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1 6//4 by. : : : : : : : : : : : : : : :. : : : : : : : : : : : : : : 3. : : : : : : : : : 4. : : : : : : : : : 3 5. : : : : : : : : : : : : : : : : : : : 3 6. : : : : : : : : : : : : 3 7. : : : : : : : : : : : 3 8. : : : : : : : : : : : : 3 9. : : : : : : : : : : 3. : : : : : : : : : : : : : : : : : : : : : : 4. : : : : : : : : : : : : : : : : : : : : : 4. : : : : : : : : : : : : : : : 4 3. : : : : : : : : : : : : : : 4 4. : : : : : : : : : : : : : : : : : : : : : : 5 5. : : : : : : : : : : : : : : : : : : : : 5 6. : : : : : : : : : : : : : : : : : : : : : : : : 5 7. : : : : : : : : : : : : : : : : : : : : : : : : 5 8. LCR : : : : : : : : : : : : : : : : : : : : : : : 6 9. LCR : : : : : : : : : : : : : : : : : : : 6 7. (A): (B): (C):

2 ... () at + b () at + bt + c (3) e at+b (4) co(t + ) (5) in(t + ) (6) co t (7) in t ( 8 a (» < at t<b) (» t<b) (8) f (t) (9) f (t) b (b» t) : (b» t) 8 8 < a at (» >< (» t<b) t<b) () f 3 (t) b () f : 4 (t) a (b» t<c) (b» t) >: (c» t) () (8) a h ; b h. () () h? (3) () a ; b d h; c d + h. h (3) (3) h? (3) (3)' d? ( (» t<b) (4) f 5 (t) a (b» t) f(t) (8) f(t) (9) f(t) () f(t) () a a a a b t b t b t b c t.. b b a b c d () () (3) (4) + a + a a (5) (6) (a 6 b) (7) ( +) ( a)( b) L[f L[f (t)]() +f () (t)]() () t + a () (t + b) (3) (t + c) 3

3 4. 4. L[f L[f (t)]() +f () (t)]() ; L[f (t)]() L[f (t)]() f () f () () co t () t in t (3) t co t (4) e at in t (5) e at co t (6) te at () y (t) + y(t) ; y() y ; y () v () y (t) (ff + fi)y (t) +fffiy(t) ; y() y ; y () v (ff 6 fi) (3) y (t) ffy (t) +ff y(t) ; y() y ; y () v (4) y (t) ffy (t) +(ff + fi )y(t) ; y() y ; y () v (fi>) (5) y (t) 3y (t) +3y (t) y(t) e t ; y() y () y () () t in t () t co t (3) t n (4) t coh at (5) t inh at (6) t e t 7. L[f (t)] F () L[tf (t)] F () F () F () tf (t) L [ F ()] f (t) 7. ()()(3) 6.()() (4)(5) () () (3) (4) log + (5) log ( + ) ( + ) ( + ) () t m e at () e at in t (3) e at co t (4) te at in t () () ( +) + ( +) + (3) + +5 (4)

4 .. (i) (ii) () ( + ) () ( + ). Λ convolution f Λ g(t) f (t fi )g(fi ) dfi: f (t) Λ g(t). () f Λ g(t) g Λ f (t) () f Λ (g + h)(t) f Λ g(t) +f Λ h(t) (3) (f Λ g) Λ h(t) f Λ (g Λ h)(t).. U () L[u(t)]() () (3) (t fi ) n u(fi ) dfi () co(t fi )u(fi ) dfi (4) in(t fi )u(fi ) dfi e t fi u(fi ) dfi 3. H() F ()G(), F () L[f (t)], G() L[g(t)] L [H()] L [F ()G()] L [L[f (t)]l[g(t)]] L [L[f Λ g(t)]] f Λ g(t) H() 3. () ( a) () ( a) (3) ( + ) (4) ( + ) (5) ( + ) (6) ( + ) (7) ( + ) 4

5 4. 4. () y(t) t ++ () y(t) + (3) y(t) +t + (4) y(t) int + (5) y(t) int + in(t fi )y(fi ) dfi co(t fi )y(fi ) dfi e t fi y(fi ) dfi in (t fi )y(fi ) dfi in(t fi )y(fi ) dfi () y (t) y(t) ++ e t fi y(fi ) dfi; y(). () y (t) t + (t fi )y(fi ) dfi; y() y () Mu (t) Ku(t) μu (t); t> u() u ; u () v M;μ;K Mu (t) Ku(t) μu (t) +K in pt; t> u() ; u () M;K;K ;μ;p 5

6 8. LCR 8. LCR I R (t) + L I (t) + I(t) ; t> LC I(t) I() I ; I () I R;L fl R L p LC q () >fl ( fl J I +fli ) q () <fl ( fl J I +fli ) (3) fl (J I + fli ) 9. LCR 9. LCR E(t) I (t) + R L I(t) + LC I(t) I() I I(fi ) dfi E(t); t> R;L fl R L p LC E(t) E ;p;a;b 8 >< (» t<a) () E(t) E () E(t) E in pt (3) E(t) E (a» t<b) >: (b» t) ( 8 E (» < E t<a) (4) E(t) (5) E(t) a t (» t<a) (a» t) : (a» t) E(t) (3) E(t) (4) E(t) (5) E E E a b t a t a t (6) E(t) ffi(t) : ( (» t<a) (7) E(t) : E (a» t) 8 < E (8) E(t) a t (» t<a) : (5)(7) : E (a» t) 6 (3): () (4): (8) (5): (9) (6): ()'(3)"

7 . () L[at + b] al[t] +bl[] a + b. () L[at + bt + c] al[t ]+bl[t] +cl[] a 3 + b + c. (3) L[e at+b ]L[e b e at ]e b L[e at ] eb a. (4) L[co(t + )] L[co t co in t in ] co L[co t] in L[in t] (5) L[in(t + )] L[in t co + co t in ] co L[in t] +in L[co t]» co t + (6) L[co t]l (L[co t] +L[]) + ( +4). (7) L[in t]l[ co t]l[] L[co t] (8) L[f (t)] (9) L[f (t)] Z b Z b ae t dt a( e b ). a b te t dt a( ( + b)e b ) b. ( +4). () (8) f (t), (9) f (t) f 3 (t) f (t) f (t) L[f 3 (t)] L[f (t)] L[f (t)] a(b +e b ) b : () L[f 4 (t)] Z c b ae t dt a(e b e c ). () e h : () e h lim h h h (3) e d e h e h (4) L[f 5 (t)] h Z b : : (3) e d e h e h lim h h ae t dt ae b. e d : (3) lim e d : d co in + : co + in + : (4)(5). L[e i(t+ ) ]e i L[e it ]. ei i (co + i in )( + i) + () (3). L[ffi(t)]. (3). L[ffi(t d)] e d L[ffi(t)] e d. 7

8 .»» b () L bl be at. + a ( a)» () L a» a + a L in at. + a a»»» b (3) L L bl a a a coh at b» a a L coh at b inh at. a a» a (4) L b c d h a i» b h» c d L +L +L i+l» 3 4» 3 4 al + bl + c» L + d» L a + 4 bt + c t + d 6 t3.»» (5) L L ( +) co t. +»» (6) L ( a)( b) a b L a b e bt ).» (7) L 4 e t co + +4 (5). A; B; C ( +) A + B + C + a b (eat p 3t p 5 p in 3t. 3 ( ) ( A) ( +) (B + C) (6) 3 ( +) 3 ( +) A + B + C D + + E 3 ( +) A + B + C 3 + D + E A ( +) + B + C D( +)+E D + 3 ( +) A B C D E ( : +) ( E E D ) 8

9 (7). 4 ( +) ( +) +3 + ( 5 +) +3 ( +) +3 ψ F p () p 5 3 p +( 3) 3 p +( 3) (?) (Π) (?) F ( +) F () F ( +) ( 9.(4) ) + +4 ± ± p 3i p 3i ( + )( ) (6)»» " L 4 L L 5 p # 3i + +4 ( + )( ) e t 5 p 3i e +t e t +» L 4 e t co p3t 5 p p in 3t (Π) x8. x9. 3. () () (3) () () L[t + a] L[] + a () L[(t + b) ] a +. L[(t + b)] + b (3) L[(t + c) 3 ] L[3(t + c) ]+c 3 6c b + L[t b + b] b c3 + 3 L[(t + c) ] c3 + 3c +. L[t + a] L[t] + L[a] L[(t + b) ]L[t ]+L[bt] +L[b ] L[(t + c) 3 ]L[t 3 ]+L[3ct ]+L[3c t]+l[c 3 ] 9

10 4. L[f (t)] L[f (t)] () L[co t] L[ cot in t] + L[ in t] + + ( +4). () L[(t in t) ] L[t in t] L[co t] L[t in t] L[t in t] + L[co t] ( + ). (3) () L[(t co t) ]L[co t] L[t in t] ( ) ( + ) L[t co t] L[(t co t) ]+ ( + ). (4) F 4 () L[e at in t], F 5 () L[e at co t] L[(e at in t) ]af 4 ()+F 5 () F 4 (). ( a)f 4 F 5. (5) (4) F 4 ;F 5 L[(e at co t) ]af 5 () F 4 () F 4 (). F 4 +( a)f 5 (). (4)(5) Fψ 4 ;F 5 ψ ψ ψ a F 4 a F 5 F 4 F 5 ψ. ( a) + a (6) F 6 () L[te at ] L[(te at ) ] L[e at + ate at ] a + af 6 (). F 6 () L[(teat ) ]+ a + af 6() F 6 () ( a). (). ) L[co t]l[] L[in t] L[in t].(7) ) co t (co t +).(6) ()(3). L[t in t] d L[in t]; d L[t co t] d L[co t] d 6.()()

11 (4)(5). ) L[e at e it ]L[e (a+i)t ] ) L[e at in t]() L[in t]( a); L[e at co t]() L[co t]( a).(7) 8.()(3) 5. L[y(t)] Y () () L[y (t)] + L[y(t)] Y () y() y () + Y () ( + )Y () y v. Y y + v () + y v y(t) L [Y ()] y co t v + in t. () L[y (t)] (ff + fi)l[y (t)] + fffil[y(t)] Y () y() y () (ff + fi)y () +(ff + fi)y() + fffiy () ( (ff + fi) + fffi)y () y v +(ff + fi)y. Y () y + v (ff + fi)y ( ff)( fi) c v fiy ff fi, c ffy v ff fi. y(t) L [Y ()] c e fft + c e fit. (3) L[y (t)] ffl[y (t)] + ff L[y(t)] c ff + c fi. ( ff + ff )Y () y() y () + ffy() ( ff) Y () y +ffy v. Y y ffy + v c c () ( ff) ff + ( ff). c y, c v ffy. y(t) L [Y ()] (c + c t)e fft. (4) L[y (t)] ffl[y (t)] + (ff + fi )L[y(t)] ( ff + ff + fi )Y () y() y () + ffy() (( ff) + fi )Y () y +ffy v. Y y ffy + v c ( ff) c () ( ff) + fi ( ff) + fi + fi ( ff) + fi. c y, c v ffy. fi y(t) L [Y ()] e fft (c co fit + c in fit). (5) L[y (t)] 3L[y (t)] + 3L[y (t)] L[y(t)] ( )Y () y() y () y () + 3(y() + y ()) 3y() ( ) 3 Y () L[e t ]. Y () ( ) ( ). 4 y(t) L [Y ()] 6 et t 3.

12 (3). 4.(6) (4)..(7) 9. (5). 8.(). y (t) +ay (t) +by(t) f (t) (Λ) dm dt m m Y () ( + a + b)y () +( ) L[f (t)]() (ΛΛ) ( ). (Λ) 3 D d dt (D + ad + b)y(t) y (t) (Λ) y Λ (t) 3 (Λ) y (t) +y Λ (t) y (t) +y Λ (t) 6. () L[t in t] d d L[in t] d d + ( + ). () L[t co t] d d L[co t] d d + ( + ). (3) L[t n ] d n d n L[] ( ) n d d ( )n ( ) n n n n+ : n+ (4) L[t coh at] d d L[coh at] d d a + a ( a ). (5) L[t inh at] d d L[inh at] d a d a (6) L[t e t ] d d L[e t ] d d ()(). 4.()(3) (6). 8.() a ( a ). ( ) () 6.() () 6.() (3) () 6.() (4)(5) d L[f (t)] L[tf (t)], d» f (t) t L d d L[f (t)]

13 () ( + ) + ( ) ( + ) + ( + ) 6.()»»» L ( + ) L L + ( + ) in t t co t (in t t co t). 3»» () L ( + ) L t in t. ( + )»»» (3) L L + L ( + ) ( + ) ( + ) t co t + (in t t co t) (in t + t co t). (4) d d log + +» L»log + t L ( co t). + t (5) d d log» L»log t L t (et ). 8. () L[t m e at ]() L[t m ]( a) () L[e at in t]() L[in t]( a) m ( a) m+. (3) L[e at co t]() L[co t]( a) ( a) +. a ( a) +. (4) L[te at in t]() d d L[eat in t]() d L[in t]( a) d d d ( a) + ( a) (( a) + ). (). 4.(6) 5.(5) 6.(6) (4). 6.() L[te at in t]() L[t in t]( a) 9. ( a) (( a) + ) : () F () F ( +). + L [F ()] co t L [F ( +)]e t co t. () F () F ( +). + L [F ()] in t L [F ( + )] e t in t. 3

14 ( +) (3) + +5 ( +) + F () F ( +). + + L [F ()] co t in t L [F ( + )] e t ( co t in t). (4) F () ( +) 5 ( +) L [F ()] co F ( +). +3 p 3t p 5 in 3 p 3t L [F (+)] e co p3t t p 5 p in 3t 3. (4)..(7). (i) (ii) () (i) L[in t] +» L» ( + ) L L[in t]»» (ii) L ( + ) L + () (i) ()» L ( co t) ( + )» L ( + ) ( co t): L» L» ( co t) in fi dfi (co t ): Z t ( co fi) dfi (t in t) (t in t): 3»» (ii) L L ( + ) + (t in t) (t in t): 3 ()(ii). A + B + C + D + : (ii) (i) 4

15 . () (f Λ g)(t) Z (?) t f (t fi )g(fi ) dfi f (r)g(t r)( dr) g(t fi )f (fi ) dfi (g Λ f )(t). (?) r t fi (dr dfi ) () (f Λ (g + h))(t) (3) ((f Λ g) Λ h)(t) f (t fi )g(fi ) dfi + fi (f Λ(gΛh))(t) r)h(r) dr dfi Z fi ρ fi f (t fi )(g(fi )+h(fi )) dfi f (t fi )h(fi ) dfi (f Λg)(t)+(f Λh)(t). (f Λ g)(t fi )h(fi ) dfi f (t fi r)g(r) dr h(fi ) dfi f (t fi )(gλh)(fi ) dfi f (t fi )g(fi r)h(r) dr dfi (i) f (t fi ) (ii) (i) (ii) ff (i) f (t fi r)g(r) dr h(fi ) dfi (ffl) (Π) (Λ) Z t Z fi t Z fi t Z ρ Z fi (.) (ii) f (t ρ)g(ρ fi ) dρ h(fi ) dfi f (t ρ)g(ρ fi )h(fi ) dρ dfi f (t ρ)g(ρ fi )h(fi ) dfi dρ f (t fi )g(fi r )h(r ) dr dfi Z fi g(fi (ffl) ρ fi + r (dρ dr) (Π) ρ fi (Λ) fi r ;ρ fi (.) r r;fi fi. (Λ) (.) (Λ) fi r;ρ fi. () () (3) (f Λ )(t) (Λ f )(t) 5

16 ? ( Λ g)(t) (g Λ )(t) 6 g(t); (f Λ f )(t) 6 (ffi). (ffi) g(t) f (t). g(t) t, f (t) co t ( Λ g)(t) (f Λ f )(t) (t fi ) dfi t 6 g(t). co fi co(t fi ) dfi (t co t + in t) t ß (f Λ f )(t) <.. () (t fi ) n u(fi ) dfi t n Λ u(t) L[t n Λ u(t)] L[t n ]L[u(t)] n U (): n+ () in(t fi )u(fi ) dfi int Λ u(t) L[in t Λ u(t)] L[in t]l[u(t)] + U (): (3) (4) co(t fi )u(fi ) dfi cot Λ u(t) L[co t Λ u(t)] L[co t]l[u(t)] e t fi u(fi ) dfi e t Λ u(t) + U (): L[e t Λ u(t)] L[e t ]L[u(t)] U (): 3.»» () L L ( a) L L [t] L e atλλ L L t Λ e atλλ a () L» ( a) (t fi )e afi dfi a (eat at ):» L a L L e atλ L e atλλ L L e at Λ e atλλ a e a(t fi ) e afi dfi te at : 6

17 . 4.(6), 5.(3) (3).()»»»» L L ( + ) in t L L [] L +» L»L Λ in t in fi dfi ( co t): (4).() (3)»»»» L L ( + ) co t L L [] L ( + )» L»L Λ co t (5) 7.() ( co fi) dfi (t in t): 3»»»»» L L ( + ) + in t in t L L L +» in t L»L in t Λ in (t fi ) in fi dfi (in t t co t): 3. in ff in fi (co(ff+fi) co(ff fi)) (6) 7.()»» L L ( + ) +» in t L»L Λ co t» L L +» in t L [co t] in (t fi )cofi dfi t in t:. in ff co fi (in(ff +fi) + in(ff fi)) 7

18 (7) 7.(3)»» L L ( + ) + L [L [co t] L [co t]] + L [L [co t Λ co t]] co (t fi )cofi dfi (t co t + in t):. co ff co fi (co(ff+fi)+co(ff fi)) 4. L[y(t)] Y () Y () Y () y(t) () Y () L[t +]+L[in t Λ y(t)] L[t] +L[] + L[in t]y () + + Y () + Y () ( + )( +) y(t) L [Y ()] + t + t + t3 6. () Y () L[]+L[co tλy(t)] L[]+L[co t]y () + Y + () Y () + ( ) + (?) ( ) y(t) L [Y ()] + te t.. (?) 4.(6) (3) Y () L[ + t]+l[e t Λ y(t)] L[] + L[t]+L[e t ]Y () + + Y () Y () ( ) + ( ) ( ) y(t) L [Y ()] e t 4 (et t ) 4 + t et.. ( ) 3.() (4) Y () L[in t] +L[in t Λ y(t)] L[in t] p +L[in t]y () L[in t] p Y () L[in t] + p +. y(t) p p L [Y ()] in t. (5) Y () L[in t]+l[in tλy(t)] L[in t]+l[in t]y () Y () L[in t] L[in t]. y(t) L [Y ()] t. 8

19 . () y(t) t++in t co fiy(fi ) dfi co t in fiy(fi ) dfi ( ) y (t) +co t co t in ty(t) y (t) in t in ty(t) t + y(t) +y(t) t +. co fiy(fi ) dfi +in t co ty(t)+in t co fiy(fi ) dfi + co ty(t) +cot in fiy(fi ) dfi ( ) in fiy(fi ) dfi + ( ) y (t) t + t + C. y(t) 6 t3 + t + C t + C. ( ) y(), ( ) y () C C. y(t) +t + t + 6 t3. 5. L[y(t)] Y () 4. () Y () y() L[y(t)] + L[] + L[e t Λ y(t)] Y ()+ + Y () y() Y () ( ) + (~) ( ) y(t) L [Y ()] t + 4 (et t ) 4 + t + 4 et.. (~) 3.() () Y () y() y () L[t] +L[t Λ y(t)] L[t] +L[t]Y () + Y () Y () 4. y(t) L [Y ()] + (inh t in t). r K 6. M, k μ M u (t) +ku (t) + u(t). U () L[u(t)] U () u v +k(u () u )+ U () ( +k+ )U () u v ku. U u + v +ku () (~). +k + k ± p k 3 9

20 (k > ). ff (k p k ) > fi (k + p k ) 5.() u(t) c e fft +c e fit. c v p +(k + k )u p ; c v +(k p k )u p. k k fi <ff< μ > 4KM, k> u (k ). ff k 5.(3) u(t) (c + c t)e fft. c u ; c v + ku. ff < μ K μ 4KM, k u c c < p 3(k <). ff ± ifi (ff k; fi k ) 5.(4) u(t) (c co fit + c in fit)e fft. c u ; c v + ku p k. ff < μ < 4KM, k< u μ k ff, fi c co t + c in t

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