FORTRAN FORTRAN ff9 JAVA file

Transcription

1 6 I (6,4, )

2 FORTRAN FORTRAN ff9 JAVA file

3 () Riemann Gauss 58

4 Legendre

5 I Ci 37 I k di dt = ki, (k > ) di I = kdt, log I = kt + c, c e c c I = ce kt t = I = I c = I I = I e kt I T I = I, I e kt = I, e kt =, ekt = T = k log = 693 k k = 4 sec T = 59 radiocarbon dating 4 C 573 T t n 4 N 4 C 4 N + n 4 C + p 4 C I I / 4

6 4 C t =, I = I 4 C 4 C 4 N + e + ν, β 4 C 4 C 4 C ẋ = 3x /3, x() = dx 3x /3 = dt, x/3 = t + c, c x = (t + c) 3 c = x = t 3 c { (t c) x(t) = (t c) 3 (t > c) ẋ = f(t, x) () f(t, x) t x Γ f(t, x) f x Γ Γ (t, x ) () x = ϕ(t) ϕ(t ) = x () () x = ϕ(t) x = ψ(t) t = t ϕ(t ) = ψ(t ) (3) 5

7 x = ϕ(t) t < t < t () ϕ(t) = f(t, ϕ(t)), ϕ(t ) = x ϕ(t) t < t < t ϕ(t) = x + t t f(τ, ϕ(τ))dτ () ϕ(t) x ϕ (t) ϕ (t) ϕ (t) ϕ n (t) {ϕ n (t)} () () ẋ = x, t =, x = x(t ) = x = ϕ(t) t ϕ(t) ϕ(t ) = ϕ(τ)dτ, ϕ(t) = + t ϕ (t) = ϕ (t) = + ϕ (t) = + ϕ 3 (t) = + t t ϕ (τ)dτ = + t t t ϕ (τ)dτ = + t +! t t t t ϕ(τ)dτ t ϕ (τ)dτ = + t +! t + 3! t3 ϕ n (t) = + t +! t + 3! t3 + + n! tn ϕ, ϕ,, ϕ n, ϕ(t) = e t ẋ i = f i (t, x, x,, x n ) ; i =,,, n (4) ẋ = f(t, x) (t, x, x,, x n ) n + Γ (4) Γ f i (t, x, x,, x n ) ; f i (t, x, x,, x n ) x j ; i, j =,,, n Γ (5) 6

8 Γ (t, a) t (4) x = ϕ(t) (6) ϕ(t ) = a (7) (4) x = ϕ(t), y = ψ(t) t ϕ(t ) = ψ(t ) = a Lipschitz f(t, x), (f i (t, x, x,, x n ) ; i =,,, n) L Γ (t, x), (t, y) f(t, y) f(t, x) L y x (8) x x = max{ x,, x n } L Lipschitz f(t, x) (t, a) Lipschitz (t, a) Γ (t, a) r >, L > Q n+ r (t, a) = {(t, x); (t, x) (t, a) r, r > } Lipschitz Lipschitz f(t, x) Γ f(t, x) Lipschitz 5 4 Lipschitz (t, a) Lipschitz ẋ = 3x /3, x() = f(x) = 3x /3 Lipschitz Lipschitz 3x /3 3y /3 L x y, (x, y ) L > y =, x 3x /3 L x 7

9 f(t, x) Lipschitz L y(t), x(t) [t, t ] 4 t t t y(t) x(t) y(t ) x(t ) exp (L(t t )) (9) u ẋ = f(t, x, u) (),(t, a, u ) f(t, x, u) f(t, y, v) k (x, u) (y, v) () ρ x a ρ, u u ρ x, u (t, x ) (t, x, u) n ẋ i = a ij (t)x j + b i (t) ; i =,,, n () j= () a ij (t) b i (t) t < t < t () t < t < t n ẋ i = a ij (t)x j ; i =,,, n (3) j= ẋ = A(t)x (4) x = ϕ(t) t ϕ(t ) = (5) 8

10 ϕ(t), t < t < t (6) ϕ (t), ϕ (t),, ϕ r (t) (7) (4) c, c,, c r (4) ϕ(t) = c ϕ (t) + c ϕ (t) + + c r ϕ r (t) (8) ϕ (t), ϕ (t),, ϕ r (t) (9) (4) c, c,, c r c ϕ (t) + c ϕ (t) + + c r ϕ r (t) () (9) t = t ϕ (t ), ϕ (t ),, ϕ r (t ) () (9) (9) t (9) (4) ϕ (t), ϕ (t),, ϕ n (t) () (4) () (4) (4) ϕ(t) c, c,, c n ϕ(t) = c ϕ (t) + c ϕ (t) + + c n ϕ n (t) (3) ẋ = A(t)x + b(t) (4) x = ψ(t) (4) x = ϕ(t) + ψ(t) (5) x = ϕ(t) (4) (4) 9

11 r = r(t) V = 4 3 πr3 S = 4πr dv dt = ks, k t = r = r dr dt = k r = kt + r m = ρv v m d(mv) dt = mg, m dv dt + dm dt v = mg, dv dt + m dm dt v = g m dm dt = dρv V ρ dt = V ks = 3k r, dv dt = dv dr dr dt = k dv dr dv dr + 3 r v = g k 5 ; v ; v v dv dr + 3 r v =, dv v = 3 r dr, log v = 3 log r + c = log r 3 + c v = c r 3, c v = Ar λ, A, λ Aλr λ + 3Ar λ = g k, A(λ + 3)rλ = g k A λ =, A = g 4k v = g 4k r v = c r 3 + g 4k r

12 t =, r = r, v = = c r 3 + g 4k r, c = g 4k r4 v = g 4k r { ( r r ) 4 } (4) () (4) x = c (t)ϕ (t) + c (t)ϕ (t) + + c n (t)ϕ n (t) (6) t x (4) ċ (t)ϕ (t) + ċ (t)ϕ (t) + + ċ n (t)ϕ n (t) = b(t) (7) ϕ (t), ϕ (t),, ϕ n (t) t (7) ċ (t), ċ (t),, ċ n (t) c (t), c (t),, c n (t) ẏ = a(t)y + b(t) (8) ẋ = a(t)x t a(τ)dτ t x = ce t a(τ)dτ t y = c(t)e ċ = b(t)e t t a(τ)dτ t t a(τ)dτ t y = e b(t)e t t a(τ)dτ dt (9) dv dr + 3 r v = g k v = c r 3, c

13 c = c(t) r = r v = ċ r 3 + 3c r 4 + 3c r 4 = g k, ċ = g k r3 c = g k r4 4 + c, c v = g 4k r + c r 3 = g 4k r + c r 3, c = g 4k r4 v = g { ( ) } 4 4k r r r y (n) + a (t)y (n ) + + a n (t)y = b(t) (3) a i (t) b(t) t < t < t x = y, x = ẏ,, x n = y (n ) (3) x, x,, x n ẋ = x ẋ = x 3 (3) ẋ n = x n (3) y = ψ(t) (3) (3) (3) ẋ n = a n (t)x a n (t)x a (t)x n + b(t) x = ψ(t) = (ψ(t), ψ(t),, ψ (n ) (t)) (33) x = ϕ(t) = (ϕ (t), ϕ (t),, ϕ n (t)) (34) (3) (3) y = ϕ (t) (35) catenary y x x + x y = f(x) x T x T x T y W T y s x + x T T x

14 s T, T W T x = T x = T T y + W = T y x T y = T y (x) T y = T y (x + x) = T y + dt y dx x + O( x ) T y T y = dt y dx x T θ T y = T x tan θ = T dy dx, T y T y = T d y dx x ρ W = ρg s = ρg ( x) + ( y) = ρg d y T x = ρg + dx d y dx = k + + ( ) dy x dx ( ) dy, k = ρg dx T ( ) dy x dx y dy dx = u(x) du dx = k + u du = k + u dx log (u + + u ) = kx + c, (c ) I x = ; dy dx = u = c = u + + u = e kx e kx = u + + u = u + + u 3

15 II x = ; dy dx = u = ekx e kx y = k (ekx + e kx ) + c y = k c = hyperbolic function sinh x = ex e x y = k (ekx + e kx ), cosh x = ex + e x, tanh x = ex e x e x + e x y = cosh kx k ds a x a s = = a a (y ) + dx = a [ sinh kx k ] a a = sinh ka k sinh kx + dx = a a cosh kxdx d x dt + ω x =, x() = a, x () = b x = Ae λt (x = a cos ωt + ω b sin ωt ) dx dt + x = et, x() = x = et (x = (e t + e t ) ) 3 t dx dt + x = t, x() = dx dt = t x + t t = t > (x = 5 t ) t 4

16 [, ) f(t), f(t) L [f(t)] = F (s) = f(t) [a, b] f(t) [a, b] e st f(t)dt () f(t) t f(t ) f(t + ) t > N f(t) < Me γt () M > γ f(t) t γ f(t) t N t > N γ F (s) s > γ N > e st f(t)dt = e st f(t)dt N N N s > γ e st f(t)dt + N e st f(t) dt e st Me γt dt = e st f(t)dt M s γ e st f(t) dt f(t)(t > ) L [f] = F (s) s = s s > s 5

17 s = s + s, s > e st f(t) e st f(t) s s s, s Re(s) > Re(s ) L [f] s > α s < α α e st f(t) s α s > α α e s t f(t) s α s e st f(t) s < α s s α e st f(t) e s t f(t) α α s f(t) F (s) L [f(t)] = F (s) f(t) F (s) f(t) = L [F (s)] t > t N (u)du = (3) N (t) null function L [N (t)] = F (s) f(t) f(t) + N (t) F (s) t N t > N f(t) F (s) L [F (s)] = f(t) f(t), g(t) f(t), g(t) L [F (s)] 3 F (s) = L [f(t)] = e st f(t)dt 6

18 F (s) = L [f(t)] f(t) = L [F (s)] λf (s) + µg(s) λf(t) + µg(t) F (λs) λ f( t λ ) F ( s λ ) λf(λt) 3 e λs F (s) f(t λ)u(t λ) : λ > 4 e λs F (s) e λs λ f(t)e st dt f(t + λ) : λ > 5 F (s λ) e λt f(t) 6 t s F (s) f(τ)dτ s n F (s) t τn τ f(τ)dτdτ dτ n (t τ) (n ) = t f(τ)dτ (n )! 7 sf (s) f(+) f (t) : f(t) sf (s) f(+) e as {f(a+) f(a )} f (t) : f(t) t = a s n F (s) f()s n f ()s n f (n ) ()s f (n ) () f (n) (t) : f(t), f (t),, f n (t) 8 F (n) (s) ( t) n f(t) 9 F (σ)dσ f(t) s t s σ n σ F (σ)dσdσ dσ n t n f(t) F (s)g(s) f(t) g(t) T e T s e st f(t)dt f(t) = f(t + T ) P (s) Q(s) n P (α k ) k= Q (α k ) eα kt P (s) : n Q(s) = (s α ) (s α n ): α i 3 lim s F (s) = 4 F (log s) π lim f(t) = lim sf (s) t + s lim f(t) = lim sf (s) : f(t) t s t u f(u) du Γ(u) e s /4u F (u)du f(t ) u F ( s) s πt F (/s) s n+ U(t) = (t ), (t > ) e u /4t f(u)du t n/ u n/ J n ( ut)f(u)du 3 f(t), g(t) t f g = t f(t τ)g(τ)dτ (t > ) (4) 7

19 f(t), g(t) f g = g f f (g + g ) = f g + f g (f g) h = f (g h) f g = = t t f(t τ)g(τ)dτ = g(t t )f(t )dt t = g f f(t )g(t t )( dt ) (t τ = t ) f (g + g ) = = t t f(t τ)(g (τ) + g (τ))dτ f(t τ)g (τ)dτ + t (f g) t = t f(t τ)g (τ)dτ = f g + f g f(t τ)g(τ)dτ = F (t) = = = (f g) h = t t τ S t t τ = (f h) g t F (t τ )h(τ )dτ f(t τ τ)g(τ)dτh(τ )dτ f(t τ τ)h(τ )g(τ)dτdτ f(t τ τ)h(τ )dτ g(τ)dτ (f g) h = (g f) h = (g h) f = f (g h) L [(f g) h)] = L [f] L [g] L [h] = L [f (g h)] τ t τ = t τ S t τ L [f g] = L [f] L [g] (5) L [f] L [g] = = e su f(u)du e sv g(v)dv e s(u+v) f(u)g(v)dudv 8

20 u + v = t, v = τ u = t τ, v (t, τ) A L [f] L [g] = = A = L [f g] e st f(t τ)g(τ)dtdτ e st { t } f(t τ)g(τ)dτ dt τ τ = t A t t e t = e t t, cos λt sin λt = t sin λt, t f(t) = f(τ)dτ [ ] L s (s + λ = t ) λ sin λt = λ (t sin λt) λ [ ] L (s + λ ) = λ ( [ ] sin λt t cos λt), s λ L (s + λ ) [ ] [ ] L (s + λ ) n, L s (s + λ ) n x λx = f(t), x() = a x(t) = ae λt + t e λ(t τ) f(τ)dτ x + λ x = f(t), x() = a, x () = b x(t) = a cos λt + b λ sin λt + λ f(t) t = t sin λt λ sin λ(t τ)f(τ)dτ 3 f(t) (, ) f(+) f (t) [, ) L [f (t)] = sl [f(t)] f(+) (6) [ ] L f (n) (t) = s n F (s) f(+)s n f (+)s n f (n ) (+) (7) f(t) [, ) α [ t ] L f(τ)dτ = s L [f(t)], (s > α) (8) [ t L τn τ ] [ t f(τ)dτdτ dτ n = L ] (t τ) (n ) f(τ)dτ = (n )! s n F (s) (9) 9

21 f() = f(+) f(t) e st [, η] f (t) se st [, η] η f(η) Me αη e st f (t)dt = [ e st f(t) ] η η ( se st )f(t)dt = e sη f(η) f(+) + s η e st f(t)dt ( < η < ) M α f(η)e sη Me (s α)η η sl [f(t)] 7 6 f(t) t = a η e st f (t)dt = [ e st f(t) ] a = e sη f(η) f(+) + s + [ e st f(t) ] η η ( se st )f(t)dt a+ η e st f(t)dt {f(a + ) f(a )}e as L [f (t)] = sl [f(t)] f(+) {f(a + ) f(a )}e as g(t) = t f(τ)dτ g(t) (, ) g(t) t f(τ) dτ M t e ατ dτ = M α (eαt ) M α eαt g(t) g(+) = g (t) = f(t) [, ) 9 [ t ] L [f(t)] = L [g (t)] = sl [g(t)] g(+) = sl f(τ)dτ g(t) = t τn τ f(τ)dτdτ dτ n 8 [ t ] [ ] (t τ) (n ) L f(τ)dτ = L t (n ) (n )! (n )! f(t) = s n F (s) n s F (σ)dσ = F (s) = d ds s e σt f(t)dtdσ = e st f(t)dt = [ f(t) ( t)e st f(t)dt = L [tf(t)] s ] e σt dσ dt = [ ] e st f(t) f(t) dt = L t t

22 [ ] P (s) L = Q(s) n k= P (α k ) Q (α k ) eα kt P (s) : n Q(s) = (s α ) (s α n ): α i P (s) Q(s) = A s α + A s α + + A n s α n s α k s α k A k = lim s α k P (s) Q(s) (s α k) = lim P (s) s α k { } s αk Q(s) = lim s α k P (s) lim s α k T f(t) f(t T ) = f(t) L [f(t)] = = P (α k) Q (α k ) { } s αk Q(s) () T e T s e st f(t)dt, f(t) = f(t + T ) () f(t) f(t T )U(t T ) = f(t){u(t) U(t T )} F (s) e T s F (s) = F (s) = L [f(t)] = = = = = T T T T e st f(t)dt T e st f(t)dt + e st f(t)dt + T T + + T e st f(t)dt T e T s e st f(t)dt (n+)t e st f(t)dt + + e st f(t)dt + nt e s(τ+t ) f(τ + T )dτ T e s(τ+nt ) f(τ + nt )dτ + T T e st f(t)dt + e T s e sτ f(τ)dτ + + e nt s e sτ f(τ)dτ + f(t)dt { + e T s + + e nt s + } = T f(t)dt e T s

23 L [f(t)] = F (s) f (t) lim F (s) = () s lim f(t) = lim sf (s) (3) t + s lim s e st f (t)dt = L [f (t)] = sf (s) f(+) s f (t) t = t (f(t + ) f(t ))e ts s s L [f (t)] lim s lim f(t) = lim sf (s), f(t) (4) t s e st f (t)dt = P f (t)dt = lim f (t)dt P = lim (f(p ) f(+)) = lim f(t) f(+)) P t L [f (t)] = sf (s) f(+) s lim f(t) f(+) = lim sf (s) f(+) t s [ L F (log s) = ] t u f(u) du = Γ(u) t = u L [ f(t ) ] = e st f(t )dt = e log s t f(t)dt = s t f(t)dt e st t u dt f(u) Γ(u) du = f(u) s u du = F (log s) (5) [ ] L e a /4t = e a s (6) πt s e su f(u) du u = e su f(u) du u

24 = π f(t) [ L πt u e s /4u F (u)du = e tu e s /4u πu dudt = ] e u /4t f(u)du 6 = = π u e s /4u e ut f(t)dtdu e st f(t) dt = L [ f(t ) ] (7) t e st /4t e u dtf(u)du πt e us f(u) s du = s F ( s) (8) 4 L [af(t) + bg(t)] = al [f(t)] + bl [g(t)] f(t) = 3t + t + 5, f(t) = cosh at L [e at f(t)] = F (s a) L [e at cos ωt] = s a (s a) + ω 3 L [f (t)] = sl [f] f(+), L [f (t)] = s L [f] sf(+) f () f(t) = t L [t ] = s L [ t ] sf() f () f (t) = t, f = L [] = s L [ t ] s, L [ t ] = s s = 3 f(t) = cos ωt f (t) = ω cos ωt s L [cos ωt] s f() f () = ω [cos ωt], (s + ω )L [cos ωt] = s, L [cos ωt] = [ ] t 4 L f(τ)dτ = s F (s) [ ] t L cos λτdτ = s [ ] (= L sin λt = λ λ s s + λ = s + λ λ s + λ ) 5 L [ tf(t)] = d ds F (s), L [( t)n f(t)] = dn ds n f(s) f(t) = sin λt, L [f(t)] = λ s + λ L [tf(t)] = d ds ( λ s + λ ) = λs (s + λ ), L [ t f(t) ] = λ(3s λ ) (s + λ ) 3 6 te t 4(s ) sin t (s s + 5) [ ] f(t) 7 L = F (σ)dσ t s [ ] L e λt e µt = t s ( σ λ σ s µ µ )dσ = log s λ s s + ω 3

25 [ ] L sin ωt = ω [tan t s σ + ω dσ = ω σ ] = π s tan ω s = tan ω s [ ] L cos ωt [ t ] ( cos ωt) L = t s ( s σ σ + ω )dσ = [ log σ log σ + ω ] s = log σ ω σ = log( + + ω s ) 8 L [f(t a)u(t a)] = e as F (s) ( < t < π) f(t) = (π < t < π) = {U(t) U(t π)} + sin (t π)u(t π) sin t (t > π) L [f(t)] = s e πs s + eπs s + s + 3 s ( + 4s + e t cos 3t + ) 3 sin 3t 3 4 e s s + 3 (s ) (s + ) e 3(t ) U(t ) ( e 9 t (3t ) + e t) s (s + 4) t sin t 4 5 s(s + λ ) λ 4 ( cos λt t sin λt) [ ] 6 L s 4 + 4a 4 = 3 (sin at cosh at cos at sinh at) 4a s 4 + 4a 4 = (s a) + a (s + a) + a = 8a 3 L [f [ (t)] = sl ] [f] f() L s s 4 + 4a 4 = sin at sinh at a { } s + a (s a) + a + s + a (s + a) + a 4

26 5 F (s) = L [f(t)] f(t) = L [F (s)] N (t) : δ(t) s U(t) = 3 s n (n =,, ) (n )! 4 (λ > ) t λ λ s Γ(λ) 5 s a e at a 6 s s + a cos at 7 a s + a sin at 8 s s a cosh at a 9 (s + a ) 3 (sin at at cos at) a s t (s + a ) sin at a a s a sinh at a e λs (λ > ) δ(t λ) 3 s e a s erfc( a t ) 4 e a s e a /4t s πt t n 5 e a s ae a /4t πt 3 π 6 /4 erfc(s/) e es t e s /4 s erfc( s ) erf(t) s erf( t) s + e a/s cos at s πt e a/s sin at s 3/ πa J (at) s + a 3 ( s + s) ν (ν >, s > ) J ν (t) s + s tj (s + a ) 3/ (at) a tj (s + a ) 3/ (at) U(t) = (t ), (t > ) 5

27 δ ϵ (t) δ ϵ (t) = ϵ < ϵ < t < ϵ ϵ < t δ ϵ (t) δ ϵ (t)dt = ϵ t f(t) f(t)δ ϵ (t)dt f() = ϵ (f(t) f())δ ϵ (t)dt = (f(t) f())δ ϵ (t)dt ϵ max ( f(t) f() ) t ϵ δ ϵ (t)dt = max ( f(t) f() ) (ϵ ) t ϵ f(t) ϵ δ(t) = lim ϵ + δ ϵ(t) f(t)δ(t)dt = f(), f(t)δ(t a)dt = f(a) δ(t) δ(t) U(t) t δ(τ)dτ = δ(τ)u(t τ)dτ = U(t), U (t) = δ(t) L [δ(t a)] = e st δ(t a)dt = e as L [ e /t t ] = = exp( s) s exp( st t t )dt = exp( τ s s τ )dτ = exp{ (τ s/τ) s}dτ s exp( s) exp{ (τ s/τ) }dτ = s exp( s) s s σ exp { ( s/σ σ) } s dσ, (σ = τ ) s ( + σ ) exp { ( s/σ σ) } dσ = exp( s) π exp( s) exp( ξ )dξ = s s [ ] L e a /4t = e a s πt s a [ ] [ L ae a /4t = e as, L erfc( a ] πt 3 t ) = e as /s 6

28 f(t) = e t f(t ) [ L e t] = π 6 d dt t e s /4u u u + du = [ L e t] π = /4 erfc(s/) es s π e x dx x + (s/) [ t ] L [erf(t)] = L π e τ dt = [ s π L e t] /4 = es s erfc(s/) [ L erf( ] t) = e st t e τ dτdt π e τ dτ = e t / t [ L erf( ] t) = πs t = x Laplace 6 st e t e dt = t s s + [ ] cos t L = t e st cos t t dt = e sx cos xdx = π s e /4s [ ] cos at L = e a/s πt s [ L sin ] [ t cos ] τ t = L τ dτ = πe /4s s 3/ 5 f(t) = a a t ( < t < T ), f(t) = f(t + nt ) (9) T F (s) = e st ( a T t)dt = a T s a s( e T s ) e st T T 3T t () 7

29 f(t) = A δ(t nt ) () n= F (s) = T T A e T s () t f(t) = A ( ) n δ(t nt ) (3) n= T T 3T 4T t F (s) = A + e T s (4) 3 f(t) f(t) = a T t + a + g(t) g(t) = a t ( < t < T ), g(t) = g(t + nt ) T (5) 3a a a T T 3T t F (s) = = e st ( a T t + a)dt + e T s a s( e T s ) T e st ( a T t)dt (6) 6 () x + x = e t, x() = 3 x = e t + e t x x 6x =, x() =, x () = x = 5 (4e t + e 3t ) 3 x 5x + 6x = e t, x() = x () = x = (e3t e t + e t ) 4 x + 6x + x =, x() = x () = x = e 3t (cos t + 4 sin t) x + 4x + 5x =, x() =, x () = x = e t (cos t + sin t) 8

30 5 x + 3x + x = te t, x() = x () = x = e t (5 t + t ) 4e t 6 x + 4x = U(t 3), x() = x () = x = ( cos (t 3))U(t 3) 4 7 x + 4x + 5x = 4 cos 5t, x() =, x () = x = e t (cos t sin t) cos 5t + sin 5t 8 x + 4x = Iδ(t a), x() = x () = x = I sin (t a)u(t a) 9 x + 4x = F U(t a), x() = x () = x = F ( cos (t a))u(t a) 4 x 3x + x =, x() = a, x () = b x = (5a b)e t + ( a + b)e t x 3x + x =, x() =, x () = x () = a s X a 3sX + X =, X = s a s a x = a(et e t ) x() = = a(e e), a = e e x = et e t e e x + λ x =, x() =, x (l) = x () = a X = a s + λ x = a sin λt λ x = a cos λt x (l) = cos λl a = cos λl x = sin λt (cos λl = ) λ cos λl { x x 3y =, x() = 3 x + y + y =, y() = { { sx X 3Y = (s )X 3Y = X + sy + Y = X + (s + )Y = { x = e t + 3e t y = e t e t { x + y + 4y =, x() = 4 x y + 4x =, y() = x = e t cos t, y = e t sin t 5 { x + y + x = e, x() = x + y + 3x + y =, y() = x = e t (cos t + sin t), y = e t ( + sin t) 9

31 7 7 m f(t) v m dv dt = f(t), v() = (7) a F f(t) = F U(t a) m dv dt f(t) F a t V (s) = F m e as s v(t) v(t) = m F (t a)u(t a) (9) a t m t I f(t) = Iδ(t a) m dv dt = Iδ(t a), v() = (3) a t V (s) = I m e as s f(t) v(t) I m v(t) = m I U(t a) (3) a t (3) t = t(> a) mv(t) mv() = t v(t) = m I, (t > a) Iδ(t a)dt ax + bx = f(t), x() = x (a, b, a ) (3) X(s) = as + b (ax + F (s)) (33) x() = x x() = F c (s) = ax f c (t) = ax δ(t) 3

32 7 t [, ) x(t) ax + bx + cx = f(t), (a, b, c, a ) (34) x() = x, x () = x (35) L [x(t)] = X(s) L [f(t)] = F (s) a[s X(s) sx x ] + b[sx(s) x ] + cx(s) = F (s) [as + bs + c]x(s) = ax s + ax + bx + F (s) Z(s) = as + bs + c, A(s) = ax s + ax + bx (36) X(s) = A(s) Z(s) + F (s) Z(s) W (s) = Z(s) 37 (37) X(s) = W (s)a(s) + W (s)f (s) (38) [ ] [ ] A(s) F (s) x(t) = L + L Z(s) Z(s) = L [W (s)a(s)] + L [W (s)f (s)] (39) Z(s) A(s) L [W (s)a(s)] L [W (s)f (s)] x() = x, A(s) = ax s + ax + bx = F c(s) x () = x f c(t) L [ax s] d ϵ (t) = (t < ) ϵ ( < t < ϵ) ϵ (ϵ < t < ϵ) (ϵ < t) d ϵ(t) ϵ ϵ ϵ t ϵ D ϵ (s) = (e ϵs ) ϵ s ϵ D ϵ (s) s D(t) = lim ϵ d ϵ (t) L [D(t)] = s D(t) D(t) δ(t) D(t) = δ (t) f c (t) = L [F c (s)] f c (t) = ax D(t) + (ax + bx )δ(t) 35 3

33 quiescent condition) x() = x () = f(t) Z(s) F (s) X(s) f(t) x(t) A(s) = [ ] F (s) x(t) = L Z(s) f(t) = δ(t) w(t) L [δ(t)] = W (s) = f(t) Z(s) w(t) = L [W (s)] = L [ Z(s) ] (4) (4) (4) X(s) = W (s)f (s) (43) x(t) = w(t) f(t) = t w(t τ)f(τ)dτ ( ) (44) δ(t) w(t) W (s) 44 f(t) f(t) f(τ i )(U(t τ i ) U(t τ i τ)) = δ(t τ i )f(τ i ) τ ( τ ) τ i δ(t τ i ) w(t τ i ) f(t) f(τ i ) τ x(t) w(t τ i )f(τ i ) τ f(t) f(τ i ) τ i w(t τ i )f(τ i ) τ t f(t) = U(t) k(t) K(s) = sz(s) = s W (s), k(t) = L [K(s)]} (45) s + sk(s) = /Z(s) k(+) = W (s) = sk(s) f(t), 6 w(t) = k (t), k(+) = (46) X(s) = W (s)f (s) = K(s)sF (s) (47) = K(s){f(+) + L [f (t)]} = f(+)l [k(t)] + L [k(t) f (t)] 3

34 f(t) x(t) = f(+)k(t) + 48 f(t) f(t) f(+)u(t) + U(t τ i ) f(τ i ) f(+)u(t) f(+)k(t) τ i τ U(t τ i ) f(τ i ) k(t τ i ) f(τ i ) x(t) f(+)k(t) + k(t τ i )f (τ i ) τ X(s) = sk(s)f (s) x(t) = d dt t t k(t τ)f (τ)dτ (48) f(t) f(τ i ) f(+) f(t) k(t τ i ) f(τ i ) τ i f(+)u(t) f(+)k(t) t t k(t τ)f(τ)dτ (49) 48 f(t) Z(s) f(t) 73 i(t) q(t) f(t) L di dt + Ri + q C = f(t) (5) f(t) L C R i = dq (5) dt q() = q, q () = i() = i (5) 5 5 I(s) = sq(s) q, Q(s) = I(s) + q s L(sI(s) i ) + RI(s) + Cs (I(s) + q ) = F (s) (Ls + R + Cs )I(s) = Li q Cs + F (53) 33

35 38 W (s) = (Ls + R + Cs ), A(s) = Li q Cs (54) I(s) = W (s)a(s) + W (s)f (s) (55) LC = ω, R = µ (56) L W (s) = s L s + µs + ω = { } s + µ L (s + µ) + ω µ µ (s + µ) + ω µ w(t) = L e µt {cos ω µ t µ ω µ sin } ω µ t (57) k(t) = K(s) = L (s + µ) + ω µ (C ) LR L ω µ e µt sin ω µ t (58) L di + Ri = f(t), i() = (59) dt f(t) L R f(t) = δ(t) W (s) = Ls + R, w(t) = L [W (s)] = L e Rt/L f(t) = U(t) K(s) = s(ls + R), k(t) = L [K(s)] = R ( e Rt/L ) f(t) = E (U(t) U(t a)), a > t x(t) = w(t) f(t) = w(t τ)f(τ)dτ = E [ ] { e R Rt/L } { e R(t a)/l }U(t a) f(t) T T f (t) = f(t)(u(t) U(t T )) 34

36 f(t) f(t) = f (t nt ), F (s) = e nt s F (s) (= F (s) e nt s ) n= I(s) = W (s)f (s) = n= e nt s W (s)f (s) n= f (t) i (t) = L [W (s)f (s)] f(t) i(t) = i (t nt )U(t nt ) n= T T f(t) { E < t < T/ f(t) = E T/ < t < T, f(t) = f(t + nt ) (6) E T T T t E F (s) = { T/ } T e T s Ee st dt + ( E)e st dt T/ = E( e T s/ ) s( + e T s/ ) ( = E s tanh T s 4 ) (6) L{sI(s) i()} + RI(s) = F (s) I(s) = E T s/ e L s(s + R/L) T s/ + e T s/ e = ( e T s/ )( e T s/ + e T s/ e 3T s/ + ) + e T s/ = (e T s/ e T s/ + e 3T s/ ) = + ( ) n e n= { } I(s) = E L s(s + R/L) + ( ) n nt s/ e s(s + R/L) n= [ ] [ ] L = L L s(s + R/L) R ( s s + R/L ) = L R ( e Rt/L ) nt s/ 35

37 i(t) = E R ( e Rt/L ) + E R ( ) n { e R L (t nt/) }U(t nt/) (6) n= E/R E/R T T 3T T/ t w(t) = L e Rt/L T F (s) = L [f (t)] = E s ( e T s/ + e T s ) f (t) ] i (t) = L [W (s)f (s)] = L [E( e T s/ + e T s ) s(ls + R) = E R ( e Rt/L ) E R ( e R(t T/)/L )U(t T/) + E R ( e R(t T )/L )U(t T ) E f (t) E/R i (t) E T T t T T 3T T/ t E/R T (6) LR () (3) i(t) = A e R L (t nt ) U(t nt ) L n= i(t) = A ( ) L n e R L (t nt ) U(t nt ) n= 36

38 8 8 x(t) X(s) = L [x(t)] tx + ( t)x x =, x() =, x () = L [tf(t)] = df ds d ds (s X s ) + sx + d (sx ) X = ds sx s dx ds x() = c = tx + x + 4tx =, x() = 3, x () = + + sx + X + sdx ds X = ( s + s) dx ds sx = ( s + ) dx ds = X, X = c s, x = e t dx ds = s X + x = cet d ds (s X sx() x ()) + sx x() 4 dx ds = ( s 4) dx ds sx = dx X + sds s + 4, X = c s + 4, x = cj (t) 6 x() = cj () = 3 c = 3 3 tx + x + tx =, x() =, x(π) = x () = c x = 3J (t) d ds (s X s c) + (sx ) dx ds = sx s dx ds (s + ) dx ds =, + + sx dx ds = X = arctan s + A dx ds = s + 37

39 s X A = π/ X = π arctan s = arctan s x(π) = x = L [ arctan s ] = sin t t 4 x tx + x =, x() =, x () = (s X s ) + sx + d ds (sx ) + X = s dx ds + (s + s )X = + s + s e (s+ s )ds = s e s / d ds (s e s / X) = ( + s + s )s e s / X = (s e s / ) ( + s + [ ] s )s e s / ds = (s e s / ) se s / + e s / + c = s + s + c s es / = s + s + c s ( s + 8 s4 + ) = s + + c s c( 8 s + ), L [ s k] =, k =,,, x = + (c + )t x () = c = x = + t 5 ( sx s dx ds tx + ( t)x + nx =, n =,, + x()) + sx x() ( X sdx ds ) + nx = (s s ) dx ds + (n + s)x = dx X = ( n s n + s )ds, X = (s )n s n+ L [ t n e t] = n! (s + ) n+, t = (tn e t ) () = (t n e t ) () = = (t n e t ) (n ) = [ L (t n e t ) (n)] = n!s n (s + ) n+ [ ] (s ) x = L n s n+ = et d n n n! dt n (t n e t ( ) n n! ) = (m!) (n m)! tm = l n (t) m= l n : 38

40 8 = sk(s) G(s) (g x)(t) = t G(s)X(s) = F (s), g(t τ)x(τ)dτ = f(t) (63) [ ] F (s) x(t) = L G(s) x(t) = L [skf ] = d dt L [KF ] = d dt (g f)(t) = d dt f() = L [sf ] = f (t) x(t) = L [KsF ] = d dt t t t k(t τ)f (τ)dτ cos (t τ)x(τ)dτ = t + s s + X(s) = s 3 + s, X(s) = + 4 s + s 4 x(t) = 3 t3 + 4t + δ(t) k(t τ)f(τ)dτ y = f(x) m (x, y) η mv = mg(y η), v = g(y η) s ds = g(y) dy y ds v = v ds dη dη = v g(η)dη y y y η g(η)dη = c, c [ ] L [g] L y = s c π, L [g] = c s s, L [g] = c πs g = ds dy = a y, + ( dx dy ) = a y, a η x y = a sin ( θ) θ = x = x = a (θ + sin θ), y = a ( cos θ) 39

41 3 x(t) x(t) + t t g(t τ)x(τ)dτ = f(t) (64) e t τ x(τ)dτ = cos t X + s X = s s + 4, X = s s + 4 x(t) = cos t sin t x (t) + x(t) + t e t τ x(τ)dτ =, x() = (65) sx + X + s X = s, X = s (s + )s = s + s + s x(t) = cos t + sin t 83 t t < t + nh x(t) a n x(t + nh) + + a (t + h) + a x(t) = f(t) (66) t + nh t < t + (n + )h x(t) t t x(t) (66) h n λ L [f(t + λ)] = e λs F (s) e λs f(t)e st dt, : λ > 3x(t + ) 4x(t + ) + x(t) =, x(t) = ( t < ), x(t) = ( t < ) (67) 3{e s X(s) e s x(t)e st dt} 4{e s X(s) e s x(t)e st dt} + X(s) = (3e s 4e s + )X(s) = 3 es + e s s 4 es s X(s) = 3e s e s s(3e s 4e s + ) = 3es + s(3e s ) = s + s e s e s /3 = s + 3 s 3 n e ns n= x(t) = + 3 n= 3 n U(t n) (68) 4

42 a n+ 3a n+ + a n = n, a = a = (69) x(t + ) 3x(t + ) + x(t) = t, x(t) = ( t < ) (7) n a n = x(n + t) ( t < ) (7) e s X(s) e s x(t)e st dt 3{e s X(s) e s x(t)e st dt} + X(s) = s 3 X(s) = = s 3 (e s )(e s ) = ( ) s 3 e s e s e s ( k+ ) s 3 e (k+)s k= x(t) = a n = ( k+ )(t (k + )) U(t (k + )) k= ( k+ )(n (k + )) U(n (k + )) k= n a n = ( k+ )(n k) (n ) (7) k= 3 a n+ 5a n+ + 6a n = 4 n, a =, a = (7) x(t + ) 5x(t + ) + 6x(t) = f(t), x(t) = ( t < ), x(t) = ( t < ) (73) f(t) r = 4 (73) f(t) = r n, n t < n +, n =,,, = r n (U(t n) U(t (n + ))) (74) L [f(t)] = = n= r n s (e ns e (n+)s ) e s s( re s ) e s X(s) es s ( e s ) 5e s X(s) + 6X(s) = e s s( 4e s ) (75) 4

43 X(s) = e s ( e s ) s(e s )(e s 3) + e s s(e s )(e s 3)( 4e s ) { } 3e s e s + s e s = e s s n t < n + { / e s / s + 3e 4e s } x(t) = 3 n n + n 3 n + 4n = (4n n ) = a n (76) 84 z t = 4xt + 3 sin x, z(x, ) = x (77) z(x, t) t L t [z(x, t)] = Z(x, s) sz(x, s) z(x, ) = 4x + 3 sin x s, Z(x, s) = 4x s s sin x s + x s 3 3{sZ(x, s) z(x, )} + z(x, t) = xt + 3 sin x t + x 3 z t + z x =, z(x, ) =, z(, t) = et (78) Z(x, s) x Z(, s) = s =, 3sZ(x, s) + Z(x, s) x Z(x, s) = s e 3sx z(x, t) = L t [Z(x, s)] = e t 3x U(t 3x) =, Z(x, s) = ce 3sx z t = k z x (79) z(, t) = f(t), x z(x, t) (8) ( ) z(x, ) =, z = (8) t z(x, t) t L t [z(x, t)] ( ) s Z(x, s) sz(x, ) z = k t Z(x, s) x 4 t= t=

44 (8) s s d s Z(x, s) Z(x, s) = Z(x, s) = A(s)e k x + B(s)e dx k (8) k x Z(, s) = L t [z(, t)] = L t [f(t)], x Z(x, s) B(s) =, A(s) = L t [f(t)] Z(x, s) = L t [f(t)] e s k x z(x, t) = f(t x k )U(t x k ) k 4 z t = k z x (8) z(, t) = a, z(x, ) = (83) z(x, t) t L t [z(x, t)] Z(x, s) x s Z(x, s) = k s s Z(x, s) = A(s) exp ( k x) + B(s) exp ( k x) x B(s) = Z(, s) = a s, [ ] [ L e a /4t = e a s L πt s [ L e a ] s = t s A(s) = a s Z(x, s) = a s exp ( s k x) ] e a /4t = πt 3 s a e a /4t dt = erf a πt 3 t e a x dx = x a e a s ( z(x, t) = a erf ) x k t 43

45 3 3 ϕ R R K D ϕ K ϕ K ϕ D R D x, x ϕ(x) = exp ( ), x < x ϵ > K f ϕ D ϕ f K T ϕ D T (ϕ) =< T, ϕ > T (ϕ + ϕ ) = T (ϕ ) + T (ϕ ) T (λϕ) = λt (ϕ), λ ϕ j j ϕ T (ϕ j ) T (ϕ) T ϕ j j ϕ ϕ j j ϕ j m j ϕ f < T f, ϕ >= T f R f(x)ϕ(x)dx f, g T f = T g f T f < f, ϕ >=< T f, ϕ > δ(x) < δ(x), ϕ >= ϕ() 44

46 δ(x a) < δ(x a), ϕ >= ϕ(a) f < f x, ϕ >= f x ϕ(x)dx = [fϕ] f ϕ ϕ dx = f, x x T T/ x T ϕ, ϕ = T, x x Heaviside U, (x < ) U(x) = +, (x = ) +, (x > ) U, ϕ = U, ϕ = U(x)ϕ (x)dx = ϕ (x)dx = [ϕ(x)] x= x= = ϕ() = δ, ϕ U (x) = δ(x), x δ, ϕ = δ, ϕ = ϕ () δ(x)dx = U(x) δ (m), ϕ = ( ) m ϕ (m) () α f αf, ϕ = (α(x)f(x))ϕ(x)dx = f(x)(α(x)ϕ(x))dx = f, αϕ T α αt, ϕ = T, αϕ αδ, ϕ = δ, αϕ = α()ϕ() = α()δ, ϕ αδ = α()δ 45

47 xδ = αδ, ϕ = δ, αϕ = (αϕ) x= = α()ϕ () α ()ϕ() = α()δ α ()δ, ϕ αδ = α()δ α ()δ xδ = δ, x δ =, xδ (m) = mδ (m ) xt = T T = Cδ, C( ) αt x = α x T + α T x T δ δ (m) T = T (m) 3 T L [T ] = T, e st L [U(t)] = s [ ] (s > ), L [δ(t)] =, L δ (m) (t) = s m (m ), L [δ(t a)] = e as L [ U(t)t λ] Γ(λ + ) = s λ+ (λ >, s > ), L [ U(t)e at t λ] Γ(λ + ) = (s a) λ+ (λ >, s > a) L [U(t) cos at] = s s + a (s > ), L [U(t) sin at] = a s + a (s > ) T, S L [T S] = L [T ] L [S] L [ e as F (s) ] = L [ e as] L [F (s)] = δ(t a) f(t)u(t) = f(t a)u(t a) L [ e as F (s) ] = = n= n= [ ( a) n ] L s n f(t)u(t) = n! n= ( a) n {f(t)u(t)} n! (n) = f(t a)u(t a) ( a) n δ n! (n) f(t)u(t) 46

48 4 4 z = x + iy f(z) z f f(z + z) f(z) (z) = lim z z δ > z z < δ z f (z) f(z) z = z f(z) = u(x, y) + iv(x, y) u x = v y, u y = v x u v 3 w x + w y =, P, Q P dx + Qdy = ( Q x P y )dxdy C f(z) 4 C A f(z)dz = f(z) a f(a) = f(z) πi z a dz 47 C

49 f n (a) = n! f(z) πi n+ dz (z a) C f(z) f(z) (Goursat 5 f(z) z = a f(z) = f(a) + f (a)! (z a) + f (a)! (z a) + + f (n) (a) (z a) n! n + z = a z = a ρ ( ) a 6 R f(z) M f(z) M f(z) f(z) f(z) (/f(z) ) z, z f(α) f(β) = α β πi C f(z)dz (z α)(z β) z = Re ϕ f(α) = f(β) z (z = /ζ ζ = ) (Liouville ) 7 8 f(z) f(z) = g(z) n, g(z) z = a (z a) f(z) z = a pole f(z) (z a) a z = a f(z) 9 a = lim z a (n )! d n dz n {z a)n f(z)} f(z) f(z)dz = πi (C ) C 48

50 f(z) = z z z branch f(z) f(z) = a n (z a) n + + a z a + a + a (z a) + (z a) z = a 4 f(t) F (s) π F (ξ + iη) = = f(t)e (ξ+iη)t dt π (f(t)u(t)e ξt )e iηt dt (4) π π F (ξ + iη) f(t)u(t)e ξt f(t)u(t)e ξt = π, s = ξ + iη π F (ξ + iη)e iηt dη f(t)u(t) = π F (ξ + iη)e (ξ+iη)t dη (4) f(t)u(t) = iπ ξ+i ξ i F (s)e st ds (43) Bromwich s integral formula (43) s = ξ (43) Bromwich contour y C F (s)e st s = ξ R ABCDE AB Γ L = R ξ R L f(t)u(t) = lim R iπ = lim R { iπ ξ+il ξ il D F (s)e st ds F (s)e st ds iπ 49 Γ E R θ B ξ + il ξ il A F (s)e st ds} (44) x

51 44 Γ R f(t)u(t) = F (s) F (s)e st (45) Γ R Γ Γ s = Re iθ F (s) < M R k k >, M (46) e st F (s) Γ lim F (s)e st ds = (47) R 46 P (s), Q(s) F (s) = P (s)/q(s) P (s) Q(s) Γ (BCDEA) BC,CD,DE,EA BC s = Re iθ,θ θ π/ I BC = I BC π/ BC e st F (s)ds = π/ θ e Rt cos θ F (Rte iθ ) Rdθ M R k θ e Rte iθ F (Rte iθ )ire iθ dθ π/ θ e Rt cos θ dθ M R k ϕ e Rt sin ϕ dϕ θ = π/ ϕ, ϕ = π/ θ = arcsin (ξ/r) sin ϕ sin ϕ = ξ/r I BC M ϕ R k e ξt dϕ = Meξt R k ϕ = Meξt arcsin ξ/r k R R lim R I BC = CD s = Re iθ,π/ θ π I CD = CD e st F (s)ds = π π/ θ = π/ + ϕ I CD M π R k e Rt cos θ dθ M π/ R k ϕ π/ sin ϕ ϕ/π I CD M R k π/ e Rteiθ F (Rte iθ )ire iθ dθ π/ e Rt sin ϕ dϕ e Rϕt/π dϕ = πm tr k ( e Rt ) 5

52 lim R I CD = DE CD EA BC lim R I DE = lim R I DE = Γ 47 P (s), Q(s) P (s) Q(s) F (s) = P (s)/q(s) 46 P (s), Q(s) m, n, (n > m) a, b = P (s) = a s m + a s m + + a m, Q(s) = b s n + b s n + + b n F (s) = = P (s) = a s m + a s m + + a m Q(s) b s n + b s n + + b n a + (a /a R)e iθ + (a /a R )e iθ + + (a m /a R m )e miθ b R n m + (b /b R)e iθ + (b /b R )e iθ + + (b n /b R n )e niθ a /a, a /a,, a m /a A b /b, b /b,, b m /b B R > A + + (a /a R)e iθ + (a /a R )e iθ + + (a m /a R m )e miθ + A R + A R + + A R m + A R < R > B + + (b /b R)e iθ + (b /b R )e iθ + + (b n /b R n )e niθ (b /b R)e iθ + (b /b R )e iθ + + (b n /b R n )e niθ ( B R + B R + + B R n ) B R A +, B + R F (s) a b R n m / M R k M 4a /b k = n m 44 F (s) s = 43 F (s) = s 3 s = 3 f(t) = L [ s 3 ] = est s 3 est = lim(s 3) s 3 s 3 = e3t 5

53 F (s) = (s ) 3 (s + ) s = 3,s = s = lim s! d [ ds ] (s ) 3 est (s ) 3 (s + ) = lim d [ s ds e st ] (s + ) = lim s {(s + ) t 4(s + )t + 6}e st (s + ) 4 = 6 (t 4t + 3)e t s = ] lim d [(s + ) s! ds est (s ) 3 (s + ) = lim s [ d ds e st ] (s ) 3 (ts t 3)e st = lim s (s ) 4 3 F (s) = = 6 ( t 3)e t f(t) = L [ (s ) 3 (s + ) ] = 6 {(t 4t + 3)e t + ( t 3)e t } a s + a e st F (s) = ae st (s ai)(s + ai) s = ai, s = ai i eiat, i e iat f(t) = L [ ] a s + a = i (eiat e iat ) = sin at 4 F (s) = e a s s f(t)u(t) = ξ+i iπ ξ i e st a s s ds (48) s = D G E F y C = e st a s iπ s ds = C AB BCD DE H EF B ξ + il x A ξ il + + F G GHA 5

54 F (s) 46 [BCD],[GHA] R R R r r f(t)u(t) = { lim e st a s iπ R,r s ds + e st a s s ds + e st a s } s ds DE F G EF [DE] s = xe i(π ), s = xi s D E x R r DE e st a s r s ds = R e xt ai x R x dx = r e xt ai x x dx [FG] s = xe i(π ), s = xi s F G x r R [EF],s = re iθ f(t)u(t) = iπ = iπ = π lim R,r R EF r R lim i R,r r F G e st a s R s ds = r e st a s s ds = i π π e xt+ai x x dx e xt (e aix e aix ) x dx iπ e xt sin a x x e xt sin a x x dx π π π dx π lim dθ = π e rteiθ a re iθ/ dθ R π lim i R,r π π π a lim re iθ/ r erteiθ e xt sin a x x dx e rteiθ a re iθ/ dθ dθ f(t)u(t) = erf(a/ t) = erfc(a/ t) 5 F (s) = s tanh T 4 s F (s) = s et s/ e T s/ + e st F (s) s F (s) T/ s = e T s/ + =, e T s/ =, T s/ = nπi = ±, ±3, ±5, s = s n i, s n = nπ/t s n i σ = s s n i s = s n i σ = e st F (s) σ /σ s = e st = e (σ+s ni)t = e s nti { + o(σ)} σ + s n i = s n i ( + σ s n i ) = { + o(σ)} s n i e T s/ = e T s/ (σ + s n i) = e T σ/ e T s ni/ = e T σ/ = { + T σ/ + o(σ )} 53

55 4i T s n e s nti, e T s/ e T s/ + = 4 T σ { + o(σ)} e st F (s) = 4i e T s snti n σ { + o(σ)} s n = nπ, n = ±, ±3, ±5, T f(t) = 4i T { = 4i T = 4 π n=,3, n=,3, n=,3, n s n e s nti + n=,3, s n e s nti } s (e snti e snti ) = 4i n T sin nπ T t n=,3, s n i sin s n t 54

56 5 5 f (x), f (x), f 3 (x) [a, b] d y dx + f (x) dy dx + f (x)y = f 3 (x) (5) d y dx + f (x) dy dx + f (x)y = (5) (5) y y = y z (5) { } d z dx + f (x) + y dz y dx = f 3(x) y (8) dz dx = y e f (x)dx { e f (x)dx f 3 (x)y dx + c } { y = y y e f (x)dx e f (x)dx f 3 (x)y dx + c } dx + c y (53) (5) y, y y = c y (x) + c y (x) dy dx = c dy dx + c dy dx + y dc dx + y dc dx c, c y dc dx + y dc dx = (54) dy dx = c dy dx + c dy dx d y dx = c d y dx + c d y dx + dy dc dx dx + dy dc dx dx 55

57 (5) y, y (5) (54) (55) dy dc dx dx + dy dc dx dx = f 3(x) (55) dc dx = f 3(x)y, dc dx = f 3(x)y, = y dy dx y dy dx (56) f3 (x)y c = dx + A f3 (x)y, c = dx + A y = A y + A y y f3 (x)y dx + y (5) (5) (5) () t = e f (x)dx dx f3 (x)y dx (57) ψ(t) = f (x)e f (x)dx, ϕ(t) = f 3 (x)e f (x)dx (5) d y + ψ(t)y = ϕ(t) dt () y = u(x)v(x) ψ(x) = f (x) f (x) 4 {f (x)}, ϕ(t) = f 3 (x)e f (x)dx (5) d u + ψ(x)u = ϕ(x) dx (5) d v + ψ(z)v = (58) dz z (5) f (z), f (z) z = a ψ(z) z = a v (z) = A + B(z a), A, B v(a) = b, v (a) = b z = a ψ(z) 5 L(u) d u dz + P (z)du + Q(z)u = (59) dz 56

58 P (z), Q(z) z = a (z a)p (z), (z a) Q(z) z = a (59) z z = a (z a)p (z) = p(z a), (z a) Q(z) = q(z a) p(z a), q(z a) z = a p(z a) = p + (z a)p + (z a) p +, q(z a) = q + (z a)q + (z a) q + (5) (59) z = a d u dz p(z a) + z a u = (z a) λ {A + du q(z a) + dz (z a) u = (5) A n (z a) n } (5) (5) (5) (5) (z a) A ϕ(λ) = A ϕ(λ + ) + A ϕ(λ) = A ϕ(λ + ) + A ϕ(λ + ) + A ϕ(λ) = A n ϕ(λ + n) + A n ϕ(λ + n ) + A n ϕ(λ + n ) + + A ϕ(λ) = n= (53) ϕ(λ) = λ + (p )λ + q, ϕ m (λ) = λp m + q m, m =,, (54) A ϕ(λ) = λ + (p )λ + q = (55) λ z = a (59) λ, λ (53) (59) u = (z a) λ { + n= A() n (z a) n } u = (z a) λ { + n= A() n (z a) n } (56) (55) λ, λ Frobenius λ (53) A, A, λ, A (5) (5) L(u) = A ϕ(λ)(z a) λ (57) ϕ(λ) = λ u u (53) A n ϕ(λ + )ϕ(λ + ) ϕ(λ + n) λ λ = m, n m λ λ λ λ A n A A = C(λ λ ) u = u(z, λ) u(z, λ ) = ku (z) (58) 57

59 (57) (λ λ ) λ λ = λ [ ] u(z, λ) v(z) = λ λ=λ = ku (z) log (z a) + (z a) λ n= ( ) An (z a) λ n (59) λ=λ ( A λ ) λ=λ v(z) z a (z a) λ u (z) 53 Riemann Gauss z = a, b, c Q(z) = d u dz + P (z)du + Q(z)u = (5) dz P (z) = z α a α + β β + γ γ z b z c { } αα (a b)(a c) (z a)(z b)(z c) z a + ββ (b c)(b a) + γγ (c a)(c b) z b z c α, α, β, β, γ, γ,α + α + β + β + γ + γ = Riemann z = a b c u = P α β γ z α β γ (5) Reimann Riemann a =, b =, c = α, β, γ, α, α, β, γ γ, β, γ α β Gauss z( z) d y + {γ (α + β + )z}dy αβy = (5) dz dz z =, z =, z = Gauss Riemann y = P α z γ β γ α β (53) 58

60 54 F (α, β, γ, x) n= + A n x n + αβ α(α + a)β(β + ) x + x! γ +! γ(γ + ) α(α + a) (α + n )β(β + ) (β + n ) x n + (54) n!γ(γ + ) (γ + n ) pf q (a,, a p ; b,, b q ; x) k= c k+ (k + a )(k + a ) (k + a p ) = c k (k + b )(k + b ) (k + b q )(k + ) c k x k (k =,, ) p F q generalized hypergeometric function F (α, β, γ, x) = F (α, β; γ; x) x η < x = n n + A n (n + )(γ + n) = A n+ (α + n)(β + n) = + + γ α β n + O( n ) + γ α β > γ > α + β γ α + β x = F (,,, x) = F (,,, x), ( + x)p = F ( p, β, β, x) log( x) x = F (,,, x), log ( + x) = xf (,,, x) df (α, β, γ, z) dz d F (α, β, γ, z) dz = e x = lim β F (, β,, x β ) cos nx = F ( n, n,, sin x) = αβ γ F (α +, β +, γ +, z) (55) α(α + )β(β + ) F (α +, β +, γ +, z) (56) γ(γ + ) F (α, β, γ, z) = F (α, β, γ, z) = Γ(γ) Γ(β)Γ(γ β) πi i i t β ( t) γ β ( tz) α dt (57) Γ(α + s)γ(β + s)γ(γ + s) ( z) s ds (58) Γ(γ s) ( ( (5)) 59

61 z = (5) (55) λ =, λ = γ λ = ϕ(λ) = λ + (γ )λ = y = A n z n (59) n= Gauss (5) {(n + )(n + γ)a n+ [n + (α + β)n + αβ]a n }z n = (53) n= A n+ = (n + α)(n + β) (n + )(n + γ) A n z < y = A F (α, β, γ, z) (53) λ = γ y = z γ η (53) z( z) d η + { γ (α + β γ + 3)z}dη (α γ + )(β γ + )η = dz dz y = z γ F (α γ +, β γ +, γ, z) (533) Gauss (5) y = AF (α, β, γ, z) + Bz γ F (α + γ, β + γ, γ, z) (534) z = ξ = z (5) ξ( ξ) dy + {α + β γ + (α + β + )ξ}dy dξ dξ αβy = y = F (α, β, α + β γ +, z) (535) y = ( z) γ α β F (γ β, γ α, γ α β +, z) (536) 6

62 3 z = z = ξ, y = ξα η ξ( ξ) dη + {α β + (α γ + )ξ}dy α(α γ + )η = dξ dξ y = z α F (α, + α γ, + α β, z ) (537) y = z β F (β, + β γ, + β α, z ) (538) = a b c P α β γ z α β γ ( ) α ( z α z γ z β z β ) γ f ( α + β + γ, α + β + γ, + α α, ) (z α)(γ β) (z β)(γ α) (f,b = z b c b ) (539) Riemann 55 Legendre (5) α = n +, β = n, γ = z( z) d y + ( z)dy + λy =, λ = n(n + ) (54) dz dz F (α, β, γ, z) z < z = γ > α + β z = γ α + β z = (54) α = n +, β = n, γ = γ = α + β n z = t = z Legendre ( t ) d y dt tdy + λy =, λ = n(n + ), n =,,, (54) dt [ d ( t dt ) dy ] + λy =, λ = n(n + ), n =,,, (54) dt,, (54) (54) y = a k t k n= [(k + )(k + )a k+ k(k )a k ka k + λa k ]t k = k= 6

63 (k + )(k + )a k+ [k(k + ) λ]a k = a k+ = k(k + ) λ (k + )(k + ) a k, (k =,,, ) a, a λ = n(n + ) n n n z = (t ) n (t ) dz dt ntz = (t ) dz(n) dt n(n + )tz (n ) = [ ] d ( t dt ) dz(n) + n(n + )z dt (n) = d n P n (t) = n n! dt n (t ) n, λ = n(n + ), (n =,, ) (Rodrigues (543) Legendre P (t) =, P (t) = t, P (t) = (3t ), P 3 (t) = (5t3 3t), P 4 (t) = 8 (35t4 3t +3), P 5 (t) = 8 (63t5 7t 3 +5t), Legendre P n (t) = F (n +, n,, ( t)/) Legendre P n (t) dt = n (n!) = ( )n (n)! n (n!) d n dt n (t ) n dn dt n (t ) n dt = ( )n n (n!) (t ) n dn dt n (t ) n dt (t ) n dt = ( )n (n)! n (n!) ( )n 4 n 3 5 (n + ) = n + t = ξ (t ) n dt = ( ) n ( ξ)n ξ / dξ = ( ) n B(n +, /) = ( ) n Γ(n + )Γ(/) Γ(n + 3/) B(n +, /) P n (t)p m (t)dt = 3 P n (t) P n ( t) = ( ) n P n (t) d n (m n) n + (m = n) P n (t) = n n! dt n [(t + ) n (t ) n ] Leibniz P n () = (544) 6

64 4 Goursat f (n) (x) = n! πi f(z) n+ dz (z x) (543) P n (x) = (z ) n πi n n+ dz, Γ x (Schläfli ) (545) Γ (z x) x > z = x + x e iϕ Γ x x P n (x) = π π (x + x cos ϕ) n dϕ (Laplace ) (546) P n (x) x (546) P n (x) ( x ) 5 Laplace z < x ± x P n (x)z n = π n= = π = π n= π [(x + x cos ϕ)z] n dϕ dϕ (x + x cos ϕ)z = πz x xz + z π ( dϕ t cos ϕ = π π t dϕ xz z x cos ϕ ) z > / z = xz + z n= P n(x)z n ( z < ) P n (x) n= z n+ ( z > ) ( x ) (547) 6 (547) z x (n + )P n+ (n + )xp n (x) + np n (x) = (n =,, ) (548) P (x) xp (x) = P n+(x) xp n(x) = (n + )P n (549) 7 P n (x) n n 56 d y dt + dy t dt ν + ( )y = (55) t 63

65 t = ϕ(λ) = λ ν = λ = ±ν λ = ν J ν (t) = k= ( ) k ( t ) ν+k (55) k!γ(ν + k + ) ν J ν (t) N ν (t) = cos νπ J ν(t) J ν (t) sin νπ (55) ν H ν () (t) = J ν (t) + in ν (t), H ν () (t) = J ν (t) in ν (t) J ν (t), N ν (t), H ν (), H ν () (t) t = J ν (t) t ν ν Γ(ν + ), t : J (t), Jν(t) (ν > ) (553) N ν (t) o( t ν ) (ν > ), N (t) π log t, t : N ν(t) (554) t y(t) t cos (t + δ) (555) d dt {tν J ν (t)} = t ν J ν (556) d dt {t ν J ν (t)} = t ν J ν (557) (556),(557) J ν(t) = ν t J ν(t) + J ν (t) J ν(t) = + ν t J ν(t) J ν+ (t) J ν+ (t) = ν t J ν(t) J ν (t) (558) ν = n = J n (t) = J n = ( ) n J n (t) k= ( ) k ( t ) n+k (559) k!(n + k)! n = J (t) = J (t) 64

66 3 e (/)t(u /u) = n= J n(t)t n u = e iθ e it sin θ = + n= J n (t)e niθ fillustexe t = kx x y = J ν (kx) J ν (kx) ν > J ν (x) d y dx + x dy dx + (k ν )y = (56) x J ν (µ i ) =, µ i > i =,, l xj ν (µ i x l )J ν(µ j x l )dx = (j i) l J ν (µ i ) = l J ν+(µ i ) (j = i) (ν > ) (56) 65

67 6 6 e x dx = π I = e x dx I = e y dy = e x dx e (x +y ) dxdy (x, y) x = r cos θ, y = r sin θ (r, θ) x =, y =, θ = π, r = I = = π/ erf(x) = x e ξ dξ π e r rdrdθ = e r d(r ) [θ] π/ = e ax dx = e x dx = π re r dr π, (a > ) a erf( ) = e ξ dξ = π erf( x) = erf(x) π/ dθ [ e r ] π = π 4 erfc(x) = erf(x) 66

68 3 Fresnel sin a a da = sin x dx = [ sin a cos x dx = π π e ax sin ada = + x 4, ] e ax dx da = π sin x dx = π x cos x dx = π x dx dx + x 4 = π e ax sin ada x x x π π/4 z = te 4 i C e z dz R e x dx + π/4 e R e iϕ (ire iϕ )dϕ + π R (cos t i sin t )e 4 i dt = R (cos t i sin t )dt = e π 4 i e x dx = i π π = ( i) y O π/4 R x C(x) = x cos t dt, S(x) = sin t dt C( ) = S( ) = π 8 c(x) = π8 C(x) = 4 x cos t dt, s(x) = π8 S(x) = sin x x dx = π ( Si(x) = x sin t dt, t si(x) = π Si(x) = Si( ) = π Ei(x) = x Ci(x) = e t t dt, x cos t dt t Li(x) = x x log t dt x sin t dt t sin t dt 67

69 5 e x x + a dx = a π ea erfc(a) a p = y I(p) = e p(x +a ) x + a dx di dp = e p(x +a ) π dx = e pa p I(p) = p = p πe pa dp = p π a y e y dy = π a erfc(a p) 6 Laplace df (b) db b = = e ax cos bxdx = = b a F (b) = πa e b 4a, (b > ) xe ax sin bxdx = [ e ax a e x cos bxdx e ax cos bxdx = b a F (b) df (b) db F (b) = ce b 4a, F () = = b a F (b) sin bx] c e ax dx = πa e ax a b cos bxdx c = πa, 7 e tx sin b x π x dx = erf(b/ t) I(b) x = y I(b) = π e ty sin by y dy I b = π e ty cos bydy = π 68 π t e b /4t = e b /4t πt

70 I() = 8 Γ(x) = t = u I(b) = b e b /4t db = b/ t πt π t x e t dt (x > ) Γ() = Γ( ) = t e t dt = t x e t dt = [ x t x e t] + x e t dt = Γ(x + ) = xγ(x) e q dq = erf(b/ t) e u du = π t x e t dt = x Γ(x + ) x = n =,, 3, Γ(n + ) = n! x = n + =, 3, 5, Γ( 3 ) = Γ( ) = π, Γ( 5 ) = 3 Γ(3 ) = 3 π,, Γ(n + ) = 4 (n)! π n n! 9 B(p, q) = tp ( t) q dt, (p >, q > ) t = cos θ B(p, q) = π/ p = m =,,, q = n =,, (cos θ) p (sin θ) q dθ B(p, q) = Γ(p)Γ(q) Γ(p + q) B(m, n) = (m )!(n )! (m + n )! Γ(p)Γ(q) = x p e x dx x = ξ, y = η y q e y dy = Γ(p)Γ(q) = 4 ξ p η q e (ξ +η ) dξdη x p y q e (x+y) dxdy 69

71 ξ = r cos θ, η = r sin θ Γ(p)Γ(q) = 4 r = s Γ(p)Γ(q) = Γ(p + q) r p+q e r dr r p+q e r dr = π/ π/ (cos θ) p (sin θ) q dθ s p+q e s ds = Γ(p + q) (cos θ) p (sin θ) q dθ = Γ(p + q)b(p, q) [ t ] L τ p (t τ) q dτ = L [ t p t q ] = Γ(p) Γ(q) s p s q = Γ(p)Γ(q) s p+q t t = Γ( ) = π τ p (t τ) q dτ = Γ(p)Γ(q) Γ(p + q) tp+q ( Γ( ) ) = B(, π/ ) = dθ = π Γ(p)Γ( p) = e x dx = π sin πp π () x = t( t) Γ(p)Γ( p) = B(p, p) = t p ( t) p dt = x p + x dx < p < z p /( + z) z = Γ z p = e (p ) log z log z r + A O D z = r z = R B C lim r,r AB z p + z dz = 7 x p + x dx

72 [BC] lim r,r CD z p + z dz = eipπ z p + z lim R e(p ) log R + z BC x p + x dx = eipπ c Rp R πr cr p, (p < ) z p + z dz = [DA] z p + z lim r e(p ) log r + z DA cr p πr ce p z p + z dz = lim r,r Γ z p + z dz = ( eiπp ) x p + x dx x p + x dx Γ z = z = e (p ) log u log z = iπ e iπ(p ) iπe iπ(p ) = ( e iπp ) x p + x dx x p iπeiπp dx = + x e iπp = π sin πp 6 J (t) J (t) t d y dt + dy dt + ty = y() = J () =, y () = J () = d ds {s Y s } + {sy } dy ds = dy ds = sy s + Y (s) = c s + lim s sy (s) = c, lim t J (t) = c = lim f(t) = lim sf (s) t s L [ J (t) ] = s + 7

73 J (t) J (t) = J (t) L [ J (t) ] = L [ J (t) ] = {sl [ J (t) ] } = s + = s + s s + 3 J n (t) L [J (t)], L [J (t)] L [J n (t)] = ( s + s) n s + s J n+ (t) = n t J n(t) J n (t) L [J n+ (t)] = n = n s s L [J n (t)] ds L [J n (t)] ( s + s) n ds ( s + s) n s + s + = ( s + s) n ( s + s) n s + = ( s + s) n s + s s + s = x 4 tj (at), tj (at) J (z) = J (z) L [tj (at)] = d ds L [J (at)] 5 J ν (t) ν >, s > = d ds s + a = s (s + a ) 3/ d da L [J (at)] = d da s + a = a (s + a ) 3/ L [tj (at)] = L [tj (at)] = a (s + a ) 3/ a (s + a ) 3/ L [J ν (t)] = ( s + s) ν s + 7

74 ν >, s > L [ ( ) ν (/) ] ( ) ν t π Γ(ν) J ν (/)(t) = s + 6 J ( ut) J ( t) = t + t t [ L J ( ] t) = s ( s + s 3! s 3 + ) = s e /s [ L J ( ] ut) = s e u/s 7 J ( ut)f(u)du [ L J ( ] ut)f(u)du = s e u/s f(u)du = F (/s)/s 63 R(x, y) x, y ϕ(x) x R(x, ϕ(x))dx (6) ϕ(x) x ϕ(x) x R(x, ϕ(x)) P (x), Q(x) x P (x) + Q(x) ϕ(x) 6 I[m] = x m dx, J[m] = ϕ(x) dx (x λ) m ϕ(x) ϕ(x) = a x 4 + a x 3 + a x + a 3 x + a 4 (m + 4)a I[m + 3] + (m + 3)a I[m + ] + (m + )a I[m + ] +(m + )a 3 I[m] + m 4 I[m ] = x m ϕ(x) mϕ(λ)j[m + ] + (m )ϕ (λ)j[m] + (m )ϕ (λ)j[m ] + m 3 ϕ 6 (λ)j[m ] + m ϕ (λ)j[m 3] = ϕ(x) (x λ) m I[m] I[], I[], I[] J[m] J[], J[], J[ ], J[ ] J[] = I[], J[ ] = I[] λi[], J[ ] = I[] λi[] + λ I[] I[], I[], I[], J[] 73

75 ϕ(x) = a x 4 + a x 3 + a x + a 3 x + a 4 a = a 4 x = /t a 4 = x = t t ϕ(x) a ϕ(x) = (ax + bx + c)(αx + βx + γ) x = At + B t + dx = dt ϕ(x) ±(t ± µ )(t ± ν ) (6) aβ bα c + (A + B) b + AB a = γ + (A + B) β + AB α = A, B A, B ξ (A + B) ξ + AB = ξ ξ c b a = γ β α aβ bα = x = t b/a t u dx = du ϕ(x) ( u )( k u ) 6 (t u) k (t µ )(t ν ) t = µ u (t < µ ), t = ν /u (t > ν ) µ /ν [µ < ν ] (t µ )(ν t ) t = ν ( k u )(µ < t < ν ) (ν µ )/ν [µ < ν ] (t + µ )(t ν ) t = ν /( u )(t > ν ) µ /(µ + ν ) (t + µ )(ν t ) t = ν ( u )(t < µ ) ν /(µ + ν ) (t + µ )(t + ν ) t = µ t /( u ) (ν µ )/µ [µ < ν ] I[] = du ( u )( k u ), I[] = u ( u )( k u ) du, I[] = J[] = u ( u )( k u ) du du (u λ)( u )( k u ) I[] v = u J[] = I[] k I[] = k u k u du u (u λ ) ( u )( k u ) du + λ du (u λ ) ( u )( k u ) 74

76 Legendre-Jacobi u = sin ϕ dϕ = du u u ϕ du ( u )( k u ) = dϕ = F (ϕ; k), (k < ) (63) k sin ϕ u k u u du = ϕ u du ( + cu ) ( u )( k u ) = ϕ k sin ϕdϕ = E(ϕ; k), (k < ) (64) dϕ ( + c sin ϕ) = Π(ϕ; c, k), (k < ) (65) k sin ϕ ϕ = π/ K(k) = F ( π, k) = π = π fillustexe [ dϕ k sin ϕ + ( ) k + ( ) 3 k ( ) ] (r )(r 3) 3 k r + r (r ) 4 64 g(x) G(x) G (x) = g(x) g(x)dx = G(x) + c (f G) = f G + f g f(x) g(x)dx = f(x) G(x) f (x) G(x)dx b f(x) g(x)dx = [f(x) G(x)] b a b a a f (x) G(x)dx sin (α ± β) = sin α cos β ± cos α sin β, sin α sin β = {cos (α β) cos (α + β)}, cos cos (α ± β) = cos α cos β sin α sin β α cos β = {cos (α + β) + cos (α β)} sin A + sin B = sin A + B cos A + cos B = cos A + B sin α cos β = {sin (α + β) + sin (α β)} cos A B, sin A sin B = cos A + B sin A B cos A B, cos A cos B = sin A + B sin A B sin x = sin x cos x, cos x = sin x = cos x 75

77 3 hyperbolic function sinh x = ex e x, cosh x = ex + e x, tanh x = ex e x e x + e x cosh ( x) = cosh x, sinh ( x) = sinh x cosh (ix) = cos x, sinh (ix) = i sin x, tanh (ix) = i tan x cos (ix) = cosh x, sin (ix) = i sinh x, tan (ix) = i tanh x cosh x sinh x =, tanh x = cosh x sinh (x ± y) = sinh x cosh y ± cosh x sinh y, cosh (x ± y) = cosh x cosh y ± sinh x sinh y sinh x = log (x + + x ), cosh x = log (x + x ) d sinh x = cosh x, d dx tanh x = log + x x cosh x = sinh x, d dx dx tanh x = cosh x 4 e px cos qxdx = epx (p cos qx + q sin qx) p + q + c, e px sin qxdx = epx (p sin qx q cos qx) p + q + c 5 p >, q > p > π/ π/ n =,, 3, π/ sin n θdθ = π/ sin p θ cos q θdθ = sin p θdθ = π/ Γ( p + )Γ(q + ) Γ( p + q + ) cos p θdθ = πγ( p + ) Γ( p + ) 4 (n ) cos n θdθ = 3 n 3 5 (n ) 4 n π (n =, 3, 5, ) (n =, 4, 6, ) 76

78 7 7 l x x f(x)dx y(x) η ζ y l R x δx δθ δx = Rδθ x η = g(ζ) η η = g(ζ) ζ x κ, R κ = [ ] d y + (dy/dx) 3/ dx (7) R = κ ( d ) = y dx (7) x x + δx δθ δx = Rδθ x η (R + η)δθ Rδθ = ηδθ = ηδx R η/r E Eη/R F = η η (73) η g(η)eη/rdη (74) η g(η)ηdη = (75) 77

79 x M I M = I = η η η 7 g(η)eη/r ηdη = E R η η g(η)η dη = EI R (76) η g(η)η dη (77) EI d y = M, ( < x < l) (78) dx x M(x) x M(x) = l y = y = y = y = y = y = x (ξ x)f(ξ)dξ (79) d M(x) dx = f(x), ( < x < l) (7) EI d4 y 4 = f(x), ( < x < l) (7) dx 7 x =, x = l W d 4 y dx 4 = W, ( < x < l) (7) EI : y() =, y () =, y(l) =, y (l) = (73) Y (s) = L [y(x)] s 4 Y s 3 y() s y () sy () y () = W EIs y () = c, y () = c 3 y() =, y () = Y = c s 3 + c 3 s 4 + W EIs 5, y(x) = c x + c 3 6 x 3 + W 4EI x4 y(l) =, y (l) = c = W l EI, c 3 = W EI y(x) = W 4EI x (l x) (74) 78

80 x =, x = l W d 4 y dx 4 = W, ( < x < l) (75) EI : y() =, y () =, y(l) =, y (l) = (76) Y (s) = L [y(x)] y () = c, y () = c 3 y() =, y () = Y = c s + c 3 s 4 + W EIs 5, y(x) = c x + c 3 6 x 3 + W 4EI x4 y(l) =, y (l) = c = W l 3 4EI, c 3 = W l EI y(x) = W 4EI (l3 x lx 3 + x 4 ) = W 4EI x(l x)(l + lx x ) (77) W δ(x l/) d 4 y dx 4 = W δ(x l/), ( < x < l) (78) EI Y = c s + c 3 s 4 + W e ls/ EI s 4, y(x) = c x + c 3 x 3 + W 6 6EI (x l/)3 U(x l/) y(l) =, y (l) = c = y(x) = 3W l 6EI, c 3 = W l EI x = l/ W 48EI {3l x 4x 3 + 8(x l/) 3 U(x l/)} (79) 3 x = x = l x = a W δ(x a) y (l) =, y (l) = d 4 y dx 4 = W δ(x a), ( < x < l) (7) EI : y() =, y () =, y (l) =, y (l) = (7) Y = c s 3 + c 3 s 4 + W e as EI s 4 y(x) = c x + c 3 6 x 3 + W 6EI (x a)3 U(x a) c = W a EI, c 3 = W EI 79

81 y(x) = a = l a l 4 W 6EI {3lx x 3 + (x a) 3 U(x a)} (7) y(x) = W 6EI (3lx x 3 ) (73) 73 x = l H = 3 W l3 EI W y / l l H = 48 W l3 EI / 4l l H = 9 W l3 EI I a b 77 I = a3 b a I = πa4 4 E = N/m 8

82 8 dx = f(x, y) dt dy = g(x, y) dt t S x ϕ t (x) : (t, x) x t S ϕ ϕ t ϕ s = ϕ t+s f, g t 8 ( ) ( ) ( ) ẋ x a b = A, A = ẏ y c b A (8) ( ) ( ) ( ) sx x() a b X = sy y() c d Y (8) (8) x() y() X = a s c b a s d s c b, Y = a s d s c x() y() b d s (83) a s b p(s) = c d s = s (a + b)s + (ad bc) (84) A exp A = e A = k= A k k! (85) 8

83 (x, y) exp A P (89) A = ( a b A = B = P AP e B = P e A P (86) ( a e A = e ai+bb = e ai e bb = e a = e a { n= (bb) n n! b AB = BA e A+B = e A e B (87) e A = (e A ) (88) ) ( ) b e A = e a cos b sin b (89) a sin b cos b ( ) B = ) b a = ai + bb, B n = ( ) n I, B n+ = ( ) n B + n= k= (bb) k k! (bb) n+ (n + )! } = ea { ( = e a {cos bi + sin bb} = e a cos b sin b n= ) sin b cos b ( ) n b n n! I + n= ( ) n b n+ B} (n + )! d dt eta = Ae ta (8) (x(), y()) (8) ( ) ( ) x = e ta x() y A P ( ) ( ) ( ) ( ) ( ) x ξ ξ ξ = P, = B, B = P λ AP = y η η η λ y() (8) A a λ b p(λ) = c d λ = (a λ)(b λ) bc = λ (a + d)λ + (ad bc) (8) p(λ) = A p(λ) (84) p(s) λ = (T r ± ), T r = a + d, D = ad bc, = T r 4D (83) > < λ = a ± bi P ( ) B = P a b AP = b a (84) 8

84 (84) λ = a ± bi P ( ) ( ) ( ) x u = P, C = P a + bi AP = y v a bi ( ) ( ) ( ) u ξ = P, P = i v η i B = = P P AP P = P CP ( ) ( ) = i a + bi i a bi ( ) i = i ( a b ) b a = ( ) P λ AP = = λi, A = λp IP = λi λ ( ) λ A = = λi λ A (8) p(λ) p(a) = A A p(λ) = µ p(λ) = (λ µ) A A = µi +N p(a) = (A µi) = N = x (x, Nx) = α u = x αnx, v = Nx ( ) N = ( ) µ B = (85) µ A (CASE ) a = b = c = d = (x, y) b, c b x = by()t, y = y() x ξ = ξ(), η = η()e µt ξ (ξ(), η()) η 83

85 CASE I A ( ) ( ) ( ) x ξ = P, B = P λ AP =, λ < < µ y η µ (8) { { ξ = λξ η = µη ξ = ξ()e λt η = η()e µt ξ η ξ η (x, y) η = p x + p y =, ξ = p x + p y = 3 A [] (86) λ <, µ < ( ) ( ) ξ lim = t η (86) λ, µ CASE II CASE III [] CASE IV ( ) ( ) ( ) x ξ = P, B = P λ AP =, λ < y η λ { ξ = ξ()e λt η = η()e λt + ξ()te λt (86) [3] CASE V λ = a ± ib, a < (84) ( ) ( ) ( ) x ξ a b = P, B =, a < y η b a (89) ( e tb = e at cos bt sin bt ) sin bt cos bt { ξ = e at (ξ() cos bt η() sin bt) η = e at (η() cos bt + ξ() sin bt) a < (86) b > b < 84

86 4 CASE VI A (86) λ >, µ > ( ) ( ) ( ) ( ) ξ ± ξ lim =, lim = t η ± t η (87) 5 CASE VII A λ = ±ib (84) ( ) ( ) ( ) x ξ b = P, B =, a < y η b e tb = ( cos bt sin bt ) sin bt cos bt π/b (ξ, η) b > b < (T r <, D > ) ( >, T r < ) D = ad bc ( <, T r < ) ( <, T r = ), ( =, T r < ) (T r >, D > ) ( <, T r > ), ( =, T r > ) = T r 4D = ( >, T r > ) dynlingfor (D < ) T r = a + d {f(x, y)} + {g(x, y)} (x, y) f(x, y) =, g(x, y) = ( x, ȳ) ( x, ȳ) ( x, ȳ) ( ) f f a b A = = x y c b g g (88) x y ( x,ȳ) 85

87 ( x, ȳ) (x(t), y(t)) lim t (x(t), y(t)) = ( x, ȳ) ( x, ȳ) U U ( x, ȳ) U (x(), y()) U (x(t), y(t)) t U ( x, ȳ) (x(t), y(t)) lim t (x(t), y(t)) = ( x, ȳ) A Liapunov ( x, ȳ) U V (x, y) V ( x, ȳ) = (x, y) ( x, ȳ) V (x, y) > ( x, ȳ) U V ( x, ȳ) ( x, ȳ) U V < ( x, ȳ) V (x, y) Liapunov Liapunov ( x, ȳ) δ U δ ( x, ȳ) U V α α > U V (x, y) < α (x, y) U U δ ( x, ȳ) ( x, ȳ) t n t n (x(t n ), y(t n )) ( x, ỹ) U δ ( x, ȳ) ( x, ỹ) = ( x, ȳ) V V (x(t n ), y(t n )) V ( x, ỹ) t > V (x(t), y(t)) > V ( x, ỹ) ( x, ỹ) ( x, ȳ) (x(t), y(t)) ( x, ỹ) τ > V (x(τ), y(τ)) < V ( x, ỹ) ( x, ỹ) V (ξ(τ), η(τ)) < V ( x, ỹ) (ξ(), η()) n (ξ(), η()) = (x(t n ), y(t n )) V (x(t n + τ), y(t n + τ)) < V ( x, ỹ) t > V (x(t), y(t)) > V ( x, ỹ) ( x, ȳ) U δ ( x, ȳ) n t n (x, y) n (x(t n ), y(t n )) (x, y) ω L ω (x, y) ω α L α (x, y) α γ γ (x, y) (x, y) L ω (x, y) L α (x, y) Poincaré-Bendixon V V 86

88 8 l, m O x ω = g l O x l mg ml d x = mg sin x (89) dt d x dt = ω sin x (8) y = dx dt dx = y dt dy (8) = ω dt sin x 8 y =, ω sin x = ( x, ȳ) = (nπ, ), n =, ±, ±, f x g x f ( ) y g = ω cos x y n = m + (m ) <, T r = n = m(m ) A = ( a c ) ( ) b = b ω >, T r = t dy dx = sin x ω y (8) (x, y ) E(x, y) = y + ω ( cos x) = y + ω ( cos x ) = c (c ) (83) c (x(t), y(t)) y + ω 4 sin x 83 V (x, y) = E(x, y) E( x, ȳ) 87

89 ( x, ȳ) = (mπ, ) V (x, y) V ( x, ȳ) = (x, y) ( x, ȳ) V (x, y) > V (x, y) = Liapunov y cos x = cos x y ω ẋ = y y > x y < x cos x y cos x y ω < (mπ, ) 83 y x y y = x y (mπ, ) kl dx dt, k > 89 ml d x dt = mg sin x kl dx dt (84) (, ) dx dt dy dt = y = ω sin x k m y (85) A = ω m k (, ) (, ) flowgfor GNUPLOT y = y = cos x x x y = ω (cos x cos x ) (86) dx dt = g cos x cos x (87) l t =, x = x = x(t) t x t = l dx g cos x cos x (x < x ) (88) 88

90 cos x = sin x k = sin x < t = l g x k dx x k sin k sin x = sin ϕ t = ϕ l dϕ g k sin ϕ 63 F (ϕ; k) = ϕ dϕ, (k < ) (89) k sin ϕ t = l g F (ϕ; k) x = x sin ϕ =, ϕ = π/, T T = 4 l g F (π ; k) (83) k = sin x F ( π π/ ; k) = ( + k sin ϕ + O(k 4 ))dϕ = π ( + 4 k + O(k 4 )) T = π l g ( + 4 k + O(k 4 )) 83 i(t), q(t) F (i(t)) L C L di(t) dt + F (i(t)) + q(t) C = e(t) LRC circuit i(t) C R L e(t) dq(t) = i(t) e(t) = L,R,C dt t = LCτ d i(τ) dτ + C L F i(i(τ)) di(τ) + i(τ) = (83) dτ 89

91 τ t,x = i f(x) = C F (i(τ)) L ẍ + f x (x)ẋ + x = (83) dx dt dy dt = y f(x) = x Lienard f(x) = λ(x 3 x) dx dt dy dt = y λ(x 3 x) = x (833) 3x x ẍ λ( x )ẋ + x = (834) Van der Pol 833 λ = 833 y f(x) = x = (835) (, ) A = ( a c ) ( ) b = b λ = ( ± 3i) (, ) A,B,C,D l ; x =, y > l ; x >, y = f(x) l 3 ; x =, y < l 4 ; x <, y = f(x) l (x, y), x =, y > x > l l 3, l 4, l, Van der Pol C y y = f(x) A l (, p) l (, σ(p)) {σ n (p)} n {σ n (p)} (, p) (, ) p σ(p) = p (, ) flowgfor van der Pol f(x) = x 3 µx µ D B x 9

92 84 y y dy dt = α y = y e αt x dy y = Cx D, C >, D > dt { x dx = A By, A >, B > dt ẋ = (A By)x, ẏ = (Cx D)y, A, B, C, D > (836) Lotka-Volterra 836 (, ), (D/C, A/B) (, ) A, D (D/C, A/B) x >, y > x = D/C, y = A/B (u, v) u > D/C >, v > A/B > A Bv = r <, Cu D = s > (x(), y()) = (u, v) (x(t), y(t)) t < τ x(t) y(t) d x log x(t) = dt x = A By r, d y log y(t) = dt y = Cx D s D C x(t) ue rt, A B y(t) vest y x < y < y = B A C D D/C x < y > A/B x = (, ) x > y < x > y > τ x(t), y(t) D/C x(t) u, A/B y(t) ve sτ x(t) t = τ x(τ) = D/C (D/C, A/B) x 9