9 5.4.3 9 49
5 9 9. 9.. z (z) = e t t z dt (9.) z z = x> (x +)= e t t x dt = e t t x e t t x dt = x(x) (9.) t= +x x n () = (n +) =!= e t dt = (9.3) z <" Rez M< e t t n dt = n! (9.4) <t je t t z je t t " (9.5) t< je t t z je t t M (9.6) e t t " dt; e t t M dt e t t z dt (9.7) (
9. 5 9. (x). t! t! " Rez M z "; M 9. Rez > (z) Rez > C Rez > I I dz(z) = dte t dzt z (9.8) C H C dztz = I C dz(z) = (9.9) C (z) Rez > 9. Rez > (z +)=z(z) (9.) (z) = (z +) z (9.) Re(z +)> z = 9. (z) > Rez > > Rez > 9. (z) > Rez > z = (z) = (z +) z(z +)
5 9 9. C. (z) z (z) = (z +3) z(z + )(z +) = = (z + n +) z(z +)(z + n) (9.) z =; ; ; n; z = n lim (z + n)(z) = z!n (n)(n +)() = () n n! (9.3) 9. 9. Rez > z 9.(a) C I(z) = = e i C z C e z d (9.4) z =e (z)(ln +i) ( ) (9.5) = C 9.(b) a I(z) = + + : (9.6) abc c
9. 53 a (r; ) a = r e z d!(z) (r! ) (9.7) = re i ; abc = = ir z d =ire i d ire (z) ln r e rei e i(z) e i d =ire (z) ln r d e rei e iz de rei e iz (9.8) Rez R > abc! (r! ) 3 = z 3 c z =e (z)(ln +i) = z e i(z) (9.9) = de i e z e i(z) r = e iz Rez > r de z! e iz (z) (r! ) (9.) I(z) =(e iz )(z) = ie iz sin z(z) (9.) I(z)=(ie iz sin z) z Rez > (z) I(z)=(ie iz sin z) (z) z (z) = eiz i sin z z =; ; ; C e z d: (9.) 9.. B(p; q) = x q dx (Rep; Req >) (9.3) ( + x) p+q
54 9 x =( t)=t B(p; q) = t p ( t) q dt (9.4) t = s p q B(p; q) =B(q; p) (9.5) t! kt(k >) (z) k z = k =+s; (p + q) ( + s) p+q = e kt t z dt (9.6) z = p + q e (+s)t t p+q dt: s p s ds = dte (+s)t t p+q s p =(p + q) e t t q dt = (p + q)b(p; q) ; s p ds ( + s) p+q e st (st) p tds =(q)(p) B(p; q) = (p)(q) (p + q) (9.7) Rep; Req > 9.7 p; q B(p; q) p; q p =z; q = z; > Rez > B( z;z) = x z +x dx = sin z 6 46 () = ( z)(z) = sin z (9.8) (9.9)
9. 55 9.9 > Rez > z (z) 9.9 z == p = (9.3) 9. 3 = p 5 ; = 3 p 7 ; = 3 5 p ; (9.3) 3 9..3 n n! n! = p nn n e n (9.3) Stirling 9. (x +)= e t t x dt t = x x > t f( )= ln (x +) = x x+ e x x d = x x+ e x e x(ln ) d: (9.33) (x +)=x x+ e x I(x) ; I(x) = e xf ( ) d (9.34) 9.34 I(x) << f( ) 9.3 x e xf ( ) = = =
56 9 9.3 f() = ln. f( ) = f () = ( ) ( )3 3 + ( )4 4 : (9.35) I(x) = +" s x expf x ( ) gd " " px= " p x= e u du s x q " x " 9.34 9.36 eu du = s x : (9.36) x=! x = n (x +) p xx x e x (x ) (9.37) n! p nn n e n (9.38) f( ) (x) x (x) p x x= e xh + x + 88x 39 584x 3 57 4883x 4 + i (9.39)
9. 57 9. 9.. g(z) z b ;b ; b n ; b j 6=) 5.4.3 C R R C R z C R i I C R g() X z d B = g(z) + j b j z B j g(z) b j j (9.4) 9.4 i I C R g() z d = i z = + I z ( z) C R g() d + z i I C R g() ( z) (9.4) d: (9.4) 9.4 z = i I C R g() d = g() + X g(z) C R lim R! i I C R g() ( z) j B j b j (9.43) d = (9.44)
58 9 9.4 9.4 9.44 g(z) =g() + X j B j f + g (9.45) z b j b j g(z) b j g(z) z = b j 6 g(z) = cot z z (9.46) z k = k(k = 6; 6; 63; ) g() =, cot z = z + B k = cos z = (9.47) d sin z=dz z=zk X k=(6=) ( z k + 9.. fa n g k )= z + X k= z z k : (9.48) 5 n= ( + a n) = ( + a )( + a )( + a 3 ) (9.49) z3.. f (z) a ;a ; lim n! a = n (9.5) f (z) f(z) =f ()e ff ()=f()gz 5 n= n ( z a n )e z=an o (9.5) f(z) a n
9.3 59 f (z) f (z)=f (z) a ; ;a n ; f(z) f (z)=f(z) a ;a ; ;a n ; 9.45 f (z) f(z) = f () f() + log f (z) = f () f () z + X n= X n= ( z a n + a n ): (9.5) flog( z a n )+ z a n g + (9.53) f(z) =Ae (f ()=f ())z 5 n= f( z a n )e z=an g: (9.54) z = = A f(z) f(z) =f()e (f ()=f ())z 5 n= f( z a n )e z=an g (9.55) 6 f(z) = sin z z (9.56) z k = k(k = 6; 6; ) f() = ;f () = sin z z =5 k= z ( ): (9.57) k 9.3 e z e z+i =e z i e z f(z) z f (z +!) =f (z) (9.58)
6 9! f(z)!!; 6!; 63!; f (z)! 6!;6!; f (z)!! ;! f(z) f(z) m ;m m! + m! (9.59) f(z)!,! f(z) z jzj < f(z) g(z) z 4 3 R q R(z; g(z))dz (9.6) g(z) = z z dz q ( z )( k z ) z s k z (9.6) dz (9.6) z dz q ; (k 6= ) 3 (9.63) (z n ) ( z )( k z ) 6 46 9.4 m = v = l! l O! g t d dt = g sin (9.64) l
9.3 6 9.4. v < 4gl sin = v 4gl (9.65) k =sin ; sin = kz (9.66) z T =4 s l g dz q ( z )( k z ) z dz u(z) = q ( z )( k z ) (9.67) (9.68) f (z) = q ( z )( k z ) (9.69) z =; ; k ; k (9.7)
6 9! a! b! c! 9.5a ; a z f (z) K = dx q ( x )( k x ) (9.7) a lim! f(z)dz = K (9.7) c z f(z) = = lim! c f (z)dz =lim abc!! a ff(z)gdz = K: (9.73) j lim f (z)dzj! abc =! (9.74)!! f(z)dz =K (9.75) 9.5b =k f (z) =k L = dz q ( z )( k z ) (9.76)! k! f(z)dz =L (9.77)
9.3 63 9.5. f(z) 9.5a 9.5b 9.5b z = f (z) f(z)dz =K L (9.78)!!! k! =k z f(z) 6K; 6L z f(z) z u(z) z u(z) = dz q ( z )( k z ) : (9.79) z f (z) z = z K u(z)! z K + u(z) u(z) z = sn(u) (9.8) K; L sn(u)
64 9 9.4 u = (9.8) u u 4 = @ @x + @ @y (9.8) u(x; y) (9.8) u(x; y) =( @4 @ 4 @x + 4 @x @y + @4 )u(x; y) (9.83) 4 @y u(x; y) =p(x; y) (9.84) p(x; y) q(x; y) f (x; y) =p(x; y) +iq(x; y) (9.85) z = x +iy (z) = 4 r(x; y);s(x; y) z f (z)dz = r(x; y) +is(x; y) (9.86) r =s = (9.87) @r @s @x = @y = 4 Re(f(z)) = p(x; y) (9.88) 4 p (x; y) =u(x; y) fr(x; y)x + s(x; y)yg (9.89)
9.4 65 p = (u rx sy) =p @r @x @s @y = (9.9) p p q (z) =p +iq (9.9) (z) (z) u(x; y) u(x; y) = (rx + sy) +p =Ref(x iy)(r +is)g +Ref g = Refz(z) + (z)g (9.9) (z); (z) u(z) =Refz(z) + (z)g (9.93) u = Ref(z) + g =Re(z) = Re h( @ @x @ i ) +zi @y @ = Re @x @ i (9.94) @y @ @x @ i = (9.95) @y u = Re u(z) S
66 9 9.6 SF (a) (b). F 9.6) F S k / T k =( kx ; ky ; kz ) T k = df ds k (9.96) k x; y; z S x x xx ; yy ; zz xy ; yz x = xx X k @ ik @x k = (9.97)
9.4 67 ik = ki (9.98) z = xz = zx = yz = zy = @ x @ @x + xy @ yx @ @x + y @y @y = = (9.99) x, y, xy 8(x; y) x = @ 8 @y ; y = @ 8 @x ; xy = @ 8 @x@y (9.) 9.99 @ @y ( x y )+ @ @x ( y x )=(+) @ xy @x@y (9.) 3 9. 9.8(x; y) 8(x; y) =( @4 @ 4 @x + 4 @x @y + @4 )8(x; y) (9.) 4 @y 8(x; y) Airy 3 Y.C. (
68 9 9.5 9.5. f (x) (; ) dxjf(x)j < (9:3) (; ) 4 F (k) = F (k) [f(x +)+f (x )] = df()e ik F[f(x)] (9:4) x f(x) f (x) = dke ikx F (k) F [F (k)] (9:5) dke ikx F (k) =F [F (k)] (9:6) 9.5 6 exp(ajxj) ; a > F (k) = dxe ajxj e ikx = f a +ik + a ik g = a (a + k : ) f (x) = dke ikx F (k) = dke ikx f i (9.7) k ia k +ia g : (9.8) k x> x< 4 39 6
9.5 69 ia ia k k (x >) (x <) ( ) f(x) = (+i)e ax : x> i (i)e ax =e ajxj : (9.9) : x< 63 exp( a x ) F (k) = 6 43 dxe a x ikx = dxe a (x+ ik a ) k a (9:) F (k) = a p k exp( a ) : (9:) 64 (d=dx)f(x) f(x) jxj! N jxj N f(x) F (k) F (k) = dxe ikx f (x) (9:) dxeikx df (x) dx = [eikx f (x)] x= = = ik x= dx(ik)e ikx f(x) dxde dx ikx f (x) dxeikx f(x) =ikf(k): (9.3) dkeikx ikf (k) = d dx dkeikx F (k) = d f(x) (9:4) dx ixf(x) dxe ikx ixf (x) = d dxe ikx f(x) dk = d F (k) (9.5) dk
7 9 64 F[f (n) (x)] = dxe ikx f (n) (x) =(ik) n F[f(x)]; (9:6) F[x n f(x)] = (i d dk )n F[f(x)]: (9:7) 9.3 9.5 n dyf(x y)g(y) (9:8) F[ = F[f(x)] = F (k); F[g(x)] = G(k) (9:9) dyf(x y)g(y)] = dte ikt f(t) dxeikx dyf(x y)g(y) dye iky g(y) =F(k)G(k): (9.) F [F(k)G(k)] = = dkeikx F (k)g(k) dk dk (k k )e ikx F (k )G(k ): F [F(k)G(k)] = (k k )= = = dy ei(k k )y dy dk dy dk e ik (xy) F (k ) (9:) dk e i(kk)y e ikx F (k )G(k ) dk e ik y G(k ) dyf(x y)g(y) (9.)
9.5 7 F (k)g(k) R R f(x) = dke ikx F (k) F (k) = dxf(x)e ikx (k) x n f(x) jxj ; (x 6= ; <<) ; (a >) x +a e ax ; (a >) sechax; (a >) (i d dk )n F (k) sin( ) () jkj a exp(ajkj) p k exp( a 4a ) k sech( a a ) sin ax x ; (a >) ( jkj <a jkj >a sin(a x ); (a >) a p k cos( + 4a ) 4 9.. 65 <x< t = x = 9.3 @ ( @t a @ )u(x; t) =(x )(t) (9:3) @x u(x; t) = ; t < (9:4) 9.3 x t u(x; t) = dk d!e ik(x) e i!t ~u (k;!) (9:5) 9.3 (x )(t) = 4 dk d!e ik(x) e i!t : (9:6)
7 9 9.3 ~u (k;!) = 4 i! + ak (9:7) k 9.7! Im! >! = iak 9.7u(x; t) 9.5! e i!t t >! t < 9.7! iak t< u(x; t) = : t<: (9:8) t>! - iak u(x; t) = dk d!e ik(x) e i!t 4 i! + ak = dke ik(x) e ak t 9. u(x; t) = x at(ki dke at ) e (x) =4at (x ) = p expf gg(x ;t); t > (9.9) at 4at t! lim G(x ;t)=(x ) (9:3) t! 9.9 t = f(x) @u @t u a@ = ; t> (9.3) @x u(x; ) = f(x)
9.5 73 9.7 ( 65. u(x; t) = dg(x ;t)f () (9:3) f(x) 9.3 9.5. y(t) () y(t) =; t < () dte t jy(t)j < ; : (9.33) Y L (p) = dte pt y(t) L[y(t)] (9:34) y(t) = i +i i dpept Y L (p) L [Y L (p)] (9:35) 9.34 y(t) 9.35 Y L (p) p 9.35 9.8
74 9 9.8. 66 (df=dx) dxe px f (x) =e px f(x) F L (p) =L[f(x)] = dxe px f(x) (9.36) +p dxe px f(x) =f() + pf L (p) lim x! e px f (x) = 66 L[f (x)] = f () + pf L (p) (9.37) L[f (n) (x)] = p n F L (p) n X r= p nr f (r) () (9.38) x df()g(x ) (9.39) L[f(x)] = F L (p); L[g(x)] = G L (p) (9:4)
9.5 75 x L[ x = dxe px = d df()g(x )] df()g(x ) = dx dxe p f()e p(x )g(x ) = x de p f()e p(x) g(x ) de p f () dye py g(y) = F L (p)g L (p): (9.4) F (k)g(k) 67 d u dx + u = f (x) ; x u() = u ()= (9.4) u(x) f (x) u L (p) = dxe px u(x) =L[u(x)]; (9:43) f L (p) = dxe px f(x) =L[f(x)] (9:44) (d u=dx ) L[u (x)] 9.38 L[u (x)] = e px u (x)dx (pe px )u (x)dx (pe px )u(x)dx = [e px u (x)] x= = [e px u (x)] +[pepx u(x)] p = p u L (p) pu() u () (9.45) e px u (x), e px u(x)! (x!) u() = u ()= 9.4 (p +)u L (p) =f L (p)
76 9 9.9 67. u L (p) = f L(p) p + (9:46) 9.4 9.46 x u(x) = n h f() L io d (9:47) p + x x =(p +) L h p + L h i= p + i +i i +i i dp p +i p i dpe px p + : (9:48) =(p +) p = 6i Rep = p 9.9 i = e px i i 9.47 x u(x) = = i [eix e +ix ]=sinx (9.49) df ()sin(x ) (9:5)
9.6 77 f(x) = i (x a) = x > R +i ( : x>a> i dpepx F L (p) F L (p) = R : x<a e pa p dxe px f(x) (+) p + e ax pa a a sin ax p +a p cos ax p +a a sinha p a p cosha p a jaj jaj 9.. 9.6 t = E P E ) P (t) = ( (" " )E : t = (t)e : t> (9.5) t E(t) P (t) =(" " )E(t) + t (t s)e(s)ds (9.5) (t s) s! E(t) P (t) E(!) = dte(t)ei!t P (!) = dtp (t)e i!t = f"(!) " ge(!) (9.53)
78 9 P (!) E(!) "(!) 9.5 "(!) =" + ds(s)e i!s (9.54) (t) t!9.54 Im! > "(!)!! +i (t) 9.54 "(!)! Im! > lim ("(!) " )= (9.55) j!j! C = =+ i ( "() "! d "(!) " = Im! > Im! < (9.56) Im! >!! Im! < "(!) " = i = i = df"() " g[! 6! ] (9.57) d! ("(! ) " )Re!! d! ("(! ) " )Im!! (9.58) 9.58 +i i + "(!) " = = i d! " (! )!! d! " (! ) "!! :Im!> :Im!> (9.59)
9.6 79 " "(!) " "(!) =" (!) +i" (!) 9.59 " (!) " = Pv " (!) = Pv d!!! " (! ) (9.6) d!!! (" (! ) " ) (9.6) "(!)
8 9 9.7 9. = sin p cos q d = p+ q+ ( )( ( p+q +) ) = sin n d = ( (n)!! n!! (n)!! n!! (n: ) (n: ). 3. dx dx dx n x ;x ; x n ; x +x ++x n = (l )(l ) (l n ) (l + l + + l n ) x l x l x ln n f (x + x + + x n ) f(u)u l +l ++l n du (l i > ) a n n n (z) = lim n! n(z) =(z) t z t n ndt (z) = z 5 m= n + m z + z m o (z) = zez 5 z m= + e m z m = lim ( + n! + + n log n) = e t log tdt =:577566
9.7 9 8 4. 9.3 (9.73)(9.75)(9.76) 5. 6.