Mathematical Physics

# Mathematical Physics - Laplace z = x iy Laplace w = f(z =...

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Unformatted text preview: Laplace December 29, 2006 z = x + iy . Laplace w = f (z) = f (x, y) = u(x, y) + iv(x, y) (1) . ( . ), z = x + iy . Laplace w = f (z) = f (x, y) = u(x, y) + iv(x, y) (1) . ( . ), z = x + iy . Laplace w = f (z) = f (x, y) = u(x, y) + iv(x, y) (1) . ( . ), . Theorem (Cauchy-Riemann )) f (z) = u(x, y) + iv(x, y) u v = , x y z ( CR Laplace u v =- y x . . (2) . Theorem (Cauchy-Riemann )) f (z) = u(x, y) + iv(x, y) u v = , x y z ( CR Laplace u v =- y x . . (2) . Theorem (Cauchy-Riemann )) f (z) = u(x, y) + iv(x, y) u v = , x y z ( CR Laplace u v =- y x . . (2) Note CR z 0 , w z , . CR z . Theorem Laplace f (z) = u(x, y) + iv(x, y) z u u v v , , x y x y z . , . Note CR z 0 , w z , . CR z . Theorem Laplace f (z) = u(x, y) + iv(x, y) z u u v v , , x y x y z . , . , , , , Laplace , . , , . . Riemann Riemann . z , Riemann , , , , Laplace , . , , . . Riemann Riemann . z , Riemann , , , , Laplace , . , , . . Riemann Riemann . z , Riemann : Laplace Theorem (( f (z) C, Laplace )Cauchy G , ) G f (z)dz = 0 C Theorem ( f (z) G n Cauchy ( ) f (z)dz i=1 Ci ) , (3) G f (z)dz = C0 Laplace C0 , C1 , C2 , ..., Cn ( ) C0 . , C1 , C2 , ..., Cn . Theorem (Cauchy C f (z) G f (a) = C ,G ( ) 1 2i ) ,a , f (z) dz z-a . (4) G . C Laplace f (n) (z) = n! 2i C f () d ( - z)n+1 (5) Theorem (Cauchy C f (z) G f (a) = C ,G ( ) 1 2i ) ,a , f (z) dz z-a . (4) G . C Laplace f (n) (z) = n! 2i C f () d ( - z)n+1 (5) Theorem ( ) , , (z - a)f (z) f (z) z=a 1 arg(z - a) 2 , |z - a| 0 k, lim a 0 Laplace f (z)dz = ik(2 - 1 ) C (6) 2 - 1 C , , , C = {|z - a| = , 1 arg(z - a) 2 } Theorem ( f (z) z z= , zf (z) lim ) , K, 1 arg z 2 , R Laplace f (z)dz = iK(2 - 1 ) CR (7) 2 - 1 CR , ,R , CR = {|z| = R, 1 arg z 2 } Theorem ( > 0, N ( ) > 0, Cauchy n>N |un+1 + un+2 + + un+p | < ) p, (8) Laplace n lim un = 0 (9) d'Alembert ( ) , Cauchy . Theorem ( > 0, N ( ) > 0, Cauchy n>N |un+1 + un+2 + + un+p | < ) p, (8) Laplace n lim un = 0 (9) d'Alembert ( ) , Cauchy . Theorem ( > 0, N ( ) > 0, Cauchy n>N |un+1 + un+2 + + un+p | < ) p, (8) Laplace n lim un = 0 (9) d'Alembert ( ) , Cauchy . , n akl = k=0 l=0 Laplace an-l,l n=0 l=0 -1 (10) akl = k=0 l=0 n=0 l=0 an+l,l + n=- l=-n an+l,l (11) Definition ( > 0, N ( ) z G, Laplace ) z . n > N( ) , (12) S(z) - k=1 k=1 uk (z) < G . uk (z) Definition ( k=1 ) uk (z) G G . , Definition ( > 0, N ( ) z G, Laplace ) z . n > N( ) , (12) S(z) - k=1 k=1 uk (z) < G . uk (z) Definition ( k=1 ) uk (z) G G . , 1 2 3 Laplace Theorem ( 1 2 ) z , t [a, b], z G, t, f (t, z) G G b f (t, z) t [a, b] b a , F (z) = Laplace f (t, z)dt , f (t, z) dt z F (z) = a (13) Theorem ( 1 2 3 0 ) t z t a, f (t, z) f (t, z)dt G , t a, z G, G , , G f (t, z) , > 0, T ( ), , T2 Laplace T2 > T1 > T ( ) f (t, z)dt < , T1 F (z) = a f (t, z)dt G , F (z) = a f (t, z) dt. z Theorem (Taylor f (z) z ) , a |z - a| < R , |z - a| < R, f (z) f (z) = n=0 Laplace an (z - a)n (14) an = f (n) (a) n! (15) Theorem (Laurent f (z) , ) R1 < |z - b| < R2 z , f (z) b f (z) = Laplace an (z - b)n n=- R1 < |z - b| < R2 (16) an = C 1 2i C f () d ( - b)n+1 . (17) m f (z) = (z - a)m (z) (z) |z - a| < R (a) = am = 0. , Laplace 1 2 m f (z) = (z - b)-m (z) (z) |z - b| < R (b) = 0. , 3 m f (z) = (z - a)m (z) (z) |z - a| < R (a) = am = 0. , Laplace 1 2 m f (z) = (z - b)-m (z) (z) |z - b| < R (b) = 0. , 3 Theorem ( G ) f1 (z) f2 (z), {zn }, f1 (zn ) = f2 (zn ). n G Laplace lim zn = a G G f1 (z) f2 (z). , b 1 Laplace . . , : b . b . b . .b . 2 d2 w dw + p(z) + q(z)w = 0 2 dz dz Laplace (18) . Theorem p(z) q(z) |z - z0 | < R , Laplace d2 w dw + p(z) + q(z)w = 0 2 dz dz w(z0 ) = C0 , w (z0 ) = C1 C0 , C1 . , w(z), w(z) Definition ( z0 . Laplace ) , (z - z0 )p(z), (z - z0 )2 q(z) , z0 . q(z) z0 p(z) z0 . Theorem d2 w dw + p(z) + q(z)w = 0 2 dz dz z0 0 < |z - z0 | < R Laplace w1 (z) = (z - z0 )1 k=0 ck (z - z0 )k 2 k=0 w2 (z) = gw1 (z) ln(z - z0 ) + (z - z0 ) z0 dk (z - z0 )k . ( - 1) + a0 + b0 = 0 (19) a0 = lim (z - z0 )p(z), Laplace zz0 b0 = lim (z - z0 )2 q(z) zz0 (20) Re1 Re2 ! 1 1 - 2 = . 1 - 2 = . 1 2 1 = 2 . 1 - 2 = m (2) cm , m 2 Laplace 0 c(2) + m l=1 m [al (m + 2 - l) + bl ]cm-l = 0 (2) [al (1 - l) + bl ]cm-l l=1 (2) = 0, = 0, cm (2) , , . . (2) cm = 0. Theorem ( C , Laplace ) ,G C bk , k = 1, 2, ..., n n , , G G f (z)dz = 2i C k=1 resf (bk ) (21) Theorem ( C , Laplace ) ,G C bk , k = 1, 2, ..., n n , , G G f (z)dz = 2i C k=1 resf (bk ) (21) 1 2 I= 0 2 R(sin , cos )d (22) + Laplace I= - 3 + R(x)dx (23) I= - R(x)eipx dx p>0 (24) + ReI = - + R(x) cos pxdx R(x) sin pxdx - (25) (26) ImI = 4 5 Laplace I= 0 xs-1 R(x)dx (27) I= 0 xs-1 lnn (x)R(x)dx , Jordan (28) , (z) = 0 e-t tz-1 dt (29) (z + 1) = z(z) sin z Laplace (z)(1 - z) = 2 (1/2) = Laplace B B(p, q) = 0 1 tp-1 (1 - t)q-1 dt (30) /2 B(p, q) = 2 0 Laplace sin2p-1 cos2q-1 d (31) B B(p, q) = (p)(q) (p + q) (32) F (p) = 0 Laplace e-pt f (t)dt (33) 1 Laplace . L {f (t)} = F (p), (34) . L {f (t)} = F (p) = F (p) p (35) L {f (t)} = pF (p) - f (0) 2 Laplace t L 3 f ( )d 0 . L {f (t)} = F (p), L -1 F (n) (p) = (-t)n f (t) (36) Laplace 4 L 5 -1 p . L {f (t)} = F (p), f (t) F (q)dq = t (37) L -1 . L -1 {F1 (p)} = f1 (t), {F2 (p)} = f2 (t), t -1 L {F1 (p)F2 (p)} = 0 f1 ( )f2 (t - )d (38) F (p) f (t) = F (p) Laplace 1 2i s+i F (p)ept dp s-i (39) , p lim F (p) = 0 . F (p) F (p) f (t) = res ept F (p) F (p) f (t) = F (p) Laplace 1 2i s+i F (p)ept dp s-i (39) , p lim F (p) = 0 . F (p) F (p) f (t) = res ept F (p) : Definition ( {n (x)} ) n Laplace lim f (x)n (x) = f (0) - (40) f (x)(x)dx lim - n f (x)n (x) = f (0) - (41) Green , Green Laplace . . , , ! Green , Green Laplace . . , , ! : 2007 1 12 2:00--4:00(4:15,4:30) : 501: 0050400100504074 503: 0050407500504150 505: 0050415100504179; 507: 005047XX; 00546XXX Laplace : (2 ) ?? ?? : 439 : 2007 1 12 2:00--4:00(4:15,4:30) : 501: 0050400100504074 503: 0050407500504150 505: 0050415100504179; 507: 005047XX; 00546XXX Laplace : (2 ) ?? ?? : 439 , ! 1 2 3 Laplace (25 ) , (15 ) (15 ) , Laplace (15 ) (15 ) (15 ) 4 5 6 , ! 1 2 3 Laplace (25 ) , (15 ) (15 ) , Laplace (15 ) (15 ) (15 ) 4 5 6 , ! 1 2 3 Laplace (25 ) , (15 ) (15 ) , Laplace (15 ) (15 ) (15 ) 4 5 6 , ! 1 2 3 Laplace (25 ) , (15 ) (15 ) , Laplace (15 ) (15 ) (15 ) 4 5 6 , ! 1 2 3 Laplace (25 ) , (15 ) (15 ) , Laplace (15 ) (15 ) (15 ) 4 5 6 , ! 1 2 3 Laplace (25 ) , (15 ) (15 ) , Laplace (15 ) (15 ) (15 ) 4 5 6 ...
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