Chapter 6 6. (5.8)(5.9) f(x) c n = a n= a a c n exp(i nx a )= [f(x +)+f(x )]; x [a; a] f (x) exp(i nx )dx (6.) a k n = n a ; k n = k n+ k n = a (6:) (6.) c n c n = k n (6.) a a df()e ikn : a [f (x +)+f(x )] = e iknx k n df()e ikn n= a a!; k n! n k n! dk [f(x +)+f(x )] = dke ikx df()e ik (6:3) 98
f(x) (; ) dxjf (x)j = (6:4) (6.3) F (k) = df()e ik F[f (x)] (6:5) [f (x +)+f(x )] = dke ikx F (k) F [F (k)] (6:6) x f(x) f (x) = dke ikx F (k) =F [F (k)] (6:7) (6.5) (6.6) (6.5) p = (6.6) p = (6.5)(6.6) 6. () exp(ajxj) ; a > () exp( a x ) (3) d f(x); f(x) jxj! dx N jxj N (6.8) (). F (k) = dxe ajxj e ikx = f dxe (a+ik)x + dxe (aik)x g a = f a + ik + a ik g = (a + k ) ; (6.9) f(x) = dke ikx F (k) = a e ikx a + k dk = i dkeikx f k ia k + ia g (6.) 99
k x> x < ( 5 ) ia ia k k (x >) (x <) ( ) f(x) = i (+i)e ax : x> (i)e ax = e ajxj : : x< () F (k) = dxe a x ikx = dxe a (x+ ik a ) k a (6:) 6. C I C dze a z = + = R dxe a x + i R R R k a dxe a (x+ ik a ) + i dye a (R+iy) dye a (R+iy) k a R!,4 dxe a x + dxe a (x+ ik a ) = ( dxe a x ) = dxdye a (x +y ) = drre a r = dte a t = a dxe a x = dxe a (x+ ik a ) = (6.) p a p a F (k) = p exp( k a a ) (6:)
// 6./////////////////////////// (3) F (k) = dxe ikx f(x) (6:3) dxeikx df(x) dx = [eikx f(x)] x= x= ikx dxde dx f (x) = dx(ik)e ikx f (x) = ik dxe ikx f(x) =ikf (k): (6.4) dke ikx ikf (k) = d dke ikx F (k) = dx d f(x) (6:5) dx ixf(x) dxeikx ixf (x) = d dk dxeikx f (x) = d F (k) (6.6) dk ( 6.4 ) 6.(3) F[f (n) (x)] = dxe ikx f (n) (x) =(ik) n F[f(x)]; (6:7) F[x n f(x)] = (i d dk )n F[f (x)]: (6:8) (6.4) (6.6) n
(x x ) F[(x x )] = lim n! = lim n! r n = lim n! r n = lim n! r n dxe ikx n (x x ) dxe ikx e n(xx) eikx dxeik(xx ) e n(xx ) eikx dxeikxnx (6.) lim n! x = r n dxe ikx n (x x )= lim n! p eikx n e k 4n = lim n! eikx e k 4n = eikx : (6.9) F[(x x )] = exp(ikx ): (6:) F[(x)] = (6:) (x x ), (x) F [ ]= dk eikx = (x) (6:) e ikx jkj! (6.) (6.), (6.) I(x) (6.) = I(x) = I(x) I n (x) =exp(x =n) (6:3) I(x) = lim n! e x =n (6:4) (6.9) I(x) I n (x) jxj! N
jxj N f(x) I(x) I(x) p n = lim p n! dxi(x)e ikx = lim n! dxi n (x)e ikx r nk n e 4 = lim = (k) (6.5) n! enk (6.5) (6.) x! k,k!x jxj! jxj! 6. ( ) dyf(x y)g(y) (6:6). F[f(x)] = F (k); F[g(x)] = G(k) (6:7) F[ dyf(x y)g(y)] = dxe ikx dyf (x y)g(y) = dte ikt f (t) dye iky g(y) =F (k)g(k): (6.8) F [F(k)G(k)] = dke ikx F (k)g(k) = dk dk (k k )e ik x F (k )G(k ): 3
(6.) (k k )= dy ei(kk)y (6:9) F [F(k)G(k)] = = = dy dk dk e i(k k )y e ik x F (k )G(k ) dy dk e ik(xy) F (k ) dk e iky G(k ) dyf (x y)g(y) (6.3) F (k)g(k) 6.,6. (x); (x);i(x) H(x) H(x) = ( : x< (6.3) : x 6.3 H(x) d H(x) =(x) (6:3) dx. jxj! f(x) dh(x) dx f(x)dx =[H(x)f (x)] x= H(x)f (x)dx = H(x)f (x)dx = f (x)dx = [f (x)] x= = f() (6.33) (6.3) 6.4 d dx f(x) f (x) =ejxj (6:34) 4
. F[f (x)] = k F[f] =k F (k) ; F[e jxj ] = eikx e jxj dx = f e ikxx dx + F (k) = f(x) = e ikx+x dxg = (k +) : dkeikx ( + k ) : +k : k = 6i x > k x< x> : f(x) = i[ d dk x< : f(x) = i()[ d e ikx (i + k) ] k=i = +x ex ; dk e ikx (i + k) ] k=i = x ex f (x) = +jxj e jxj (6:35) 6. 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 ) a sech(k a ) sin ax x ; (a >) ( jkj <a jkj >a sin(a x ); (a >) a p k cos( + 4a ) 4 6.5 <x< t = x = ( @ @t a @ )u(x; t) =(x )(t) (6:36) @x 5
(6.36) u(x; t) = ; t < (6:37).(6.36) x t u(x; t) = dk dwe ik(x) e i!t ~u (k;!) (6:38) (6.36) (x )(t) = 4 (6.36) dk d!e ik(x) e i!t : (6:39) ~u (k;!) = 4 i! + ak (6:4) k (6.4)! Im! > (! = iak ) (6.4) u(x; t) (6.38)! e i!t t >! t < ( 6.)! iak t < u(x; t) = : t<: (6:4) t>! - iak u(x; t) = dk d!e ik(x) e i!t 4 i! + ak = dke ik(x) e ak t (6.) u(x; t) = = p at x at(ki dke at ) e (x) =4at ) expf(x gg(x ;t); t > (6.4) 4at t! lim G(x ;t)=(x ) (6:43) t! 6
(6.4) // 6./////////////////////////// t = f (x) ( @u @t a @ u = ; t> @x (6.44) u(x; ) = f(x) u(x; t) = dg(x ;t)f() (6:45) f(x) (6.45) (6.45) (6.45) (6.44) ( @ @t a @ @x )u = df ()( @ @t a @ )G(x ;t) @x = df ()(x )(t) =f(x)(t) (6.46) G(x ; t) (6.36) u(t) t 6= (6.44) t! (6.43) lim u(x; t) = df()lim G(x ; t) = df()(x ) =f(x) (6:47) t! t! u(x; t) (6.44) 6. y(t) () y(t) =; t < 7
() dte t jy(t)j < ; : (6.48) y(t) Y F (!)! Y F (!) = e i!t y(t)dt; (6:49)! =! i (! ; : ) Y F (!) = e i!t e t y(t)dt: (6:5) (6.5) Im! <! =! i ( = Im! ) Y F (! )= e i!t e t y(t)dt Y F (!) (! =! i) Y F (!) = e i!t y(t)dt = e i! t e t y(t)dt: (6:5) (6.5) y(t) =(t<) (6.5) Im! = < (6.5) + y(t) =e t d! e i! t YF (!) = +i d!e i!t Y F (!) (6:5) i ( > )!! = i +i (6.5),(6.5) i! = p Y L (p) =Y F (!) = dte pt y(t) L[y(t)] (6:53) y(t) = +i dpe pt Y L (p) L [Y L (p)] (6:54) i i p = i!! 9 p (Y L (p) ) 6.3 p = p = (6.53) y(t) (6.54) p Re p = Y L (p) Y L (p) (> ) (6.54) 8
// 6.3/////////////////////////// 6.6 df dx. F L (p) =L[f(x)] = dxe px f (x) (6.55) dxe px f (x) =e px f (x) +p dxe px f (x) =f () + pf L (p) f() = 6.6 L[f (x)] = f () + pf L (p) (6.56) L[f (n) (x)] = p n F L (p) n r= p nr f (r) () (6.57) 6.7 x. df()g(x ) (6.58) L[f(x)] = F L (p); L[g(x)] = G L (p) (6:59) x L[ df ()g(x )] x x = dxe px df()g(x ) = dx de p f ()e p(x) g(x ) = d dxe p f()e p(x )g(x ) = de p f () dye py g(y) = F L (p)g L (p): (6.6) 9
F (k)g(k) 6.8 ( d u + u = f(x) dx ; x u() = u () = (6.6).u(x) f(x) u L (p) = dxe px u(x) =L[u(x)]; (6:6) f L (p) = dxe px f(x) =L[f(x)] (6:63) d u L[u (x)] (6.57) dx L[u (x)] = e px u (x)dx = [e px u (x)] x= (pe px )u (x)dx = [e px u (x)] +[pepx u(x)] p (pe px )u(x)dx = p u L (p) pu() u () (6.64) e px u (x), e px u(x)! (x!) u() = u () = (6.6) (p +)u L (p) =f L (p) u L (p) = f L(p) p + (6.6) (6.65) (6:65) x u(x) = f ()fl [ p + ]g xd (6:66) x =(p +) L [ p + ]= +i dpe px i i p + : (6:67)
=(p +) p = 6i Rep = p 6.4 L [ p + ] = +i i dp( i p + i p i )epx ( i ) = i [eix e +ix ]=sinx (6.68) (6.66) x u(x) = df()sin(x ) (6:69) // 6.4/////////////////////////// 6. R R f(x) = +i i i dpepx F L (p) F L (p) = dxe px f (x) ( : x>a> e (x a) = pa p : x<a 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 6.3
f(x) () a [; a] [; a] N f(x) = n= c n = a a a c n e i nx a nx i f(x)e a dx (6.7) = x <x <x <...<x N <x N =a; x k = ak ; k =; ;...;N (6.7) N (6.7) c n = N N k= f( ak nk N )ei N (6:7)! = exp i N ; f k = f(x k );k =; ;...;N c n = N N k= f k (!) nk ;n=; ; ;...;N (6.73) ff ;f ;...f N g fc ;c ;...;c N g ( ) (6.7) f k = (6.73) N n= c n! nk (k =; ; ;...;N ) (6:74) N n= c n! nk = N N k = f k N n=! n(kk) = N N k = (k 6=k) f k! N (kk )! (kk ) + f k! N (! N =) (6.74) ( ) (6.73) n k N N ( N N (N ) ) N N 3
(FFT=Fast Fourier Transform) FFT! N = N = p p =(N = ) (6.73) (!) 4 = 4c = f (!) + f (!) + f (!) + f 3 (!) ; 4c = f (!) + f (!) + f (!) + f 3 (!) 3 ; 4c = f (!) + f (!) + f (!) + f 3 (!) ; 4c 3 = f (!) + f (!) 3 + f (!) + f 3 (!) ; (6.75) f () (; ) = f ; f () (; ) = f ; f () (; ) = f ; f () (; ) = f 3 : (6.76) f () (k ;k ) f k k k =k + k ; f () (k ;k )=f k ; k =k + k : (6.77) p> f () () p f () (; ) = f () (; )(!) + f () (; )(!) f () (; ) = f () (; )(!) + f () (; )(!) f () (; ) = f () (; )(!) + f () (; )(!) f () (; ) = f () (; )(!) + f () (; )(!) (6.78) f () (k ;k ) k =; ; f () (n ;k )=f () (;k )(!) + f () (;k )(!) n : (6.79) 3 f () (n ;k ) k =; 4c = f () (; ) = f () (; )(!) + f () (; )(!) 4c = f () (; ) = f () (; )(!) + f () (; )(!) 4c = f () (; ) = f () (; )(!) + f () (; )(!) 4c 3 = f () (; ) = f () (; )(!) + f () (; )(!) 3 (6.8) 3
c n (n =n + n ) f () (n ;n ) ; c n+n = f () (n ;n )=f () (n ; )(!) + f () (n ; )(!) n+n : (6.8) (6.78) (6.8) (!) = N =4 N =4 (6.78) (6.8) p = 8=4=Np N = p k; n k = p k p + p k p +...+k + k n = p n p + p n p +...+n + n f () (k p ;k p ;...;k )=f k (6.8) f (p) (n ;n ;...;n p )=Nc n (6.83) (!) nk nk nk = n p k p + n p k p + + n k = p n k p + p (n + n )k p + +( p n p + p n p + + n )k : (mod p ) (6.84) f (p) (n ;n ; ;n p ) = k k f () (k p ;k p ; ;k )(!) p n k p k p (!) (p n p+ p n p++n )k (!) p (n +n )k p (6.85) (6.84) mod( p = N)! N = 6.85 N (6.85) (6.76)(6.8) k p =; k p =; p c n (n =; ;...;N) Np = N log N FFT P N h n = f m= mg nm 4
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