos nx + b n sin nx) a n = 1 f(x) cos nx dx (n =, 1, 2, ) b n = 1 f(x) sin nx dx (n = 1, 2, ) x f(x) f(x) = a 2 + a n cos nx, a n = 2 f(x) cos nx dx (n =, 1, 2, ) x f(x) f(x) = b n sin nx, b n = 2 f(x) sin nx dx (n = 1, 2, ) x x, L L [ 2L f(x) ] L f(x) = a (a n cos nx L a n = 1 L L L b n = 1 L L L f(x) cos nx L f(x) sin nx L + b n sin nx L ) dx (n =, 1, 2, ) dx (n = 1, 2, ) L x L f(x) f(x) = a 2 + a n cos nx L, a n = 2 L L f(x) cos nx L dx (n =, 1, 2, ) L x L f(x) f(x) = b n sin nx L, b n = 2 L L f(x) sin nx L dx (n = 1, 2, )
1.7 [] a x b a = x < x 1 < x 2 < < x n = b x 1, x 2,, x n 1,,, lim f(x i + e) = f(x i + ) (e >, i =, 1, 2,, n) e lim f(x i e) = f(x i ) (e >, i =, 1, 2,, n) e, f(x) a x b 1, 2 () x i [] x i, x i+1 f(x) xi+1 e 2 xi+1 lim f(x) dx = f e 1,e 2 i (x) dx x i +e 1 x i, f i (x), f(x) [x i, x i+1 ], f(x i + ) fi (x) = f(x) f(x i+1 ) (x = x i ) (x i < x < x i+1 ) (x = x i+1 ), f(x) b a f(x) dx = x1 x f (x) dx + x2 x 1 f 1 (x) dx + + xn x n 1 f n 1(x) dx,,,
[] f(x) f (x) a x b, f(x) [] f(x) 2, f(x) [] f(x) x, f(x) x f(x), x f(x + ) + f(x ) 2
[ ( )] P.17[ 2](2), f(x) = x ( x < ) f(x) = a (a n cos nx + b n sin nx) = 2 [ ] 2 cos nx {( 1)n 1 = 2 4 [ = 2 4 ( cos x + cos 3x + 3 2 cos (2n 1)x ] (2n 1) 2 n 2 cos 5x 5 2 +, f(x) x <, x = 2 4 ( ) cos 3x cos 5x cos x + + + 3 2 5 2, x =, f() = =,, cos = 1, = 2 4 (1 + 1 3 2 + 1 5 2 + ) 2, 4 1 + 1 3 + 1 2 5 + + 1 2 (2n 1) + = 2 2 8, ) P.29 1-6 2