Transcription

1 00 3 9

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3 log I II III II i

4 ) I II III ii

5 . x n, exp x, sin x, cos x d dx xn = nx n d exp x = exp x dx d sin x = cosx dx d cos x = sin x dx x n x n = x m+n exp(x + y) = exp x exp y sin(x + y) = sin x cos y + cos x sin y cos(x + y) = cos x cos y + sin x sin y = cos x + sin x +x x n = nx n +x exp x = exp x +x sin x = cos x +x cos x = sin x x n, exp x, sin x, cos x x m x n x m+n exp (x + y) = exp x exp y sin (x + y) = sin x cos y + cos x sin y cos (x + y) = cos x cos y sin x sin y cos x + sin x x m x n = x m+n = cos x + sin x )

6 ... df dx (operator) (advanced,forwaord) +x, (central) x, (backward,retarded) x f(x + ɛ) f(x) +x f(x) =, ɛ (.) x = f(x + ɛ ) f(x ɛ ), ɛ (.) f(x) f(x ɛ) x f(x) =. ɛ (.3) ɛ ɛ 0 ( ) lim f(x + ɛ) f(x) +xf(x) = lim ɛ 0 ɛ 0 ɛ = f (x) f(x + ɛ lim xf(x) = lim ) f(x ɛ ) ɛ 0 ɛ 0 ɛ f(x + ɛ = lim ) f(x) + f(x) f(x ɛ ) ɛ 0 ɛ = ɛ lim f(x + ) f(x) + ɛ 0 lim ɛ = f (x) + f (x) = f (x) df f(x + ( ɛ )) f(x) ɛ 0 lim f(x) f(x ɛ) xf(x) = lim ɛ 0 ɛ 0 ɛ = lim ɛ 0 = f (x) f(x + ( ɛ)) f(x) ɛ ɛ dx ( ) (averaging operator)

7 f(x + ɛ) + f(x) M +x f(x) = M x f(x) = f(x + ɛ ) + f(x ɛ ) f(x) + f(x ɛ) M x f(x) = (.4) (.5) (.6) (.7) f(x) ɛ 0 ( ) x = +x x M x = + ɛ 4 x E(shift operator) E +x f(x) = f(x + ɛ) E x f(x) = f(x ɛ) +x = E +x ɛ +x = E +x + f(x) ( ) f(x) f(x) = [x] x = ɛ 0 f(x ɛ ), f(x ɛ) M x f(x), M x f(x) (ɛ 0) f(x) = [ x] x = ɛ 0 f(x + ɛ) M +x f(x) (ɛ 0) f(x) ( ) 3

8 ( ) +x +x f(x) + g(x) f(x + ɛ) + g(x + ɛ) f(x) + g(x) = ɛ f(x + ɛ) f(x) g(x + ɛ) g(x) = + ɛ ɛ = +x f(x) + +x g(x) +x (αf(x)) αf(x + ɛ) αf(x) = ɛ f(x + ɛ) f(x) = α ɛ = α +x f(x) ( ) x, x... +x g(x)h(x) = [ +x g(x)]h(x + ɛ) + g(x)[ +x h(x)] (.8) x g(x)h(x) = [ x g(x)]h(x + ɛ/) + g(x ɛ/)[ x h(x)] (.9) x g(x)h(x) = [ x g(x)]h(x) + g(x ɛ)[ x h(x)] (.0). n n +xg(x)h(x) = n xg(x)h(x) = n xg(x)h(x) = n ( n i i=0 n ( n i i=0 n ( n k i=0 ) [ n i +x g(x)][ i +xh(x + (n i)ɛ)] (.) ) [ n i x g(x iɛ/)][ i xh(x + (n i)ɛ/)] (.) ) [ n i x g(x iɛ)][ i xh(x)] (.3) (.) 4

9 . n = g(x + ɛ)h(x + ɛ) g(x)h(x) +x g(x)h(x) = ɛ g(x + ɛ)h(x + ɛ) g(x)h(x + ɛ) + g(x)h(x + ɛ) g(x)h(x) = ɛ = [ +x g(x)]h(x + ɛ) + g(x)[ +x h(x)] (.). n = k (k =,, 3, ) (.) n +xg(x)h(x) = k i=0 ( ) k [ k i +x g(x)][ i i +xh(x + (k i)ɛ)] +x [ k +xg(x)h(x)] [ k ( ) ] k = +x [ k i +x g(x)][ i i +xh(x + (k i)ɛ)] i=0 k ( ) k [ = +x [ k i +x g(x)][ i i +xh(x + (k i)ɛ)] ] i=0 k ( ) k [[ k+ i = +x g(x)][ i i +xh(x + (k + i)ɛ)] + [ k i +x g(x)][ i+ +x h(x + (k i)ɛ)] ] = i=0 k i=0 [( ) k + i + ( )] k [ +x [ k+ i +x g(x)][ i+ +x h(x + (k + i)ɛ)] ] i k+ ( ) k + = [ k+ i +x g(x)][ i i +xh(x + (k + i)ɛ)] i=0 n = k + (.) (.) n..3 f = f(x), x = x(t) t δ,x ɛ f(x(t)). f(x(t)) +t f(x(t)) = f(x(t + δ)) f(x(t)) δ (.4) 5

10 x(t) (.4) +x x(t) = x(t + δ) x(t) δ +t f(x(t)) = f(x(t) + +xx(t)) f(x(t)) δ = +x x(t) f(x(t) + +xx(t)) f(x(t)) δ +x x(t) x ɛ( 0) ɛ = δ +x x(t) = x(t + δ) x(t) f(x(t) + ɛ) f(x(t)) +t f(x(t)) = +x x(t) (.5) ɛ = +x x(t) +x f(x) (.6) +x f(x). f(x(t)) f(x(t) ɛ) t f(x(t)) = x x(t) (.7) ɛ = x x(t) x f(x) (.8) ɛ = δ x x(t) = x(t) x(t δ) 3. f(x(t)) t f(x(t)) = f(x(t + δ/)) f(x(t δ/)) δ (.9) x(t) x x(t) = x(t + δ/) x(t δ/) δ 6

11 M t x(t) (.9) x(t + δ/) + x(t δ/) M t x(t) = x(t + δ/) x(t δ/) + x(t δ/) = = δ xx(t) + x(t δ/) x(t + δ/) + x(t δ/) M t x(t) = x(t + δ/) (x(t + δ/) x(t δ/)) = = x(t + δ/) δ xx(t) f(x(t + δ/)) f(x(t δ/)) t f(x(t)) = δ = [ ( f M t x(t) + δ ) ( δ tx(t) f M t x(t) δ )] tx(t) = t x(t) f( M t x(t) + δ tx(t) ) f ( M t x(t) δ tx(t) ) δ t x(t) ɛ = δ t x(t) = x(t + δ/) x(t δ/) t f(x(t)) = t x(t) f( M t x(t) + ɛ/ ) f ( M t x(t) ɛ/ ) (.0) ɛ = t x(t) x f(x) (.).3 d dx xn = nx n (.) x 3 +x x 3 = (x + ɛ)3 x 3 ɛ = x3 + 3ɛx + 3ɛ x + ɛ 3 x 3 ɛ = 3x + 3ɛx + ɛ 7

12 ( ɛ x x 3 = 3x + ) x x 3 = 3x 3ɛx + ɛ ɛ 0 3x x (.) x x 3 x 3 = x(x ɛ)(x ɛ) +x x 3 = (x + ɛ)x(x ɛ) x(x ɛ)(x ɛ) ɛ x n = x(x ɛ)(x ɛ) {x (n )ɛ} = 3x(x ɛ) (.3) (.3) +x x 3 = x x 4, x 5 (.) +x x n = nx n (.4) ( x 0 = ) (.4) (.4) Proof. (i) n= = +x x (x + ɛ) x = = ɛ = x 0 =... = (ii) n= = +x x (x + ɛ)x x(x ɛ) = = x ɛ = x = x... = 8

13 (iii) n +x x n = x n n + = +x x n+ = +x {x n (x nɛ)} = [ +x x n ]{x (n )ɛ} + x n [ +x (x nɛ)] = (nx n ){x (n )ɛ} + x n = (n + )x n =... = (.4) x { } { } { } { x ((n)) = x x x + x + (n )ɛ (n 3)ɛ x n = x(x + ɛ)(x + ɛ) {x + (n )ɛ} (.) (n 3)ɛ (n )ɛ x x ((n)) = nx ((n )) n : (.5) x x n = nx n n : (.6) x ((0)) =, x 0 = } n < 0, n = 0 n {x (k )ɛ} (n > 0) k= x n = (n = 0) n (n < 0) (x + kɛ) k= n { ( ) } n + x k ɛ (n > 0) x ((n)) k= = (n = 0) (n < 0) x (( n )) n {x + (k )ɛ} (n > 0) k= x n = (n = 0) n (n < 0) (x kɛ) k= (.7) (.8) (.9) 9

14 n k= F k (x) = F (x) F (x) F 3 (x) F n (x) F n (x) x n, x ((n)), x n d dx xn = nx n +x x n = nx n x n n- n P n (x) = x n n =, n = P (x) = x + a P (x) = x + bx + c +x P (x) = P (x) a, b, c = +x P (x) = (x + ɛ) x ɛ = x + ɛ + b = P (x) = (x + a) = x + a + b x + ɛ x ɛ + c 0... a = ɛ + b 0

15 a, b = a ɛ, c a = c = 0 b = ɛ P (x) = x = x P (x) = x = x(x ɛ) P 3 (x) = x 3 + px + qx + r 8 (.4) p, q, r r = 0 p = 3ɛ, q = ɛ, r P 3 = x 3 = x(x ɛ)(x ɛ) n P n (x) = x n = x(x ɛ)(x ɛ) {x (n )ɛ} (n > 0) n = 0 P 0 (x) = C C +x P (x) = (x ɛ) x ɛ =... C =... P 0 (x) = x 0 = n < 0 P = x + a P = x + bx + c +x P (x) = P (x) a, b, c a, b = a + ɛ, c = a(a + ɛ) (.30) a = ɛ b = 3ɛ, c = 3ɛ P = x = x + ɛ P = x = (x + ɛ)(x + ɛ) P n (x) = x n = n k= (x + kɛ) (n < 0)

16 a = 0 b = ɛ, c = 0 P n (x) = n k= P = x P = x(x + ɛ) {x + (k )ɛ} (n < 0) (.3) n > 0 9 (.7) (.3) (.30) a +x x n = nx n x x ((n)) = x ((n )) x x n = x n.4 log d dx log x = x log (.3) log x log x log +x g(x) = f(x) (.33) g(x + ɛ) g(x) = f(x) ɛ x x = nx (n = 0,,, 3, ) x ɛ g((n + )ɛ) g(nɛ) = ɛf(nɛ)

17 n 0 n g(ɛ) g(0) = ɛf(0) g(ɛ) g(ɛ) = ɛf(ɛ) g(3ɛ) g(ɛ) = ɛf(ɛ). g(nɛ) g((n )ɛ) = ɛf((n )ɛ) n g(nɛ) = g(0) + ɛ f(kɛ) (.34) k=0 (.34) x n g(x) = g(0) + ɛ f(kɛ) (.35) (.33) k=0 (.33) f(x) g(x) (.35) f(x) g(x) f(x) +x g(x) (.35) g(x) = Σ +x f(x), g(0) = C n Σ +x f(x) = C + ɛ f(kɛ) (.36) k=0 C = g(0) C Σ +x f(x) C log dx = C + log x (.37) x x = 0 log x = n Σ +x x = C + ɛ kɛ Σ +x x = C n C n log (x) k= Log (x) = n (.38) (.39) (.40) (.4) 3

18 .5.5. exp x.. 3. d exp x = exp x dx exp x x=0 = exp x = n= n! xn 4. exp(x + y) = exp x exp y,,. exp x exp x = n= n! xn,. exp x d exp x = exp x dx exp x = n= n! xn,,.5. +x exp x exp x.. +x exp x = exp x (.4) exp x x=0 = 4

19 (.4), q +x φ(x) = qφ(x) (.43) x x = mɛ ( m=0,,,3 m ) x φ(mɛ) φ m (.43) φ m+ φ m ɛ = qφ m φ m+ = ( + qɛ)φ m (.44) φ 0 φ = ( + qɛ)φ 0 φ = ( + qɛ)φ φ 3 = ( + qɛ)φ... φ m = ( + qɛ)φ m+ m φm (.43) exp x exp x a k exp x exp x = a k x k (.43), x n k=0 k=0 +x x n = nx n a k kx k = a k x k k=0 [a k+ (k + ) a k ]x k = 0 k=0 x k 0 a k+ (k + ) a k = 0 exp x x=0 = a 0 = a k = k! 5

20 exp x = k=0 k! xk exp x exp x = φ m, x = mx ( φ 0 = ) φ m+ φ m ɛ = φ m (.45) φ m = ( + ɛ)φ m exp x x φ m = ( + ɛ) m exp x = ( + ɛ) x/ɛ exp x (.45 ) x = mɛ, m, exp x ( + ɛ) x/ɛ = k=0 k! xk x = mɛ, y = nɛ, m, n, exp x = φ m exp y = φ n exp x exp y = φ m,n exp x + y = φ m+n. φ m+n n (.44) φ m+n+ φ m+n ɛ = φ m+n proof) +m f(l(m)) = +m l(m) +l f(l) l = n + m +m l(m) = +l f(l) = +m f(m + n) l φ l = φ l = φ m+n 6

21 . φ m,n n (.44 ) φ m+,n φ m,n ɛ = φ m,n ( n m ) 3. φ m+n φ m,n m = 0 φ 0,n = φ 0+n = φ n (.44 ) φ m+n = φ m,n exp x + y = exp x exp y trivial t=0 t! xt s=0 s! ys = k=0 (x + y)k k!.5.3 x q e qx.. x e qx = qe qx (.46) e qx x=0 = x x = mɛ(m = 0,,, 3,...) x m (.46) e qx ϕ m ϕ m ϕ m ɛ = qϕ m ϕ m = ϕ m ( qɛ) 7

22 ϕ 0 ϕ = ϕ 0 ( qɛ) ϕ = ϕ ( qɛ) ϕ m = ϕ m ( qɛ) m ϕ m qɛ (.46) e qx e q x a k e q x e qx = x n k=0 x x n = nx n a k kx k = q a k x k k=0 k=0 [a k+ (k + ) a k q]x k = 0 k=0 x k (k = 0,,,...) 0 a k+ (k + ) a k q = 0 (e x x=0 = ) a 0 = a k = qk k! e qx = k! xk q k e qx e qx = ϕ m, x = mɛ ϕ 0 = k=0 ϕ m ϕ m ɛ = qϕ m 8

23 qɛ ϕ m = ( qɛ) m e qx = ( qɛ) x ɛ e qx x = mɛ, m ( qɛ) x ɛ = k=0 k! xk q k e qx x = mɛ y = nɛ m n e qx = ϕ m, e qy = ϕ n, e qx e qy = ϕ m,n, e qx+y = ϕ m+n. ϕ m+n n (.46) ϕ m+n ϕ m+n ɛ = qϕ m+n. ϕ m,n n (.46) ϕ m,n ϕ m,n ɛ = qϕ m,n 3. ϕ m+n ϕ m,n m = 0 ϕ 0,n = ϕ 0+n = ϕ n (.46) ϕ m+n = ϕ m,n e qx+y = e qx e qy 9

24 .6.7 a) M x x ((k)) = xx ((k )) k > 0 M x x ((k)) =M x ( k {x ( k + s)ɛ}) s= = k { [x + ɛ (k + k s)ɛ] + [x ɛ (k + s)ɛ]} s= s= = {(x + k k ɛ) [x ( k s)ɛ] + (x k k ɛ) [x ( k + s)ɛ]} =xx ((k )) s= b) s= x f(x)g(x) = [ x f(x)][m x g(x)] + [M x f(x)][ x g(x)] = x f(x)g(x) = [ x f(x)]g(x + ɛ ) + f(x ɛ )[ xg(x)] = ɛ (f(x + ɛ )g(x + ɛ ) f(x ɛ )g(x + ɛ ) + f(x ɛ )g(x + ɛ ) f(x ɛ )g(x ɛ )) = ɛ {f(x + ɛ )g(x + ɛ ) f(x ɛ )g(x ɛ )} = [ x f(x)][m x g(x)] + [M x f(x)][ x g(x)] = ɛ {f(x + ɛ ) f(x ɛ )}{g(x + ɛ ) + g(x ɛ )} + ɛ {f(x + ɛ ) + f(x ɛ )}{g(x + ɛ ) g(x ɛ )} = ɛ {f(x + ɛ )g(x + ɛ ) f(x ɛ )g(x ɛ )} = ɛ {f(x + ɛ )g(x + ɛ ) f(x ɛ )g(x ɛ )} = 0

25 c ) kx [ xx ((k)) ] = [x ((k )) ] = kx x[x ((k)) ] = kx {[ xx ((k)) ][M x x ((k)) ] + [M x x ((k)) ][ x x ((k)) ]} = kx x[x ((k)) ][M x x ((k)) ] (a) = kx [kx((k )) ][x x ((k )) ] = [x ((k )) ] =.8 c ) (k + l)x x[x ((k)) x ((l)) ] = x ((k )) x ((l )) = (k + l)x x[x ((k)) x ((l)) ] = (k + l)x {[ xx ((k)) ][M x x ((l)) ] + [M x x ((k)) ] x [x ((l)) ]} = (k + l)x {[kx((k )) ][x x ((l )) ] + [x x ((k )) ][l x ((l)) ]} = x ((k )) x ((l )) = a) = +x (f(x)g(x) f(x ɛ)g(x ɛ) ɛ = (f(x + ɛ)g(x + ɛ) f(x)g(x) f(x)g(x) + f(x ɛ)g(x ɛ)) ɛ = {(f(x + ɛ) f(x) + f(x ɛ))g(x) ɛ f(x + ɛ)g(x) + f(x ɛ)g(x)f(x + ɛ)g(x + ɛ) + f(x ɛ)g(x ɛ)} = [ +x x f(x)]g(x) + (f(x + ɛ)g(x + ɛ) f(x + ɛ)g(x) f(x ɛ)g(x) + f(x ɛ)g(x ɛ)) ɛ = [ +x x f(x)]g(x) + {f(x)(g(x + ɛ) g(x) + g(x ɛ)) ɛ f(x)g(x ɛ) + f(x)g(x) f(x)g(x ɛ) + f(x + ɛ)g(x + ɛ) f(x + ɛ)g(x) f(x ɛ)g(x) + f(x + ɛ)g(x + ɛ)} = [ +x x f(x)]g(x) + f(x)[ +x x g(x)] + {f(x + ɛ)(g(x + ɛ) g(x)) f(x)(g(x + ɛ) g(x)) ɛ + f(x)(g(x) g(x ɛ)) f(x ɛ)(g(x) g(x ɛ))} = [ +x x f(x)]g(x) + f(x)[ +x x g(x)]

26 + {(f(x + ɛ) f(x))(g(x + ɛ)g(x)) + (f(x) f(x ɛ))(g(x) g(x ɛ))} ɛ = [ +x x f(x)]g(x) + [ +x f(x)][ +x g(x)] + [ x f(x)][ x g(x)] + f(x)[ +x x g(x)] b) = [ +x ρ(x) ɛ (y k(x) y k (x ɛ))]y l (x) y k (x)[ +x ρ(x) ɛ (y l(x) y l (x ɛ))] = ɛ {ρ(x + ɛ)(y k(x + ɛ) y k (x)) ρ(x)(y k (x)y k (x ɛ))}y l (x) y k (x) ɛ {ρ(x + ɛ)(y l(x + ɛ) y l (x)) ρ(x)(y l (x)y l (x ɛ))} = ɛ {ρ(x + ɛ)(y k(x + ɛ)y l (x) y k (x)y l (x) y k (x)y l (x + ɛ) + y k (x)y l (x)) ρ(x)(y k (x)y l (x) y k (x ɛ)y l (x) y k (x)y l (x) + y k (x)y l (x ɛ))} = ɛ {ρ(x + ɛ)(y k(x + ɛ)y l (x) y k (x)y l (x + ɛ)) ρ(x)(y k (x)y l (x ɛ) y k (x ɛ)y l (x))} = +x ɛ ρ(x)(y k(x)y l (x ɛ) y k (x ɛ)y l (x)) = +x ρ(x) ɛ {(y k(x) y k (x ɛ))y l (x) y k (y l (x) y l (x ɛ))} = +x ρ(x){[ +x y k (x)]y l (x) y k (x)[ +x y l (x)]} b) (ρ(x)y k(x)) y l (x) y k (ρ(x)y l(x)) = (ρ y k + ρy k)y l y k (ρ y l + ρy l ) = ρ (y ky l y k y l) + ρ(y ky l y k y l ) = ρ (y ky l y k y l) + ρ(y ky l y k y l) = {ρ(y ky l y k y l)}.7 sin mx, cos mx π π d dx sin mx + m sin mx = 0 d dx cos mx + m cos mx = 0 0, m n, m, n cos mx cos nx dx = π, m = n, m π, m = n = 0

27 π π 0, m n, m, n sin mx sin nx dx = π, m = n 0, m π π cos mx sin nx dx = 0, m, n W N (m, k) = e imk π N N m, k i N k=0 0, m n mod N W N (m, k)w N ( n, k) = N, m = n mod N W N (m, k) k.7. sin x,cos x. d dx sin x = cos x (.47) d dx cos x = sin x sin x x=0 = 0 cos x x=0 = ( ) k sin x = (k + )! xk+ k=0 ( ) k cos x = (k)! xk k=0 (.48) (.49) sin x + cos x = (.50) sin (x + y) = sin x cos y + cos x sin y cos (x + y) = cos x cos y sin x sin y (.5) (.49), (.50), (.5) (.47) (.48) 3

28 (.49) (.47) a k, b k (k = 0,,, ) sin x = a k x k k=0 (.5) cos x = b k x k k=0 (.47) k=0 d dx xk = kx k a k kx k = b k x k, k=0 k=0 b k kx k = a k x k k=0 x (a k+ (k + ) b k )x k = 0, (.53) k=0 (b k+ (k + ) + a k )x k = 0 (.54) k=0 x 0 k = 0,,, a k+ (k + ) b k = 0, b k+ (k + ) + a k = 0 a k, b k b k a k+ = (k + ), a k b k+ = (k + ) sin x x=0 = 0, cos x x=0 = (.5) a 0 = 0, b 0 = k = 0,,, a k+ = ( )k (k + )!, a k = 0 b k+ = 0, b k = ( )k (k)! 4

29 (.5) (.49) ( ) k sin x = (k + )! xk+ k=0 ( ) k cos x = (k)! xk k=0 (.50) (.50) x () d dx (sin x + cos x) = d sin x dx sin x + d cos x dx cos x = (cos x sin x sin x cos x) = 0 sin x + cos x = const. const. = (.50) sin x + cos x = (.5) (.47) sin x cos x d φ(x) + φ(x) = 0 (.55) dx φ(x) φ(x) x=0 = c 0, d dx φ(x) x=0 = c φ(x) x=0 = sin y, d dx φ(x) x=0 = cos y (.56) φ (x, y) = cos y sin x + sin y cos x φ (x, y) = sin (x + y) φ (x, y) φ (x + y) (.55) (.56) φ (x, y) = φ (x, y) sin (x + y) = cos y sin x + sin y cos x (.57) sin x (.57) y cos x cos (x + y) = cos y cos x sin y sin x 5

30 (.47) (.48) sin (x + y) = sin x cos y + cos x sin y cos (x + y) = cos x cos y sin x sin y d φ(x) + φ(x) = 0 dx φ(x) φ(x) x=0 = c 0, d dx φ(x) x=0 = c ( ) y + y = 0, y(0) = c 0, y (0) = c z + z = 0, z(0) = c 0, z (0) = c (y z ) + (y z) = 0, (y z)(0) = 0, (y z )(0) = 0 ω = y z ω + ω = 0, ω(0) = ω (0) = 0 d dx (ω + ω ) d dx (ω + ω ) = ω ω + ωω = ω (ω + ω) = 0 ω (x) + ω(x) = ω (0) + ω(0) = 0 ω + ω = 0 ω = 0, ω = 0 y z = 0 y = z ( ).8.8. x x = nɛ, n x, x n, x n, x n+,,( n ) f(x) ɛ n = x n+ xn n ( n ) +n, n, M +n +n f(x n ) = f(x n+ ) f(x n ) n f(x n ) = f(x n ) f(x n ) M +n f(x n ) = [f(x n+) + f(x n )] 6

31 f(x n+ ) f(x n ) x n+ x n = ɛ n +n f(x n ) f(x n ) f(x n ) = x n x n ɛ n n f(x n ) exp < iλ, x n > ɛ n +n exp < iλ, x n > = iλm +n exp < iλ, x n > (.58) exp < iλ, x n > n=0 = exp < iλ, x n > n=n = (.59) λ exp < iλ, x n+ > exp < iλ, x n > = iλɛ n [exp < iλ, x n+ > + exp < iλ, x n >] ( ) ( λɛ n exp < iλ, x n+ > = + ) λɛ n exp < iλ, x n > exp < iλ, x n+ > = + λɛ n λɛ exp < iλ, x n > n exp < iλ, x n > = + λɛ n λɛ exp < iλ, x n > n = + λɛ n + λɛ n λɛ n λɛ exp < iλ, x n > n = + λɛ n λɛ + λɛ 0 n λɛ exp < iλ, x 0 > 0 = n j=0 + λɛ j λɛ j exp < iλ, x n > exp < iλ, x n >, exp < iλ, x n > +n exp < iλ, x n > exp < iλ, x n > +n exp < iλ, x n > exp < iλ, x n > = [ +n exp < iλ, x n >][M +n exp < iλ, x n >] + [M +n exp < iλ, x n >][ +n exp < iλ, x n >] = i(λ λ )ɛ n [M +n exp < iλ, x n >][M +n exp < iλ, x n >], ( (.58) ) 7

32 n 0 N exp < iλ, x n > exp < iλ, x n > N n=0 N = i(λ λ ) ɛ n [M +n exp < iλ, x n >][M +n exp < iλ, x n >] n=0 (.59) λ λ N n=0 θ j (λ) ɛ n [M +n exp < iλ, x n >][M +n exp < iλ, x n >] = 0 tan θ j(λ) = λɛ j exp < iλ, x n > exp < iλ, x n > = = = n j=0 n j=0 n j=0 cos θ j(λ) + i sin θ j(λ) cos θ j(λ) sin θ j(λ) exp [i θ j(λ)] exp [ i θ j(λ)] exp [iθ j (λ)] n = exp i θ j (λ) j=0 exp < iλ, x n > sin < λ, x n >, cos < λ, x n > n exp < iλ, x n > = exp i θ j (λ) n sin < λ, x n > = sin i θ j (λ) j=0 j=0 n cos < λ, x n > = cos i θ j (λ) j=0 (.58) +n sin < λ, x n > = λɛ n M +n cos < λ, x n > +n cos < λ, x n > = λɛ n M +n sin < λ, x n > 8

33 +n sin < λ, x n > = λɛ n M +n cos < λ, x n > +n cos < λ, x n > = λɛ n M +n sin < λ, x n > +n sin < λ, x n > = λɛ n M +n cos < λ, x n > +n cos < λ, x n > = λɛ n M +n sin < λ, x n > +n [sin < λ, x n > cos < λ, x n > + cos < λ, x n > sin < λ, x n >] = [ +n sin < λ, x n >] [M +n cos < λ, x n >] + [M +n sin < λ, x n >] [ +n cos < λ, x n >] + [ +n cos < λ, x n >] [M +n sin < λ, x n >] + [M +n cos < λ, x n >] [ +n sin < λ, x n >] = (λ + λ )ɛ n [M +n cos < λ, x n >] [M +n cos < λ, x n >] (λ + λ )ɛ n [M +n sin < λ, x n >] [M +n sin < λ, x n >] (.60) λ λ +n [sin < λ, x n > cos < λ, x n > cos < λ, x n > sin < λ, x n >] = (λ λ )ɛ n [M +n cos < λ, x n >] [M +n cos < λ, x n >] + (λ λ )ɛ n [M +n sin < λ, x n >] [M +n sin < λ, x n >] (.6) (.60) (.6) n 0 N sin < λ, x n > n=0 = 0, sin < λ, x n > n=n = 0 sin < λ, x n > n=0 = 0, sin < λ, x n > n = N = 0 λ = λ N n=0 N n=0 ɛ n [M +n sin < λ, x n >] [M +n sin < λ, x n >] ɛ n [M +n cos < λ, x n >] [M +n cos < λ, x n >] 9

34 .8. sin qx, cos qx +x sin qx, cos qx { +x sin qx = q cos qx +x cos qx = q sin qx (.6) { sin qx x=0 = 0 cos qx x=0 = (.63) a k, b k sin qx, cos qx sin qx = a k x k q k k=0 cos qx = b k x k q k (.6) k=0 +x x n = nx n a k kx k q k = q b k x k q k k=0 k=0 b k kx k q k = q a k x k q k k=0 k=0 a k, b k a k+ = b k (k + ) b k+ = a k (k + ) sin qx, cos qx { a 0 = 0 b 0 = a k+ = ( )k (k + )!, a k = 0 b k+ = 0, b k = ( )k (k)! 30

35 sin qx, cos qx ( ) k sin qx = (k + )! xk+ q k+ cos qx = k=0 k=0 ( ) k (k)! xk q k sin qx cos qx +xϕ(x) + q ϕ(x) = 0 (.64) x = nɛ (n = 0,,, ) ϕ((n + )ɛ) = ϕ((n + )ɛ) ( + q ɛ )ϕ(nɛ) ϕ(x) ϕ(x) x=0 = c, ϕ(x + ɛ) x=0 = c +x ϕ(x) x=0 = [(ϕ(x + ɛ) ϕ(x))/ɛ] x=0 = (c c )/ɛ ϕ(x), ϕ(x + ɛ) ϕ(x), +x ϕ(x) ϕ (x, y) = cos qy sin qx + sin qy cos qx ϕ (x, y) = sin q(x + y) ϕ (x, y), ϕ (x, y) +xϕ (x, y) = +x [ +x {cos qy sin qx + sin qy cos qx} ] = q +x [ cos qy cos qx sin qy sin qx ] = q [ cos qy sin qx + sin qy cos qx ] = q ϕ (x, y) ] +xϕ (x, y) = +x [ +x sin q(x + y) = q +x cos q(x + y) = q sin q(x + y) = q ϕ (x, y) ϕ (x, y), ϕ (x, y) (.64) ϕ (x, y) x=0 = ϕ (x, y) x=0 = sin qy +x ϕ (x, y) x=0 = +x ϕ (x, y) x=0 = q cos qy 3

36 x = mɛ, y = nɛ, m, n ϕ (x, y) = ϕ (x, y) sin q(x + y) = cos qy sin qx + sin qy cos qx (.65) cos q(x + y) = cos qy cos qx sin qy sin qx (.66) sin qx + cos qx = (.67) +x = 0 0 +x (sin qx + cos qx) = [ +x sin qx] sin q(x + ɛ) + sin qx[ +x sin qx] +[ +x cos qx] cos q(x + ɛ) + cos qx[ +x cos qx] = q cos qx sin q(x + ɛ) q sin qx cos q(x + ɛ) 0 sin qx + cos qx (.66) cos q(x + y) = cos qy cos qx sin qy sin qx y = x = sin qx + cos qx (.66) m, n y = x.8.3 y = x sin qx, cos qx ( ) k sin qx = (k + )! xk+ q k+ k=0 cos qx = k=0 ( ) k (k)! xk q k 3

37 x x ( ) k sin q( x) = (k + )! ( x)k+ q k+ k=0 cos q( x) = k=0 sin q( x) = k=0 cos q( x) = k=0 ( ) k (k)! ( x)k q k ( ) k (k + )! xk+ q k+ ( ) k (k)! xk q k sin qx, cos qx ( ) k sin qx = (k + )! xk+ q k+ k=0 cos qx = k=0 ( ) k (k)! xk q k sin q( x) = sin qx cos q( x) = cos qx x x n = nx n sin qx, cos qx x sin qx = q cos qx x cos qx = q sin qx (.68) sin qx x=0 = 0 cos qx x=0 = (.68) sin qx sin q(x ɛ) = qɛ cos qx cos qx cos q(x ɛ) = qɛ sin qx x ɛ sin q(x + ɛ) sin qx = qɛ cos q(x + ɛ) cos q(x + ɛ) cos qx = qɛ sin q(x + ɛ) (.68) +x sin qx = q cos q(x + ɛ) +x cos qx = q sin q(x + ɛ) 33

38 (.67) sin qx + cos qx = (.66) y = x cos q(x + y) = cos qy cos qx sin qy sin qx cos q(x + ( x)) = cos q( x) cos qx sin q( x) sin qx = cos qx cos qx + sin qx sin qx (.67).8.4 e x x ix e x = e ix = k= k= k! xk (i) k k! xk D f(z) D D D f (z) D f(z) = f (z) f (z) f(z) (analytic continuation) f f(x) = c n (x a) n Taylar k=0 f(x) = e x R C R C C f (x) = e x = k= k! xk R f(x) = f (x) f (z) f(z) e ix = cos x + i sin x e ix = cos x i sin x 34

39 sin x = ex e ix cos x = ex + e ix q exp(iqx) = cos qx + i sin qx exp( iqx) = cos qx i sin qx sin qx = [exp(qx) exp( iqx)] cos qx = [exp(qx) + exp( iqx)].9 d sin x = cos x dx d cos x = sin x dx +x sin qx = cos qx +x cos qx = sin qx (.69) sin x + cos x = (.70) sin qx sin qx + cos qx cos x = (.70) (.70) sin, cos (.69) (.70) sin q < x >, cos q < x > (.70) sin q < x > + cos q < x > = g (x) + h (x) ɛ +x [g (x) + h (x)] = g (x + ɛ) + h (x + ɛ) g (x) h (x) = [g(x + ɛ) g(x)][g(x + ɛ) + g(x)] + [h(x + ɛ) h(x)][h(x + ɛ) + h(x)] (.7) 35

40 0 g(x + ɛ) g(x) ɛ h(x + ɛ) + h(x) h(x + ɛ) h(x) ɛ (.7) g(x + ɛ) + g(x) ɛ 0 (.73) +x g(x) M +x h(x) (.73) +x h(x) M +x g(x) (.74) g(x + ɛ) g(x) h(x + ɛ) + h(x) lim lim ɛ 0 ɛ ɛ 0 ɛ g (x) h(x) (.74) h (x) g(x) g(x) = sin q < x >, h(x) = cos q < x > (sin q < x >) cos q < x > (cos q < x >) sin q < x > sin, cos (.73)) (.74) g(x) = sin q < x >, h(x) = cos q < x > +x sin q < x > M +x cos q < x > +x cos q < x > M +x sin q < x > (.75) +x sin q < x > M +x cos q < x > = +x cos q < x > M +x sin q < x > q +x sin q < x > = qm +x cos q < x > (.76) +x cos q < x > = qm +x sin q < x > (.77) +x [sin q < x > + cos q < x >] = +x sin q < x >M +x sin q < x > + +x cos q < x >M +x cos q < x > = qm +x cos q < x >M +x sin q < x > qm +x sin q < x >M +x cos q < x > = 0 36

41 +x [sin q < x > + cos q < x >] = 0 (.78) sin q < x > + cos q < x > = constant constant : (.79) constant sin, cos sin q < x > x=0 = 0 cos q < x > x=0 = (.7) (.7) g(x + ɛ) g(x) + ɛ) h(x) = h(x h(x + ɛ) + h(x) g(x + ɛ) + g(x) = a g(x + ɛ) g(x) = ah(x + ɛ) + h(x) h(x + ɛ) h(x) = ag(x + ɛ) + g(x) a = ɛ g(x + ɛ) g(x) = ɛ h(x + ɛ) + h(x) h(x + ɛ) h(x) = ɛ g(x + ɛ) + g(x) +x g(x) = M +x h(x) +x h(x) = M +x g(x) g(x) = sin q < x >, h(x) = cos q < x > sin, cos d sin q < x > = q cos q < x > dx d cos q < x > = q sin q < x > dx (.75) q, (.76), (.77) sin q < x > + cos < x > = (.80) 37

42 e iθ = cos θ + i sin θ exp(iq < x >), exp( iq < x >) (.80) sin q < x > + cos < x > exp(iq < x >) = cos q < x > +i sin q < x > exp( iq < x >) = cos q < x > i sin q < x > sin q < x >= [exp(iq < x >) exp( iq < x >)] i cos q < x >= [exp(iq < x >) + exp( iq < x >)] = 4 [exp(iq < x >) exp( iq < x >)] + [exp(iq < x >) + exp( iq < x >)] = exp(iq < x >) exp(iq < x >) exp(iq < x >) exp(iq < x >) = (.8) (.8) +x exp(iq < x >) exp( iq < x >) = [ +t exp(iq < x >)][M +t exp( iq < x >)] (.76), (.77) 0 = +[M +t exp(iq < x >)][ +t exp( iq < x >)] +t exp(iq < x >) = iqm +t exp(iq < x >) (.8) +t exp( iq < x >) = iqm +t exp( iq < x >) (.83) exp(iq < x >) x=0 = (.84) exp( iq < x >) x=0 = (.85) φ(n) = exp(iq < x >), φ (n) = exp( iq < x >) (.8), (.83) ( x = nɛ, n ) φ(n + ) φ(n) ɛ φ(n + ) + φ(n) = iq φ(n + ) = + iqɛ iqɛ φ(n) ( + iqɛ = iqɛ φ (n + ) φ (n) ɛ ) n+ = iq φ (n + ) + φ (n) φ (n + ) = iqɛ + iqɛ φ (n) ( iqɛ = + iqɛ ) n+ 38

43 exp(iq < x >) = ( ) +iqɛ n exp( iq ( n iqɛ < x >) = +iqɛ iqɛ) ( x = nɛ, n ), exp(iq < x >), exp( iq < x >) x θ qɛ/ = tan(ɛθ) ( ) n + iqɛ/ exp(iq < x >) = iqɛ/ ( ) n + i tan(ɛθ/) = i tan(ɛθ/) ( ) n cos(ɛθ/) + i sin(ɛθ/) = cos(ɛθ/) i sin(ɛθ/) = (cos(ɛθ/) + i sin(ɛθ/)) n = e iθx ( ) n iqɛ/ exp( iq < x >) = + iqɛ/ ( ) n i tan(ɛθ/) = + i tan(ɛθ/) ( cos(ɛθ/) i sin(ɛθ/) = cos(ɛθ/) + i sin(ɛθ/) ) n = (cos(ɛθ/) + i sin(ɛθ/)) n = e iθx exp(iq < x >) exp( iq < x >) = exp(iq < x >) = e iθx exp( iq < x >) = e iθx (.9), (.9) sin < x > = sin θx cos < x > = cos θx ( qɛ = tan(ɛθ/)) sin q < x > + cos q < x >=.0 3 A = (a, a, a 3 ) B = (b, b, b 3 ) ( A B) ( A B) a b + a b + a 3 b 3. ( A B) 0 A = (a, a, a 3 ) B = (b, b, b 3 ) 3 e, e, e 3, e = (, 0, 0), e = (0,, 0), e 3 = (0, 0, ) ( e e ) = ( e e 3 ) = ( e e 3 ) = 0 39

44 3 A = (a, a, a 3 ) A = a e + a e + a 3 e 3 3 A = a j e j j= A a, a, a 3 e, e, e 3 A a = ( A e )/( e e ), a = ( A e )/( e e ), a 3 = ( A e 3 )/( e 3 e 3 ), a j = ( A e j )/( e j e j ), j =,, 3 3 n (n =,, 3, 4, ) n A = (a, a, a 3,, a n ) B = (b, b, b 3,, b n ) ( A B) ( A B) n a j b j j= ( A B) 0 n A B n n A = (a, a, a 3,, a n ) n e = (, 0, 0,, 0), e = (0,, 0,, 0), e 3 = (0, 0,,, 0),, e n = (0, 0, 0,, ) A = n a j e j j= A a, a, a 3,, a n e, e, e 3,, e n A a j = ( A e j )/( e j e j ), j =,,, n. A = n j= A e j e j e j e j a p k, (k = 0,, ) 3 P 0, = (,, ), P, = (, 0, ), P, = (,, ). 40

45 b P k,3 (k = 0,,, 3) 4 P 0,3 = (,,, ), P,3 = (, 3, 3, ), P,3 = (,,, ), P 3,3 = (, 3, 3, ). N + P k,n N k = 0,,,, N n + n P k,n (n) P k,n (n) k P 0,N (n) =, P,N (n) = n N, P,N (n) = 6 n N P 3,N (n) = n N. + 6 n(n ) N(N ), n(n ) n(n )(n ) N(N ) N(N )(N ), P k,n (n) = k ( )( ) k k + r n(n ) (n r + ) ( ) r r r N(N ) (N r + ) r=0 P k,n P l,n N P k,n (n)p,n (n) = 0, k l n=0 P k,n Legendre N y k,n N k =,,, N n y k,n (n) = sin Knɛ, ɛ = π N y k,n y l,n N n= y k,n (n)y l,n (n) = 0, k l (mod N) y k,n (n) sin kx N ( A B) N a n b n n= 4

46 ɛ = a/n, a nɛ = x Nɛ = a, f n = f(x), ɛ dx lim (a/n) N f n = n n= a 0 f(x)dx. P k,n (n) = k ( )( ) k k + r n(n ) (n r + ) ( ) r r r N(N ) (N r + ) r=0 nɛ = x, Nɛ = lim N n(n ) (n r + ) N(N ) (N r + ) = lim ɛ 0 x(x ɛ) (x (r )ɛ) ( ɛ) ( (r )ɛ) =x r P k,n (n) = k ( )( ) k k + r ( ) r x r r r r=0 k P 0 (x) =, P (x) = x, P (x) = 6x + 6x, P 3 (x) = x + 30x 0 x 3, Pk (x) k Legendre /N lim n n= N P k P l = ( P k P l ) 0 P k (x)p l (x)dx = 0, k l. y k,n (x) = sin knɛ nɛ = x, Nɛ = π lim y k,n (x) = sin kx N 4

47 π/n ( y k y l ) π 0 sin kx sin lxdx = 0, k l, strum-liouvile Pk sin kx(k = 0,,, ) φ k (x)(k =,, 3, ) ω(x)φk (x)φ l (x)dx = 0, k l ω(x) > 0 Legendre ω(x) = N A N e k A = N a k e k k= k a k e k A a k = ( A e k ) ( e k e k ) N f(x) φ k (x) f(x) = a k φ k (x) a k f(x) φ k (x) ω(x)f(x)φk (x)dx a k = ω(x)φk (x)φ k (x)dx k= f(x) f(x) f f k f = {f(ɛ), f(ɛ),, f(nɛ)} P k (x) = k ( )( ) k k + r ( ) r x r (.86) r r Legendre r=0 d d x( x) dx dx P k (x) + k(k + )Pk (x) = 0 (.87) 43

48 k ( )( k k + r (.86) = (x )k(k + ) + ( ) r r r r= k ( )( k k + r + k(k + ) ( ) r r r r= k ( )( k k + r = ( ) r r r = = r= ) x r r{r (r + )x} ) x r x + k(k + ){ k(k + )x} ) x r [r{r (r + )x} + k(k + )x] + k(k + ){x + k(k + )x} k ( )( ) k k + r ( ) r x r [r {r(r + ) k(k + )}x] + k(k + ){ k(k + )}x r r k ( )( ) k k + r ( ) r r x r r r k ( )( ) k k + r ( ) r {r(r + ) k(k + )}x r + k(k + ){ k(k + )}x r r r= r= r= x l ( l k ) ( )( ) k k + r ( )( ) ( ) r r x r r=l+ k k + r ( ) r {r(r + ) k(k + )}x r r=l r r r r [ ( )( ) ( )( ) ] k k + l + k k + l = ( ) l+ (l + ) ( ) l {l(l + ) k(k + )} l + l + l l ( )( ) [ ] k k + l (k l)(k + l + ) = ( ) l+ l l (l + ) (l + ) + l(l + ) k(k + ) x l ( )( ) k k + l = ( ) l [(k l)(k + l + ) + l(l + ) k(k + )]x l l l ( )( ) k k + l = ( ) l [k + (l + )k lk l(l + ) + l(l + ) k(k + )]x l l l = 0 x l x l 0 x ( )( ) k k + ( ) x + k(k + ){ k(k + )}x [ ] k(k ) (k + )(k + ) = 4 + k(k + ){ k(k + )} x = k(k + ){(k )(k + ) + k(k + )}x = k(k + )(k + k + k k)x = 0 44

49 x k (.86) ( )( ) k k ( ) k {k(k + ) k(k + )} = 0 k k N ω(n)φ k (n)φ l (n) = 0, k l (.88) n=0 Strum-Liouville +n p(n) n φ k (n) q(n)φ k (n) + λ k ω(n)φ k (n) = 0, k = 0,,, (.89) φ k (n) φ k (n) φ l (n) (.89) +n p(n) n φ k (n) q(n)φ k (n) + λ k ω(n)φ k (n) = 0 +n p(n) n φ l (n) q(n)φ l (n) + λ k ω(n)φ l (n) = 0 [ +n p(n) n φ k (n)]φ l (n) φ k (n)[ +n p(n) n φ l (n)] = (λ k λ l )ω(n)φ k (n)φ l (n) +n {p(n){[ n φ k (n)]φ l (n) φ k (n)[ +n p(n) n φ l (n)]}} = (λ k λ l )ω(n)φ k (n)φ l (n) n 0 n n p(n){[ n φ k (n)]φ l (n) φ k (n)[ +n p(n) n φ l (n)]} n + n 0 = (λ k λ l )ω(n)φ k (n)φ l (n) 0 φ k (n) φ l (n) φ k (n) P k,n (n) = k ( )( ) k k + r n(n ) (n r + ) ( ) r r r N(N ) (N r + ) r=0 +n n(n + n) n P k,n (n) + k(k + )P k,n (n) = 0 45

50 p(n)[ n φ k (n)]φ l (n) φ k (n)[ n φ l (n)] n+ n 0 (.90) = (λ k λ l ) n n=n 0 ω(n)φ k (n)φ l (n) n(n + n)[ n P k,n (n)]p l,n (n) P k,n (n)[ n P l,n (n)] n+ n 0 = [k(k + ) l(l + )] n n=n 0 P k,n (n)p l,n (n) n 0 = 0, n = N 0 N P k,n (n)p l,n (n) = 0, k l n=0 φ k (n) sin θ k n φ k (n) = sin θ k n, θ k sin θ k (n + ) ( cos θ k ) sin θ k n + sin θ k (n ) = 0 +n n sin θ k n + ( cos θ k ) sin θ k n = 0 () {[ n sin θ k n] sin θ l n sin θ k n[ n sin θ k n]} n + n 0 = (cos θ k cos θ l ) n n=n 0 sin θ k n sin θ l n n 0 = 0, n = N, θ k = kπ/n sin θ k N = sin θ l N = 0(k, l ) 0 N n=0 sin θ k n sin θ l n = 0, k l mod N φ k (n) W N (k, n) = exp(inkπ/n) φ k (n) = exp(inkπ/n) k, n, N 46

51 e i(n+)kπ/n e inkπ/n + e i(n )kπ/n (e ikπ/n + e ikπ/n )e inkπ/n = 0 +n n exp(inkπ/n) + λ k exp(inkπ/n) = 0 λ k = sin kπ/n () {[ exp(inkπ/n)] exp( inlπ/n) exp(inkπ/n)[ n exp( inlπ/n)]} n+ n 0 = (λ k λ l ) n n=n 0 exp(in(k l)π/n) n 0 = 0, n = N exp(ikπ) = exp(ilπ) = (k, l ) 0 N exp(in(k l)π/n) = 0, k l mod N n=0 N n=0 W N (k, n)w N ( l, n) = 0, k l mod N 47

52 .0 ε x v v = x ε (.) ε (.) ε x v = lim ε 0 ε (.) ε t x s t t s s (.) v = lim t 0 s t s = v t t 0 s 0 t dt s ds (.3) s v = lim t 0 t = ds dt (.4) t s a a = dv dt (.5) t v m v mv C = mv 48

53 F t 0 F = lim t 0 F = (mv) t (mv) t = d(mv) dt (.6) (.6) F = m d(v) dt (.7) (.5) F = ma (.8). m 0 κ v(t) q d q(t) dt + ω 0q(t) = 0, ω 0 = κ m 0 (.9) dq(t) p(t) p(t) = m 0 v(t) p(t) = m 0 dt dp(t) d q(t) = m 0 dt dt = κq(t), (.0) dq(t) = p(t) dt m 0 (.) n n 49

54 E (t) E E (t) E (t) = m 0v (t) = m 0 ( dq(t) dt ) = m 0 p (t) (.) E (t) E (t) = m 0κq (t) (.3) H(t) H(t) = E (t) + E (t) = m 0 p (t) + κq (t) (.4) H(t) H(t) = C 0 dh(t) = d ( p (t) + ) dt dt m 0 κq (t) = κ m 0 p(t)q(t) + κ m 0 q(t)p(t) = 0 H(t) q(t) = ( q(t) m 0κq (t) + ) m 0κq (t) = κq(t)... H(t) q(t) = dp(t) dt (.0) H(t) p(t) = ( p (t) + ) p(t) m 0 m 0κq (t) = m 0 p(t)... H(t) p(t) = dq(t) dt (.) (.0) (.) (.9) 50

55 I d q(t) (.9) dt = q(t) (.9) q (t) + ω 0q(t) = 0 (.5) q dq(t) (t) = Q(t) = Q(t) Q(t) q(t) dt Q(q(t)) Q(t) dq(t) dq(t) + ω 0q(t) = 0 Q dq(t) dq(t) dq + q(t) dq = s Q(t) + ω 0q(t) = s Q(t) dq, q (t) + ω 0q(t) = dq(t) dt q(t) dq, = dq(t) dq(t) + ω 0q(t) dq(t) dt + ω 0q(t) = Q(t) dq(t) dq(t) + ω 0q(t) 0 dq q 0 dq Q(t) = ± s ω0 q(t) dq(t) = ± s ω0 dt q(t) dt dq(t) = ± s ω 0 q(t) dt dq(t) dq = ± s ω 0 q(t) dq t + r = ± ω 0 arcsin ω 0q(t) s t ( r dq, dq s ω 0 q(t) (.).. s. q(t) = ± sin ω 0 (t + r) ω 0 dq(t) dt = ± d dt ( ) s sin ω 0 (t + r) ω 0 = ± s cos ω 0 (t + r) = m 0 p(t)... p(t) = ±m 0 s cos ω0 (t + r) 5

56 s ± = A, r = t 0 ω 0 { q(t) = A sin ω0 (t t 0 ) (.6a) p(t) = m 0 ω 0 A cos ω 0 (t t 0 ) (.6b) (.9) (.9) (.6) (.9) d dt A sin ω 0(t t 0 ) + ω 0 A sin ω 0 (t t 0 ) = d dt{ ω0 A cos ω 0 (t t 0 ) } + ω 0 A sin ω 0 (t t 0 ) = ω 0A sin ω 0 (t t 0 ) + ω 0 A sin ω 0 (t t 0 ) = 0 (.6) (.9) (.6) (.9) (.9) u(t) (.5) q (t) + ω 0 q(t) = 0 u (t) + ω 0 u(t) = 0 (.7) (.7),.8 q(0) = C 0 q (t) = C u(t) = C 0 u (t) = C (.8) {q(t) u(t)} + {q(t) u(t)} = 0, q(0) u(0) = 0, {q(0) u(0)} = 0 w(t) = q(t) u(t) w (t) + w(t) = 0, w(0) = 0, w (0) = 0 {w (t) + w(t) } {w (t) + w(t) } = w (t)w (t) + w(t)w (t) = w (t){w (t) + w(t)} = 0... w (t) + w(t) = C C : w (t), w(t) (.8) w (t) + w(t) = w (0) + w(0) = 0 w (t) 0, w(t) 0 5

57 w(t) = q(t) u(t) = 0... q(t) = u(t) (.6) (.9) II q(t) = e at (.9) q(t) = e at d dt eat + ω 0e at = 0 a e at + ω 0e at = 0 (a + ω 0)e at = 0 e at 0 a + ω 0 = 0 a = ±iω 0 q(t) = e ±iω 0t (.9) n n (.9) q(t) = ae iω 0t + be iω 0t (.0) q(t) = a{cos ω 0 t + i sin ω 0 t} + b{cos ω 0 t i sin ω 0 t} = (a + b) cos ω 0 t + i(a b) sin ω 0 t (a + b) cos ω 0 t = (a + b) sin ω 0 (t π ω 0 ) π (a + b) = A, ω 0 = t 0 q(t) = A sin ω 0 (t t 0 ) p(t) = m 0 ω 0 A cos ω 0 (t t 0 ) (.9) (.6) 53

58 III II q(t) = e iω0t z(t) (.) (.9) (.) d dt {eiω 0t z(t)} + ω 0e iω 0t z(t) = 0 {iω 0 e iω0t z(t) + e iω0t z (t)} + ω 0e iω0t z(t) = 0 B {iω 0 z (t) + z (t)}e iω 0t = 0 {iω 0 z (t) + z (t)}e iω 0t = 0 e iω0t z (t) dt, d dt eiω 0t z (t) = 0 d dt {eiω 0t z (t)} dt = e iω 0t z (t) = B 0 dt 0 dt t z (t) = Be iω 0t z (t) dt = B e iω0t dt z(t) = B iω e iω0t + b t b z (t) dt, B (.) e iω0t dt q(t) = e iω 0t z(t) = e iω 0t { B iω 0 e iω 0t } + be iω 0t = B iω 0 e iω0t + be iω0t (.9) ( q(t) = b cos ω 0 t + B ) ( ) B sin ω 0 t + i cos ω 0 t + B sin ω 0 t ω 0 ω 0 a sin x + b cos x = a + b cos (x + α) q(t) = b + b cos ω 0 t + α) 4ω 0 = b + b sin ω 0 (t π α ) 4ω 0 ω 0 54

59 b + b 4ω 0 = A, π α ω 0 = t 0 q(t) = A sin ω 0 (t t 0 ) (.9) p(t) = m 0 ω 0 A cos ω 0 (t t 0 ) (.9) (.6).. t t = mδ m δ (), () +t p(t) δ [p(t + δ) p(t)] +t q(t) δ [q(t + δ) q(t)] +t p(t) = κq(t) +t q(t) = m 0 p(t) t = mδ ( m ) p m+ = p m δκq m (.) q m+ = q m + δ m 0 p m (.3) p m, q m m p, q (.), (.3) c, c Ω 0 Ω m p m = c Ω m (.4) q m = c Ω m (.5) c Ω m+ = c Ω m δκc Ω m c Ω m+ = c Ω m + δ m 0 c Ω m (Ω )c + δκc = 0 δ m 0 c + (Ω )c = 0 p, q m 55

60 c, c 0, c, c 0 Ω δκ δ m 0 Ω = 0 (Ω ) + κ δ = 0 m 0 Ω Ω ( ) / κ Ω = ± i δ m 0 ) / Ω = ( + δ κm0, δ > 0 Ω (.4), (.5) p, q m, H(t) = m 0 p(t) + κq(t) t +t H(t) = m 0 +t p(t)p(t) + κ +tq(t)q(t) = m 0 [ +tp(t)][p(t) + p(t + δ)] + κ[ +t q(t)] [q(t) + q(t + δ)] = [ κq(t)][p(t) δκq(t)] + κ[ p(t)][q(t) + δ p(t) ] m 0 m 0 m 0 = δ κ m 0 H(t) > 0 ( +t f(t)g(t) = [ +t f(t)]g(t + δ) + f(t)[ +t g(t)] ), +t H(t) = 0 (0, 0).. (.0), (.) t p(t) δ [p(t) p(t δ)] t q(t) δ [q(t) q(t δ)] 56

61 t p(t) = κq(t) t q(t) = m 0 p(t) t = mδ ( m ) p m = p m δκq m (.6) q m = q m + δ m 0 p m (.7) p, q (.6), (.7) c, c Ω 0 Ω m p m = c Ω m (.8) q m = c Ω m (.9) c Ω m = c Ω m δκc Ω m c Ω m = c Ω m + δ m 0 c Ω m (Ω )c + δκc Ω = 0 δ m 0 c Ω + (Ω )c = 0 c, c 0, c, c 0 Ω δκω δ m 0 Ω Ω = 0 (Ω ) + κ m 0 δ Ω = 0 Ω Ω ( ) /δ κ ± i m 0 Ω = + δ κ m 0 ) / Ω = ( + δ κm0 Ω δ > 0 (.8), (.9) p, q m, 57

62 H(t) = m 0 p(t) + κq(t) t t H(t) = m 0 t p(t)p(t) + κ tq(t)q(t) = m 0 [ tp(t)][p(t δ) + p(t)] + κ[ t q(t)] [q(t δ) + q(t)] = [ κq(t)][p(t) + δκq(t)] + κ[ p(t)][q(t) δ p(t) ] m 0 m 0 m 0 = δ κ m 0 H(t) < 0, t H(t) = 0,..3 I ( δ ) t p(t) (δ) [p(t + δ) p(t δ)] t q(t) (δ) [q(t + δ) q(t δ)] t p(t) = κq(t) t q(t) = m 0 p(t) t = mδ ( m ) p m+ = p m δκq m (.30) q m+ = q m + δ m 0 p m (.3) p, q p m = c Ω m (.3) q m = c Ω m (.33) 58

63 (.30), (.3) (Ω )c + δκωc = 0 δ m 0 Ωc + (Ω )c = 0 c, c 0, c, c 0 Ω δκω δ m 0 Ω Ω = 0 (Ω ) + 4 κ m 0 δ Ω = 0 Ω Ω ( ) / κ Ω = ± i δ m 0 ) / Ω = ( + δ κm0, δ > 0 Ω (.4), (.5) p, q m,..4 II +t p(t) = kq(t), (.34) t q(t) = m 0 p(t) (.35) (.35) +t q(t) = m 0 p(t) p m+ = p m δkq m q m+ = q m + δ m 0 p m+ p m = c Ω m q m = c Ω m 59

64 ( ) Ω δ k Ω + = 0 m 0 ω 0 ω 0 = k m 0 Ω Ω ( ) ω0δ Ω + = 0 Ω = ω 0δ ± ω 0 δ [ ( ω 0δ ) ] [ ( ) Ω = ( ω 0 δ ) + ω0 δ ω 0δ ] = ω 0δ + 4 ω4 0δ ω4 0δ 4 ω 0δ = ω 0δ + ω4 0δ 4 ω4 0δ 4 ω0δ 4 ω 0δ ω 0δ (ω 0 > 0, δ > 0 ) /δ +t p(t) = kq(t) t m 0 p(t) H(t) = p (t) δk p(t)q(t) + m 0 m 0 kq (t) = m 0 [p(t) δk q(t) ] + k [ ( ) ] ω 0δ q (t) (.36) +t H(t) = 0 [ (.36) ω 0δ ] p(t), q(t) ω 0δ 60

65 t p(t) = kq(t) +t q(t) = m 0 p(t) +t p(t) = kq(t + δ) Ω ( ) ω0δ Ω + = 0..5 III..4 δ +t H(t) = 0 δ H(t) = m 0 P (t) + κq (t) +t H(t) = m 0 [ +t p(t)][p(t + δ) + p(t)] + κ [ +tq(t)][q(t + δ) + q(t)] +t H(t) = 0 +t p(t), +t q(t) +t p(t) = [q(t + δ) + q(t)] (.37) κ +t q(t) = m 0 [p(t + δ) + p(t)] (.38) δ 0 (.0) (.) p m+ = ( δ κ 4m 0 )p m δκq m (.39) + δ κ 4m 0 q m+ = ( δ κ 4m 0 )q m + δ m 0 p m (.40) + δ κ 4m 0 p m = c Ω m q m = c Ω m (.39)(.40) 6

66 c ( + δ κ 4m 0 ( δ κ { 4m 0 ) }ω ( + δ κ 4m 0 c ω = ( δ κ 4m 0 )c δκc (.4) + δ κ 4m 0 c ω = ( δ κ 4m 0 )c + δ m 0 c (.4) + δ κ 4m 0 ) ( c ω = δ κ 4m 0 ) ω = ( δ κ 4m 0 ) c + δ κ c m 0 ( δ κ 4m 0 ) ( + δ κ 4m 0 )ω ) ( δ ) κ { }ω + δ κ 4m 0 m 0 ω ( ) ( + δ κ δ ω ) ( κ { }ω + δ κ δ ) κ δ κ 4m 0 4m 0 4m 0 4m 0 4m 0 ( ) ( + δ κ δ ω ) ( ) κ { }ω + + δ κ = 0 4m 0 4m 0 4m 0 ω { ( δ κ 4m 0 ) } ω + = 0 ( + δ κ 4m 0 ) D/4 = { ( δ κ 4m 0 ) } ( + δ κ 4m 0 ) 4 = ( δ κ 4m 0 ) ( + δ κ 4m 0 ) ( + δ κ 4m 0 ) = 4 δ κ 4m 0 ( + δ κ 4m 0 ) < 0 ω = δ shin..6 II t H(t) = 0 t H(t) = [ t p(t) ] + m 0 κ[ tq(t) ] = [ t p(t)][p(t + ɛ) + p(t ɛ)] + m 0 κ[ tq(t)][q(t + ɛ) + q(t ɛ)] 6

67 t H(t) = 0 t p(t), t q(t) t p(t) = κ[q(t + ɛ) + q(t ɛ)] t q(t) = [p(t + ɛ) + p(t ɛ)] m 0 p m+ p m = δκ(q m+ + q m ) p m+ + p m = m 0 δ (qm+ q m ) p m+, q m+ p m+ = ( δ κ m 0 )p m δκq m + δ κ m 0 q m+ = ( δ κ m 0 )q m δκp m + δ κ m 0 p m = c Ω m q m = c Ω m ( + δ κ ) Ω 4 ( δ4 κ )Ω + ( + δ κ ) = 0 m 0 m 0 m 0 Ω =...5 tanchu ,. 63

68 ..7 δ +t p(t) = κ [q(t + δ) + q(t)] (.43) +t q(t) = [p(t + δ) + p(t)] m 0 (.44) () +t (.44) +tq(t) = m 0 [ +t p(t + δ) + +t p(t)] +tq(t) = ω 0 [q(t + δ) + q(t + δ) + q(t)] 4 (ω0 = κ/m 0 ) +t q(t + δ) q(t) δ = ω 0 [q(t + δ) + q(t + δ) + q(t)] 4 δ [ +tq(t + δ) +t q(t)] = ω 0 [q(t + δ) + q(t + δ) + q(t)] 4 q(t + δ) q(t + δ) + q(t) δ = ω 0 [q(t + δ) + q(t + δ) + q(t)] 4 q(t + δ) q(t + δ) + q(t) = δ ω0 [q(t + δ) + q(t + δ) + q(t)] 4 ( + δ ω 0 4 )q(t + δ) ( δ ω 0 4 )q(t + δ) + ( + δ ω 0 4 q(t) = 0 q(t + δ) δ ω /4 + δ ω q(t) + q(t δ) = 0 /4 φ 0 tan (δφ 0 /) = δω 0 / q(t) δ ω 0/4 + δ ω 0 /4 = cos φ 0δ q(t) q(t + δ) tan ( δφ 0 ) + tan ( δφ + δ) + q(t) = 0 0 )q(t q(t + δ) cos ( δφ 0 )( tan ( δφ 0 ))q(t + δ) + q(t) = 0 q(t + δ) cos ( δφ 0 ( δφ0 )cos ) sin ( δφ 0 ) cos ( δφ 0 ) q(t + δ) cos(φ 0 δ)q(t) + q(t δ) = 0 (.45) (.9) (.45) A, t 0 q = A sin (φ 0 (t t 0 )) (.46) p = m 0 ω 0 A cos (φ 0 (t t 0 ) (.47) 64

69 q(t + δ) + q(t δ) = A(sin (φ 0 (t t 0 ) + φ 0 δ) + sin (φ 0 (t t 0 ) φ 0 δ)) = A cos (φ 0 δ) sin (φ 0 (t t 0 )) = cos (φ 0 δ)q(t) (.45) H H = κa [sin (φ 0 (t t 0 ) + cos (φ 0 (t t 0 ))] (.48) (.46) (.47) (.6), q(t) = A sin (ω 0 (t t 0 ) p(t) = m 0 ω 0 A cos (ω 0 (t t 0 )) t ω 0 ω 0 t = mδ (.46) (.47) (.6a), (.6b) (.43),(.44) tan (δφ 0 /) = δω 0 / δ 0 φ 0 = ω 0 (.46) (.47) (.6a) (.6b). dn dt = αn( λn), α, λ (.49) α λ t dn(t) ( N(t) + dn(t) = αn(t)( λn(t)) dt d N(t) dx N(t)( λn(t)) = α N(t)( λn(t)) = α dt ) λ = α(t + c) λn(t) log N(t) log λn(t) = α(t + c) N(t) = exp α(t + c) λn(t) 65

70 c 0 = exp α(c + t 0 ) exp α(t + c) N(t) = + λ exp α(t + c) exp α(t t 0 ) = c 0 + λ exp α(t t 0 ), (0 < N(t) λ ) N(t 0 ) = c 0 +λ t N( ) = λ N f, g N = g f (.49) d g dt f = α g f ( λ g f ) (.50) dg dt f g df dt f = α g ( λ g ) (.5) f f dg dt f g df = αg(f λg) (.5) dt F (f, g), h s.t F (fh, gh) = F (f, g) h (.5 check dgh fh ghdfh = αgh(fh λgh) ( dt dt dg dt h + g dh ) ( df fh gh dt dt h + f dh ) = h αg(f λg) dt dg dt f g df = αg(f λg) dt (.5) (.5) (.5) ( ) ( ) dg df dt αg f = αg dt αλg (.53) β df αλg = βf dt dg dt αg = βg (.54) 66

71 (.53) f, g t f(t) f(t) exp g(t) g(t) exp (.54) d t t t [f(t) exp βdt] αλg(t) exp βdt = βf(t) exp βdt dt d t t t [g(t) exp βdt] αg(t) exp βdt = βg(t) exp βdt dt t t t df dt exp βdt + f(t) d dt exp βdt αλg(t) exp βdt = βf(t) exp t t t dg dt exp βdt + g(t) d dt exp βdt αg(t) exp βdt = βg(t) exp t [ df dt αλg(t)] exp βdt = 0 0 t [ dg dt αg(t)] exp βdt = 0 0 df dt αλg(t) = 0 dg dt αg(t) = 0 0 t 0 βdt βdt t 0 t 0 βdt βdt f = c 0 + λ exp α(t t 0 ) g = exp α(t t 0 ) (.55) N N = g f = exp α(t t 0 ) c 0 + λ exp α(t t 0 ) (.56).. d N(t) = αn(t)[ λn(t)] dt d dt +t δ +t N(t) = αn(t)[ λn(t)] N(t + δ) N(t) = δαn(t)[ λn(t)] 67

72 N(t + δ) a = + δα δαλ = + δλ t = nδ N(t + δ) = N(t) + δαn(t)[ λn(t)] N(t + δ) = { + δα[ λn(t)]} N(t) N(t + δ) = [ + δα λδαn(t)]n(t) [ N(t + δ) = ( + δα)n(t) λδα ] + δα N(t) N(nδ) = N n N n+ = an n ( N n ) a 4 a (0, y) y a a =.5.4 (0, 0.5) (0, 0.6) (0, 0.8) a =

73 .4 (0,0.5) (0,0.6) (0,0.8) dg(t) f(t) g(t) df(t) = αg(t)[f(t) λg(t)] dt dt d dt +t [ +t g(t)]f(t) g(t)[ +t f(t)] = αg(t)[f(t) λg(t)] [g(t + δ) g(t)]f(t) g(t)[f(t + δ) f(t)] = δαg(t)[f(t) λg(t)] g(t + δ)f(t) g(t)f(t + δ) = δαg(t)[f(t) λg(t)] (.57) f(t) f(t)h(t) h(t) g(t) g(t)h(t) (.57) g(t + δ)h(t + δ)f(t)h(t) g(t)h(t)f(t + δ)h(t + δ) = δαg(t)h(t)[f(t)h(t) λg(t)h(t)] h(t + δ)h(t)[g(t + δ)f(t) g(t)f(t + δ)] = {δαg(t)[f(t) λg(t)]} h (t) h (t) h(t + δ)h(t) δ 0 I (.57) g(t) g(t + δ) g(t + δ)f(t) g(t)f(t + δ) = δαg(t + δ)[f(t) λg(t)] h(t + δ)h(t)[g(t + δ)f(t) g(t)f(t + δ)] = {δαg(t + δ)[f(t) λg(t)]} h(t + δ)h(t) 69

74 II (.57) g(t), f(t) g(t + δ), f(t + δ) g(t + δ)f(t) g(t)f(t + δ) = δαg(t)[f(t + δ) λg(t + δ)] III (.57) g(t + δ)f(t) g(t)f(t + δ) = δαg(t)[f(t) λg(t)] = { } δαg(t)[f(t) λg(t)] + { } δαg(t)[f(t) λg(t)] (t) (t + δ) g(t + δ)f(t) g(t)f(t + δ) = { } δαg(t + δ)[f(t) λg(t)] + { } δαg(t)[f(t + δ) λg(t + δ)] g(t + δ)f(t) g(t)f(t + δ) = δα[g(t + δ)f(t) + g(t)f(t + δ)] δαλg(t)g(t + δ) I II III g(t + δ)f(t) g(t)f(t + δ) (.58) = δαg(t + δ)[f(t) λg(t)] g(t + δ)f(t) g(t)f(t + δ) (.59) = δαg(t)[f(t + δ) λg(t + δ)] g(t + δ)f(t) g(t)f(t + δ) = (.60) δα[g(t + δ)f(t) + g(t)f(t + δ)] δαλg(t)g(t + δ) I I g(t + δ)f(t) g(t)f(t + δ) = [g(t + δ) g(t)]f(t) g(t)[f(t + δ) f(t)] I [g(t + δ) g(t)]f(t) g(t)[f(t + δ) f(t)] = δαg(t + δ)[f(t) λg(t)], [f(tδ) f(t)] = δαλg(t + δ) [g(t + δ) g(t)] = δαg(t + δ) (.6) t = nδ ( n ) f n+ f n = δαλg n+ (.6) g n+ g n = δαg n+ 70

75 (.6) +t f(t) = λ +t g(t) f(t) = c 0 + λg(t) (.6) g n+ = g n = δα g n = ( δα) g n g n δα = = ( δα) n g 0 = ( δα) (t t 0 ) δ g(t) = ( δα) (t t 0 ) δ f, g f(t) = c 0 + λg(t) g(t) = ( δα) (t t 0 ) δ (.63) II II g(t + δ)f(t) g(t)f(t + δ) = [g(t + δ) g(t)]f(t) g(t)[f(t + δ) f(t)] g(t + δ)f(t) g(t)f(t + δ) = [g(t + δ) g(t)]f(t + δ) g(t + δ)[f(t + δ) f(t)] II [g(t + δ) g(t)]f(t) g(t)[f(t + δ) f(t)] = δαg(t)[f(t + δ) λg(t + δ)] [g(t + δ) g(t)]f(t + δ) g(t + δ)[f(t + δ) f(t)] = δαg(t)[f(t + δ) λg(t + δ)] [f(t + δ) f(t)] = δα +δαλg(t + δ) [g(t + δ) g(t)] = δαg(t) 7

76 [f(t + δ) f(t)] = δαλg(t) [g(t + δ) g(t)] = δαg(t) I f(t) = c 0 + λg(t) g(t) = ( δα) (t t 0 ) δ (.64) III III g(t + δ)f(t) g(t)f(t + δ) = [g(t + δ) g(t)]f(t + δ) g(t + δ)[f(t + δ) f(t)] δα[g(t + δ)f(t) + g(t)f(t + δ)] δαλg(t)g(t + δ) = δα[g(t) + g(t + δ)]f(t + δ) g(t + δ)[δαλg(t) δαf(t) + δαf(t + δ)] III [g(t + δ) g(t)]f(t + δ) g(t + δ)[f(t + δ) f(t)] = δα[g(t + δ) + g(t)]f(t + δ) g(t + δ)[δαλg(t) δαf(t) + δαf(t + δ)] [f(t + δ) f(t)]( δα) = δαλg(t) [g(t + δ) g(t)] = δα[g(t) + g(t + δ)] I, II f(t) = c 0 + λg(t) g(t) = [ + δα δα ] (t t0)/δ (.65) I, II, III N(t) = g(t) f(t) I II III N(t + δ) N(t) = δαn(t + δ)[ λn(t)] N(t + δ) N(t) = δαn(t)[ λn(t + δ)] { } N(t + δ) N(t) = δα [N(t) + N(t + δ)] λn(t)n(t + δ) t = nδ 7

77 N n I N n+ = δα( λn n ) II N n+ = ( + δα)n n + δαλn n ) III N n+ = (.63), (.64), (.65) I N(t) = ( + δα)n n δα + δαλn n) ( δα) (t t0)/δ c 0 + λ( δα) (t t 0)/δ ( + δα) (t t0)/δ II N(t) = c 0 + λ( + δα) (t t 0)/δ III N(t) = c 0 + λ ( ) + (t t0)/δ δα δα ( + δα δα ) (t t0 )/δ I = exp (δ ˆα) δα II III + δα = exp (δ ˆα) + δα + = exp(δ ˆα) δα N = g f = exp (ˆα(t t 0 )/δ) c 0 + λ exp(ˆα(t t 0 )/δ) I, II, III, I N n+ N n N n+ N n 0 N n+ N n = N n δα( λn n ) N n = N n N n ( δα( λn n )) δα( λn n ) = N nδα N nδαλ δα( λn n ) = N nδα( N n λ) δα( λn n ) 0 N(t) λ N nδα( N n λ) 0 ( δ α δα( λn n ) 0 δ α( λn 0 N(t) n) λ δ I, II, III 73

78 logistic "logio5" "logio5" "logi3o5" I, III.3 p, q r(x) x d dx y + px d y + qy = r(x). dx r(x) = 0 x d dx y + px d y + qy = 0 (.66) dx (.66) D = x d dx x k k Dx k = x d dx xk = kx k D = x d dx x d d = x dx dx + x d dx (.66) D y + (p )Dy + qy = 0. y = c x k + c x k 74

79 k j + (p )k j + q = 0, j =, (.66) D = x d dx D t t t = log x x d dx y = dy dt x d dx y = d y dt dy dt (.66) d y dt dy dt + pdy dt + qy = 0 y y = c e kt + c e kt p, q r(x). x d y dx x dy 3y = 0, dx D y Dy 3y = 0 (D 3)(D + )y = 0 Dy 3y = 0 x dy dx 3y = 0 y 3 y dx = x log y = log x 3 y = x 3 Dy + = 0 y = x y = ax 3 + b (.67) x 75

80 . x d y dy 4x + 6y = x, dx dx x = r(x) = 0 y = ax + bx 3 + Y Y Y = Ae t d dt (Aet ) 5 d dt (Aet ) + 6Ae t = e t Ae t 5Ae t + 6Ae t = e t Ae t = e t, A = e t = x y = ax + bx 3 + x 3. x d y dx x dy dx + y = 0, y = x D y Dy + y =D x u(x) + Dx Du(x) + x D u(x) Dx u(x) x Du(x)+x u(x) 0 y = x u(x) Du(x) = v(x) Dx v(x) + x Dv(x) x v(x) = 0(Dx = x x ) x Dv(x) = 0 x du dx = b du dx dx = v(x) = c 0 ( ) b x dx u(x) = b log x + a y = (a + b log x)x 4. x d y dx x dy + 5y = 0, dx 76

81 D = ± i R,±I y = ce (R+iI)t + de (R ii)t = (ce iit + de iit )e Rt (â = c + d, ˆb = i(c d) ) = [â cos It + ˆb sin It]e Rt ( R =, I = ) = [âe i log x + ˆbe i log x ]x = [â cos(log x ) + ˆb sin log x ]x..3. D = x d dx D = x d dx D +x = x +x D +x k x k x k = x(x + ɛ)(x + ɛ) (x + (k )ɛ) k D +x x k = kx k. D +x = x +x + x +x D +x D +xy + (p )D +x y + qy = r(x) x +ty + px +x y + qy = r(x) x +ty x +x y3y = 0 D +x D +xy D +x y 3y = 0 (D +x 3)(D +x + )y = 0 77

82 (D +x 3)y a = 0 (D +x + )y b = 0 y a = x 3 y b = x ɛ y = ax 3 + b x ɛ a, b (.67) ɛ 0 (.67).4 dy dx + p(x)y = q(x)ym+ (.68) (p(x), q(x) x ) m =, 0 m m (m, 0) f, g α y ( ) α g y = (.69) f (.68) dg dx α f g df dx f + p(x) g f = q(x) α = m ( ) mα+ g (.70) f (.7) (.70) dg dx f g df dx mp(x)gf = mq(x)f (.7) f(x) f (x)h(x) = f(x) (.73) g(x) g (x)h(x) = g(x) (.74) [ ] [ ] dg df dx + mq(x)f f g dx + mp(x)f = 0 (.75) 78

83 β(x) h(x) df + mp(x)f = β(x)f (.76) dx dg + mq(x)f = β(x)g (.77) dx dh(x) dx (.76) = β(x)h(x) (.78) d(f h) + mp(x)f h = β(x)f h dx df dx h + f dh dx + mp(x)f h = β(x)f h df dx h + f β(x)h + mp(x)f h = β(x)f h df dx + mp(x)f = β(x)f β(x)f df dx + mp(x)f = 0 (.79) β(x) 0 (.77) dg dx + mq(x)f = 0 (.80) (.79) df dx f = mp(x) log f = ( mp(x)) dx (.80) [ f = exp m dg dx = mq(x)f ] p(x) dx (.8) g = m q(x)f dx (.8) (.68) y = u m (.83) 79

84 d dx u m + p(x)u m = q(x)u m du u m m dx + p(x)u m = q(x)u m du + p(x)u = q(x) m dx du mp(x)u = mq(x) (.84) dx du h dx mp(x) = u h (.85) [ u h (x) = exp m ] p(x) dx (.86) u(x) = C(x)u h q(x) u(x) = m dxu h (x) (.87) u h (3.90) dg dx f g df mp(x)gf = mq(x)f dx,. [g(x + ɛ) g(x)]f(x) g(x)[f(x + ɛ) f(x)] ɛmp(x)g(x)f(x + ɛ) = ɛmq(x)f(x)f(x + ɛ). [g(x + ɛ) g(x)]f(x) g(x)[f(x + ɛ) f(x)] ɛmp(x)g(x + ɛ)f(x) = ɛmq(x)f(x)f(x + ɛ) f(x + ɛ) f(x) = ɛmp(x)f(x + ɛ) (.88) g(x + ɛ) g(x) = ɛmq(x)f(x + ɛ) (.89) f(x + ɛ)[g(x + ɛ) g(x)] [f(x + ɛ) f(x)]g(x + ɛ) = ɛmp(x)g(x + ɛ)f(x) ɛmq(x)f(x)f(x + ɛ) 80

85 f(x + ɛ) f(x) = ɛmp(x)f(x) (.90) g(x + ɛ) g(x) = ɛmq(x)f(x) (.9) x = nɛ(n ) f(x), g(x), p(x), q(x) f n, g n, p n, q n (.88) (.89) f n+ = f n + ɛmp n, g n+ = g n ɛmq n f n+ n f n = f 0 k=0 + ɛmp k, n g n = g 0 ɛmq k f k+ (.90) (.9) k=0 f n+ = f n ( ɛmp n ) g n+ = g n ɛmq n f n n f n = f 0 ( ɛmp k ) k=0 n g n = g 0 ɛmq k f k k=0 g(x) = u(x)f(x) +x u(x) mp(x)u(x) = mq(x) +x u(x) mp(x)u(x + ɛ) = mq(x) (3.90) du mp(x)u = mq(x) dx 8

86 u(x) = y m (x) d dx y α = α dy y α+ dx (.9) dy dx + p(x)y = q(x)ym+ (.93) y(x) (.9) (.93) m y m y(x)y(x + ɛ) y(x + (m )ɛ), m y m (x) =, m = 0 y(x ɛ)y(x ɛ) y(x m )ɛ), m m mɛ +x m +mx y(x) = y(x + mɛ) y(x) mɛ y(x m ɛ) y(x) +mx y(x) = m ɛ m +x y m (x) = ɛ [ y(x + ɛ)y(x + ɛ) y(x mɛ) y(x)y(x + ɛ) y(x + (m )ɛ) ] m = [y(x) y(x + mɛ)] y(x)y(x + ɛ) y(x + mɛ) mɛ = m y m+ (x) +mxy(x) m +x y m (x) = [y(x)y(x ɛ) y(x ( m )ɛ) y(x ɛ)y(x ɛ) y(x m ɛ)] ɛ = y(x ɛ)y(x ɛ) y(x ( m )ɛ) m [y(x) y(x m ɛ)] mɛ = m y m+ (x) +mxy(x) 8

87 +x y m (x) = m y m+ (x) +mxy(x) u(x) = y m (x). +x u(x) mp(x)u(x) = mq(x). +x u(x) mp(x)u(x + ɛ) = mq(x). +mx y(x) + p(x)y(x + mɛ) = q(x)y m+ (x). +mx y(x) + p(x)y(x) = q(x)y m+ (x) p(x) =, q(x) = shin ber I, ber II shin ber ber

88 .4... [g(x) g(x ɛ)]f(x) g(x)[f(x) f(x ɛ)] ɛmp(x)g(x)f(x ɛ) = ɛmq(x)f(x)f(x ɛ) (.94). [g(x) g(x ɛ)]f(x) g(x)[f(x) f(x ɛ)] ɛmp(x)g(x ɛ)f(x) = ɛmq(x)f(x)f(x ɛ) (.95) g(x) g(x ɛ) = ɛmq(x)f(x ɛ) (.96) f(x) f(x ɛ) = ɛmp(x)f(x ɛ) (.97) g(x) g(x ɛ) = ɛmq(x)f(x) (.98) f(x) f(x ɛ) = ɛmp(x)f(x) (.99) x = nɛ(n ) f(x), g(x), p(x), q(x) f n, g n, p n, q n. (.96),(.97) f n = ( ɛmp n )f n g n = g n ɛmq n f n n f n = f 0 ( ɛmp k ). (.98) (.99) k= n g n = g 0 ɛmq k+ fk k=0 f n f n = + ɛmp n g n = g n ɛmq(x)f n n f n = f 0 k= n g n = g 0 + ɛmp k ɛmq k+ f k+ k=0 84

89 g(x) = u(x)f(x). x u(x) mp(x)u(x) = mq(x). x u(x) mp(x)u(x ɛ) = mq(x) m y m (x) y(x)y(x ɛ) y(x (m )ɛ), m y m (x) =, m = 0 y(x+ɛ)y(x+ɛ) y(x+ m )ɛ), m m mɛ mx mx = m y(x) y(x mɛ) mɛ y(x) y(x+ m ɛ) mx y(x) = m ɛ m x y m (x) = ɛ [ y(x)y(x ɛ) y(x (m )ɛ) y(x ɛ)y(x ɛ) y(x m ) ] = y(x mɛ) y(x) ɛ y(x)y(x ɛ) y(x mɛ) m = [y(x mɛ) y(x)] y(x)y(x ɛ) y(x mɛ) mɛ = m y m+ (x) mxy(x) m x y m (x) = [y(x + ɛ)y(x + ɛ) y(x+ m ɛ) y(x)y(x + ɛ) y(x+ m )ɛ] ɛ = y(x + ɛ)y(x + ɛ) y(x + ( m +)ɛ)[y(x+ m ɛ) y(x)] ɛ = m y m+ (x) mxy(x) 85

90 x y m (x) = m y m+(x) mxy(x) u(x) = y m (x),,. mx y(x) + p(x)y(x mɛ) = q(x)y m+ (x). mx y(x) p(x)y(x) = q(x)y m+ (x) [g(x + ɛ ) g(x ɛ )]f(x + ɛ ) g(x + ɛ )[f(x + ɛ ) f(x ɛ )] ɛmp(x)g(x + ɛ )f(x ɛ ) = ɛmq(x)f(x + ɛ )f(x ɛ ). [g(x + ɛ ) g(x ɛ )]f(x ɛ ) g(x ɛ )[f(x + ɛ ) f(x ɛ )] ɛmp(x)g(x ɛ )f(x + ɛ ) = ɛmq(x)f(x + ɛ )f(x ɛ ) g(x + ɛ ) g(x ɛ ) = ɛmq(x)f(x + ɛ f(x + ɛ f(x ɛ ) = ɛmp(x)f(x + ɛ (.00) (.0) g(x + ɛ ) g(x ɛ ) = ɛmq(x)f(x + ɛ ) (.0) f(x + ɛ ) f(x ɛ ) = ɛmp(x)f(x + ɛ ) (.03) x = nɛ(n ) f(x), g(x), p(x), q(x) f n, g n, p n, q n. (.00) (.0) f n+ = ( ɛp n )f n g n+ = g n ɛq n f n 86

91 n f 0 ( ɛp k ) (l =,, ) k=l f n = n f ( ɛp k ) (l =,, ) k=l g 0 g n = g k k=l+ ɛq k f k (l =,, ) k ɛq k f k (l =,, ) k=l. (.0) (.03) f n+ = f n ɛmp n f n+ g n+ = g n ɛmq n f n+ n f 0 (l =,, ) ɛmp k k=l+ f n = n f 0 (l =,, ) ɛmp k k=l n g 0 ɛmq k f k (l =,, ) k=l+ g n = n g ɛmq k f k (l =,, ) k=l, g(x) = u(x)f(x) x u(x) mp(x)u(x + ɛ ) = mq(x) m y ((m)) (x) y ((m)) (x) = y(x (m )ɛ )y(x (m 3)ɛ ) y(x + (m 3)ɛ )y(x + (m )ɛ ), m, m = 0 y(x (m )ɛ )y(x (m 3)ɛ ) y(x+ (m 3)ɛ )y(x+ (m )ɛ mx mx y(x) = mɛ mɛ y(x + ) f(x ) mɛ 87, m )

92 m mx y(x) = m x y ((m)) (x) m ɛ m ɛ y(x ) y(x + ) m ɛ = ɛ [ y(x (m )ɛ )y(x (m 4)ɛ ) y(x + (m )ɛ )y(x + mɛ y(x mɛ (m )ɛ )y(x ) y(x + (m 4)ɛ )y(x (m )ɛ ) ] = ɛ y(x mɛ (m )ɛ )y(x ) y(x + (m )ɛ = m y ((m)) (x) mxy(x) m x y ((m)) (x) = ɛ (m )ɛ [y(x )y(x (m )ɛ )y(x y(x mɛ = (m )ɛ y(x )y(x ɛ = m y ((m)) (x) mxy(x) (m 4)ɛ ) y(x + (m 4)ɛ ) y(x + (m 4)ɛ ) y(x + )y(x mɛ ) (m )ɛ )y(x + mɛ ) (m )ɛ )y(x )] (m 4)ɛ )y(x + x y ((m)) (x) = m y ((m)) (x) mxy(x) mɛ mɛ ) y(x + )[y(x )] (m )ɛ )[y(x + mɛ mɛ ) y(x )] u(x) = y ((m)) (x), x u(x) mp(x)u(x + ɛ ) = mq(x) mx y(x) + p(x)y(x mɛ ) = q(x)y((m+)) (x) by.5 du dt = a(t) + b(t)u + c(t)u (.04) 88

93 (Riccati) a(t), b(t), c(t). (.04) u (t) du (t) dt du(t) dt + dv(t) dt dv(t) dt u(t) = u (t) + v(t) = a(t) + b(t)(u (t) + v(t)) + c(t)(u (t) + v(t)) = a(t) + b(t)(u (t) + v(t)) + c(t)(u (t) + u (t)v(t) + v(t) ) = du (t) dt + [b(t) + c(t)u (t)]v(t) + c(t)v(t) = [b(t) + c(t)u (t)]v(t) + c(t)v(t) v(t) d dt v(t) = [b(t) + c(t)u (t)] v(t) c(t) /v(t) u (t) /v(t). (.04) u (t), u (t) u(t) = g(t) f(t) (.05) (.04) dg dt f g df dt = a(t)f + b(t)fg + c(t)g α(t) { } { } dg df a(t)f [b(t) + α(t)]g f g + [b(t) α(t)]f + c(t)g = 0 dt dt β(t) df + [b(t) α(t)]f + c(t)g = β(t)f dt dg a(t)f [b(t) + α(t)]g = β(t)g dt df + b(t)f + c(t)g = [α(t) + β(t)]f dt dg a(t)f b(t)g = [α(t) + β(t)]g dt 89

94 { } f(t) f(t) exp [α(t) + β(t)dt] { } g(t) g(t) exp [α(t) + β(t)dt] df + b(t)f + c(t)g = 0 dt (.06) dg a(t)f b(t)g = 0 dt (.07) {f, g }, {f, g } c, c f = c f + c f g = c g + c g u = g/f c 0 (= c /c ) u = g c 0 g f c 0 f c 0 = uf g uf g (.08) u, u (.05) u = g f u = g f (.08) c 0 = u u f (.09) u u f (.06) df dt = f [ b(t) c(t)u ] df dt = f [ b(t) c(t)u ] f df dt = b(t) c(t)u f df dt = b(t) c(t)u d dt log f = b(t) c(t)u d dt log f = b(t) c(t)u 90

95 [ ] d dt log f = c(t)(u u ) t [ ] f = exp c(t)(u u )dt f f (.09) u(t) c 0 = u u [ ] exp c(t)(u u )dt (.0) u u 3. (.04) u, u, u 3 u(t) u(t) u (t) u 3 (t) u (t) u(t) u (t) u 3 (t) u (t) = C (.) C (.06), (.07) df + b(t)f + c(t)g = 0 dt dg a(t)f b(t)g = 0 dt {f, g }, {f, g } {f, g} c, c f = c f + c f g = c g + c g {f 3, g 3 } c 3, c 4 f 3 = c 3 f + c 4 f g 3 = c 3 g + c 4 g u = g/f c 0 (= c /c ) u = g c 0 g f c 0 f u 3 = g 3 /f 3 ĉ 0 (= c 4 /c 3 ) u 3 = g ĉ 0 g f ĉ 0 f 9