数学物理方法讲义七.pdf

1 lC 8 1 3 1 1 lC 3 1 2 K 4 1 3 x 5 5 1 4 7 1 5 X 8 1 6 n 8 2 9D 9 2 1 lC K 9 2 2 x 5 10 2 3 12 2 4 X 12 2 5 n 12 3 g g 13 3 1 g gz 13 3 2 g m 16 3 3 x J 18 3 4 gz 20 4 21 5 4 IX 24 5 1 25 5 2 X 27 c 1992 2011 nX S n d H N 28 5 4Poisson n 30 6 5 532 6 1 5 32 6 2 5 34 SK36 1 3 m ATP g k keZ N 3 gC L m 5 u U U aq g k k 5 k 4 K lC u3 ye gd u x t K 2u t2 a2 2u x2 0 0 x 0 1a u x 0 0 u x l 0 t 0 1b u t 0 x u t t 0 x 0 x l 1c x x O u 5 C y3 e A u x t X x T t 2 1a X00 x X x T 00 t a2T t 0 L g gC t m x A x t P u T 00 t a2T t 0 3 X00 x X x 0 4 1 4 2 1b 1c O X 0 T t X l T t 0 5 X x T 0 x X x T 0 0 x 6 kw 6 U k U 6 x x X x l x x U g XJ x x X x K7 v 4 4 U n e 0 e 1b u 1c 2 K k X x 4 X 5 g N L a X 0 c0 X0 0 c15 y3 N 7 3 e T 3 4 X A S 4 X 3 lC gC x t N K I 3 L J 4 k v 7 T r 4 7 3e X00 x X x 0 8a X 0 0 X l 0 8b K A k Nk K 3 T eigenvalue A T eigenfunction 3 o K J L N K un k e K 8 1 XJ 0 K 8a X x Cx D C D 8b C D 0 l X x 0 T 0 2 XJ 0 K 8a X x C cosh x Dsinh x 1 5 C D 8b d X 0 0 C 0 2d X l 0 Dsinh l 0 l 0 sinh l 6 0 D 0 u X x 0 T 0 2 0 K 8a X x C cos x Dsin x C D 8b d X 0 0 C 0 2d X l 0 Dsin l 0 T 7L sin l 0 9 XdK D 9 N n n l n N 5 0 u n n l 2 Xn x sin n l x n N 10 D 1 n 3 Tn t un x t Xn x Tn t Xn x Tn t D Tn t 3 w n x 5 3UY c k0 x Xn x sin n x l n 1 5 u 7 k T x3 m 0 l Xm x Xn x Z l 0 Xm x Xn x dx Z l 0 sin m l xsin n l xdx 0 m 6 n 11 p Xm x Xn x L Xm x Xn x S d 1 Xm x Xn x N N d O y m e z n sin m l xsin n l x 1 2 cos m n l x cos m n l x g y T x3 m 0 l f x 3 m 0 l k 5 k Y Y 5 8 d K m f x X n 1 ansin n l x x 0 l 12 mX an 2 l Z l 0 f x sin n l xdx n N 13 1 6 51 3 x K e X 3 n m A i j k n 5 m e k i j K 2 f x x k m 3 U u8 x kaq 3 x m 12 cJe N mX 13 12 I n k sin n x l x l 0 l Z l 0 f x sin n l xdx Z l 0 X k 1 aksin k l xsin n l xdx X k 1 ak Z l 0 sin k l xsin n l xdx A U d 5 x 5 m k k n T kXn x k2 Xn x Xn x Z l 0 sin2 n l xdx l 2 14 dd 13 11 14 Xm x Xn x Z l 0 sin m l xsin n l xdx l 2 mn m n N 15 mn 1 m n 0 m 6 n 16 Kronecker AT G K u Sturm Liouville a K x XJ E K 5 J d O y v y u 5 y J y u 3 0 l f x r l l 2 2l P f x 3 0 l f x f x 3 f x m e Fourier f x b0 X n 1 bncos n l x ansin n l x x 17 mX b0 1 2l Z l l f x dx bn 1 l Z l l f x cos n l xdx an 1 l Z l l f x sin n l xdx n N 18 1 7 du f x l l b0 0 bn 0 n N u m z f x X n 1 ansin n l x x 19 X an L an 2 l Z l 0 f x sin n l xdx 2 l Z l 0 f x sin n l xdx n N 20 x 0 l 19 z 12 X 20 13 K w m 12 Fourier X u K x n du 5 5 5 m 2 Fourier Ud n Fourier G n Fourier A 3 Q d x 5 12 un Fourier 5 w H g o 10 3 T 00 n t n a l 2 Tn t 0 21 Tn t Ancos n a l t Bnsin n a l t 22 An Bn u X v 1a 1b A un x t Xn x Tn t Ancos n a l t Bnsin n a l t sin n l x n N 23 u v 1c x x sin n x l y3 5 1a 1b g A U E v u e u x t X n 1 un x t X n 1 Ancos n a l t Bnsin n a l t sin n l x 24 Lu3 F L An Bn 1c v 5 v K 1 k 51 24 k lC 2 AT v N v v v XJ k UC o I l m 5 k 1 8 U v L J An Bn we 3 J A U E v p S b 3 C 8 An Bn d x x we 8 x x 5 N 2 8 b k X X 24 1c X n 1 Ansin n l x x X n 1 Bn n a l sin n l x x 25 y3 K An Bn U dc x sin n x l n 1 5 L 5 x x k 5 K k Y 12 13 N An 2 l Z l 0 x sin n l xdx Bn 2 n a Z l 0 x sin n l xdx n N 26 x x N O X 24 K 1 Y du u A 2 k lC x t l z 24 X A U T lC Fourier 3 18 V k Fourier S 24 x u Fourier d 24 z 7 e T q 7 A 2 7 u 5 lC L 1 g U P AT 24 A u vk AT e K 1 gC L 5 J 3 e 2 K U 9D Tn t k a k E L e SN w f gC J 5 2 SK aq K L S S 2 aq SN S 8 n d 24 w u X U z e un x t Ancos n a l t Bnsin n a l t sin n l x Cnsinknxcos nt n 27 2 9D 9 kn n l n kna Cn pA 2 n B2n n arctan Bn An 28 27 L 7 u n u 3 e K n uX 5 u Cn nK k n 1 1 a l du n n 1 n 1 n g 7 27 k node v knx n x l m x m n l m N 29 n g k n 1 vk J antinode du m7k J n g k n J u g 5U SK l 5 3 FUs e gd Fe v0 2 9D 9D 1 K lC K 9 9D K u x t K u t a2 2u x2 0 0 x 0 30a u x x 0 0 u x x l 0 t 0 30b u t 0 x 0 x l 30c x 5 C y3 e A u x t X x T t 31 30a 30b 1 T 0 t a2T t 0 32 2 9D 10 K X00 x X x 0 33a X0 0 0 X0 l 0 33b 3 lC e k K 33 1 XJ 0 K 33a X x C cosh x Dsinh x C D 33b D 0 C sinh l 0 0 l 0 C D 0 u X x 0 T K O XJ x 3 XJ 0 2 0 K 33a X x C cos x Dsin x C D 33b d X0 0 0 D 0 2d X0 l 0 C sin l 0 T 7L sin l 0 XdK C N n n l n N u n n l 2 Xn x cos n l x n N 35 C 1 XJ3 n 0 K 34 J r k n n l 2 Xn x cos n l x n N 36 5 3 n 0 n 6 0 I m x 5 1 aq k x Xn x cos n x l n 0 5 k T x3 m 0 l Xm x Xn x Z l 0 cos m l xcos n l xdx 0 m n N m 6 n 37a 2 9D 11 X0 x Xn x Z l 0 cos n l xdx 0 n N 37b d O y m e z n cos m l xcos n l x 1 2 cos m n l x cos m n l x g T x3 m 0 l f x 3 m 0 l k 5 x k K m f x X n 0 ancos n l x a0 X n 1 ancos n l x x 0 l 38 mX a0 1 l Z l 0 f x dx an 2 l Z l 0 f x cos n l xdx n N 39 5 a0 an n N L 3 an L n 0 U a0 3 38 L a0 5 rN 3 x m 38 cJe N mX 39 38 c L I n k cos n x l x l 0 l Z l 0 f x cos n l xdx X k 0 ak Z l 0 cos k l xcos n l xdx n N 5 x 5 m k k n T kXn x k2 Xn x Xn x Z l 0 cos2 n l xdx l n 0 l 2 n 6 0 40 dd 39 u 3 0 l f x r l l 2 2l P f x 3 0 l f x f x f x m e Fourier f x a0 X n 1 ancos n l x bnsin n l x x 41 mX a0 1 2l Z l l f x dx an 1 l Z l l f x cos n l xdx bn 1 l Z l l f x sin n l xdx n N 42 du f x l l bn 0 n N u m z f x a0 X n 1 ancos n l x x 43 2 9D 12 X L a0 1 l Z l 0 f x dx 1 l Z l 0 f x dx an 2 l Z l 0 f x cos n l xdx 2 l Z l 0 f x cos n l xdx n N 44 x 0 l 43 z 38 X 44 39 K m 38 Fourier X n 36 32 N T0 t A0 Tn t Anexp na2t Anexp n a l 2 t n N 45 A0 An u X v 30a 30b A u0 x t X0 x T0 t A0 un x t Xn x Tn t Anexp na2t cos n l x n N 46 A U v u x t X n 0 un x t A0 X n 1 Anexp na2t cos n l x 47 L 9 A0 An v 30c o X 47 30c A0 X n 1 Ancos n l x x 48 dc x cos n x l n 0 x k 5 38 39 N A0 1 l Z l 0 x dx An 2 l Z l 0 x cos n l xdx n N 49 x N O X 47 K 30 Y n 5 47 1 k P f exp na2t K A Ny L 5 1 K 1 a g Ak 3 g g 13 t N u x t A0 1 l Z l 0 x dx 50 5 3 47 XJ r A0 5 0 0 K N u x t 0 J 49 A0 L An n N r A0 5k 50 n L m u 9 9D K cJ 9 K k1 a g vk9 u 9 U3S 6 d9D Fourier 9 o l p 6 7 u 2dU U SK 9 x 0 l y 0 d 9 f x y u0 x l u0 t 4 3 g g c 0 lC u g g N S K k g g K w 7 XJ K k g g U lC m 5 XJ g K g kAT gz XJ 5 g gz E g 5 g gz C g gz m 5 u g K g K k m duz U 0 AT3 k 2 X J U K v g gz u K x 0 x l Asin t A u u K 2u t2 a2 2u x2 0 0 x 0 51a u x 0 0 u x l Asin t t 0 51b u t 0 0 u t t 0 0 0 x l 51c gz g E U u0 x t u x t v u x t v x t u0 x t 52 3 g g 14 v x t w v g v u u x t u K u0 x t x l Asin t 53 K v x t v K 2v t2 a2 2v x2 2A l xsin t 0 x 0 54a v x 0 0 v x l 0 t 0 54b v t 0 0 v t t 0 A l x 0 x l 54c gz C g L e 0 g K XJU 3 gz g5 u K u0 x t Asin x a sin l a sin t 55 N w v 51b v 51a 5 v x t K k g 2v t2 a2 2v x2 0 0 x 0 56a v x 0 0 v x l 0 t 0 56b v t 0 0 v t t 0 Asin x a sin l a 0 x l 56c 1 L K A 24 r 3e v x t X n 1 Ancos n a l t Bnsin n a l t sin n l x 57 56c X n 1 Ansin n l x 0 X n 1 Bn n a l sin n l x Asin x a sin l a 58 N An 0 Bn 2 A la n 1 a 2 n l 2 59 u x t A sin l a sin tsin x a 2 A la X n 1 n 1 a 2 n l 2 sin n a l tsin n l x 60 3 g g 15 N w XJ n a l m uu u d 60 k 1C S 55 U 1 C L 53 vk K p 0 e 55 L k J d 51b e k 5 k u0 x t F x sin t 61 v 51a 51b F00 x 2 a2 F x 0 62a F 0 0 F l A 62b 62a F x C cos x a Dsin x a 62b C 0 D A sin l a Xd F x Asin x a sin l a 61 55 k g 9D K X J 52 u0 x t 3 gz g5 XJ 5 g gz XJ 5 g AT U gz g K 3 e u0 x t v XJT E o v x t K kE 5 O v x t k X u0 x t gz 0 X E u0 x t XJ J u0 x t K y 5 S y J AT J U u0 x t 2 f u 1 K 2u t2 a2 2u x2 f x t 0 x 0 63a u x 0 t u x l t t 0 63b u t 0 x u t t 0 x 0 x l 63c t t x x gz u0 x t t x l t t 64 3 g g 16 K v x t K 2v t2 a2 2v x2 f x t 00 t x l 00 t 00 t 65a v x 0 0 v x l 0 65b v t 0 x 0 0 0 l x v t t 0 x 0 0 0 0 0 0 l x 65c p gz u K 5vk E J u0 x t t x2 l2 t t 66 5 gz w X 64 5 g m b gz Ku 1 K 2u t2 a2 2u x2 f x t 0 x 0 67a u x 0 0 u x l 0 t 0 67b u t 0 x u t t 0 x 0 x l 67c 65 k w u K y3 K u x t x t C XJ t o x m x 5 Xc y x sin n x l n 1 cos n x l n 0 t Cz mX A 3 t A T m L mX t u x t X n 1 Tn t sin n l x 68 u x t X n 0 Tn t cos n l x 69 m g L J x Tn t m v K 67 Xd K m 69 w v 67b v m U k J y3 AT A g K x5 m u x t XJ A g K G A Tk ATk f x t 0 1 7 X I x x 2 aqu 68 m u8c K 3 g g 17 m 68 T J v 67b y3AT 67a 67c x Tn t AT v d r m A m f x t X n 1 fn t sin n l x fn t 2 l Z l 0 f x t sin n l xdx n N 70a x X n 1 nsin n l x n 2 l Z l 0 x sin n l xdx n N 70b x X n 1 nsin n l x n 2 l Z l 0 x sin n l xdx n N 70c p mX fn t n n f x t x x N K L O 5 m 68 67a 67c X n 1 T 00 n t n a l 2 Tn t sin n l x X n 1 fn t sin n l x 71a X n 1 Tn 0 sin n l x X n 1 nsin n l x 71b X n 1 T 0 n 0 sin n l x X n 1 nsin n l x 71c d 5 N mX AT O X sin k x l l 0 l k T 00 n t n a l 2 Tn t fn t 72a Tn 0 n T 0 n 0 n 72b X K U Tn t ncos n a l t l n n a sin n a l t l n a Z t 0 fn sin hn a l t i d 73 J 68 K 67 XJ f x t 0 K k fn t 0 u z 1 22 X g gz g u n9D aq K u 2 51 73 uw J A g J 1 3 SO B 7 P4 J u N K 73 fn t K A 5 Xe fn t c KN w k A c l n a 2 u Tn t Ancos n a l t Bnsin n a l t c l n a 2 3 g g 18 2 72b An Bn fn t ct 5 k A ct l n a 2 N X q X fn t ccos t dsin t u A n a l n 72a b A C cos t Dsin t C D l Tn t Ancos n a l t Bnsin n a l t ccos t dsin t n a l 2 2 X An Bn 2 XJ G Laplace C K K 72 A N Laplace C u K 0 n x J y32 ww u m 68 m U A g K XJ 67a k u x X a xk o 68 v Ec L J K E u K b u K 67 AT x 2u t2 K 2u x2 F x t 0 x 0 74a u x 0 0 u x l 0 t 0 74b u t 0 x u t t 0 x 0 x l 74c p ru K y3 68 m F x t X n 1 Fn t sin n l x Fn t 2 l Z l 0 F x t sin n l xdx n N 75 N X n 1 x T00 n t K n l 2 Tn t sin n l x X n 1 Fn t sin n l x 76 du kx x d aqu 72a J 3 g 3 sin m x l l0 l x 5 mn 2 l Z l 0 x sin m l xsin n l xdx 77 X n 1 mnT00 n t K m l 2 Tm t Fm t m N 78 u x Tm t m 1 72b K S J 38c e AT m 68 AT x sin n x l n 1 3 g g 19 y3 e m u x t X n 1 Tn t Xn x 79 74a 74b X n 1 x T00 n t Xn x KTn t X 00 n x F x t 80 X n 1 Tn t Xn 0 X n 1 Tn t Xn l 0 81 XJX00 n x x Xn x Xn 0 Xn l 0 o K k S XJ 3 K 74 F x t 0 lC u x t X x T t oX x v K KX00 x x X x 0 82a X 0 0 X l 0 82b A T t K v T00 t T t 0 L e I T x p 1 k JK IO Sturm Liouville K c K y x weight K UC x X e Sturm Liouville K T K N 1 k n 0 2 3 1 2 n n 1 20 m 79 74c 3 N Tn 0 Rl 0 Xn x x x dx Rl 0 X2 n x x dx T0 n 0 Rl 0 Xn x x x dx Rl 0 X2 n x x dx 87 x N K 82 Xn x 2d 85 87 Tn t m 79 K 74 Y L x E K 82 J d w m g K AT A g K x5 m u x t c rN o gz 9D K u x u x 0 t u x u x l t 88 0 2 26 0 2 26 0 t t 5 1 na X x 0 e 0 K 1 a e 0 K 1 a e 6 0 K 1na du 2 26 0 c e U3 gz g5 X J 5 g gz XJ 5 g U gz g K gz g E U u0 x t u x t v U 52 u x t v x t u0 x t 5 Kz v x t K w v g K X E u0 x t U 88 u0 x t t 7 t t k 88 9 x b e x 5 u0 x t A t xB t 89 88 B t A t XJ 1 a Kdu u0 x t x B t Us t t b e x g u0 x t xA t x2B t 90 2 88 B t A t 1 a 5 89 we 89 88 B t A t t 91a l B t A t t 91b 4 21 uB t A t 5 X 1 l l 92 T1 B t A t du y k K m T1 u d1 0 l 0 2d1 0 k 0 K 6 0 0 1 a 89 U 90 e A f 11 a u x 0 t u x l t 89 A t t A t lB t t dd B t A t u0 x t t x l t t d c L 64 21 a u x x 0 t u x x l t c d AT 90 A t t A t 2lB t t dd B t A t u0 x t x t x2 2l t t 3 u x 0 t u x x l t 89 A t t B t t u0 x t t x t SK l m m 4 k N u0 S SvkT T 3 S u x t 4 c A 9D K 9 gC K 9 gC d K q K k vk k K lC e K 2u x2 2u y2 0 0 x l 0 y 9 o L K J y3 e A u x y X x Y y 94 93a 93b 1 Y 00 y Y y 0 95 K X00 x X x 0 96a X 0 0 X0 l 0 96b 3 lC e k K 96 1 XJ 0 K 96a X x C cosh x Dsinh x C D 96b C 0 Dcosh l 0 cosh l 1 0 D 0 u X x 0 T 0 2 0 K 96a X x C cos x Dsin x C D 96b d X 0 0 C 0 2d X0 l 0 Dcos l 0 T 7L cos l 0 XdK D N n 2n 1 2l n N u n 2 n 2n 1 2l 2 Xn x sin nx sin 2n 1 2l x n N 97 D 1 51 XJ n n 2n 1 2l KAT n N 7L5 2 2 ww v O L X K XJ cos nx o L y uy 3 B o U O G L O I3 v 4 23 7 5 X 1 2 I 0 vk T G e 101 x 5 d Sturm Liouville K y 5 y S N y Z l 0 sin mxsin nxdx Z l 0 sin 2m 1 2l xsin 2n 1 2l xdx l 2 mn m n N 98 97 95 Y 00 n y 2 nYn y 0 n N 99 5 exp ny exp ny cosh ny sinh ny e O B 5 sinh n d y sinh ny Yn y Ansinh n d y Bnsinh ny n N 100 An Bn u e v 93a 93b u x y X n 1 Ansinh n d y Bnsinh ny sin nx 101 93c X n 1 Ansinh ndsin nx x X n 1 Bnsinh ndsin nx x 102 du x 5 X An Bn d 98 An 2 lsinh nd Z l 0 x sin nxdx Bn 2 lsinh nd Z l 0 x sin nxdx 103 x x N O X 101 K p g u K g g u X 9D K X Laplace 7 k g K k T e 2 g Poisson K 2u x2 2u y2 f x y 0 x l 0 y E g m 5 L du K k r Kz K5 2u1 x2 2u1 y2 f x y 0 x l 0 y d 105a u1 x 0 0 u1 x l 0 0 y d 105b u1 y y 0 x u1 y y d x 0 x l 105c 2u2 x2 2u2 y2 0 0 x l 0 y 106b X S L 5 4 IX Xc lC k H5 u gC K X n m a K lC U u G 3n m N n N I X IX r I I G 3 AT A IX X K AT IX XJ IX K lC lC I 9 K c A N n 3 K y G u K XJu o A 5 4 IX 25 K J u IX K E XJ 3N K y X 9 l X S K ow AT 4 IX 3 4 IX 9D K 9A K I U IX K K 9 gC c A aq K k A AT5 3 4 IX Laplace 2u 1 u 1 2 2u 2 0 109 Laplace d IX L IC x cos y sin g O uO IX Laplace c X d 0 6 k e b d AT5 e u N y3 e A u R 110 109 1 2R00 R0 R 0 111 00 0 112 3 lC A 5 3 IX I A A IX X 4 IX I 2 L A n 3 A ATk AT T K 7L u 2 u 113 110 R 2 0 R 2 114 d n AT v 5 I A5 g natural boundary conditions g k q 5 112 K e K 5 4 IX 26 1 XJ 0 K 112 Ce De C D 114 C e 2 1 e D e 2 1 e 0 du k 7L 7Lk C e 2 1 0 D e 2 1 0 0 e 2 1 6 0 l C D 0 u 0 T 0 2 0 K 112 Cei De i C D 114 C ei 2 1 ei D e i 2 1 e i 0 du k 7L 7Lk C ei 2 1 0 D e i 2 1 0 D ei 2 1 0 XJ ei 2 1 6 0 K C D 0 l 0 T T 7L ei 2 1 XdK C D N m m N u m m2 m cosm sinm m N 116 y3 Au k Au y degeneration degree of degeneracy p 2 5 u 1 3 O V n J N uy 38c e B XJ3 116 m 0 K 115 J r k m m2 m cosm sinm m N 117 5 4 IX 27 5 3 m 0 m 6 0 I m m 0 S k 1 m 6 0 2 117 x3 m 0 2 S IO Fourier m 111 2R00 m R 0 m m 2Rm 0 118 3 L Euler X N gC C et z X d2Rm dt2 m2Rm 0 dd Rm emt e mt m 6 0 R0 1 t C 5 gC Rm m m m N R0 1 ln 119 R k 118 k m L m 0 U 118 R00 0 0 u R00 B0 l R0 A0 B0ln A0 B0 J 117 119 109 u A0 B0ln X m 1 m Amcosm Bmsinm X m 1 m Cmcosm Dmsinm 120 X S Laplace K 2u 1 u 1 2 2u 2 0 121b A0 X m 1 Amcosm Bmsinm F 123 du x 5 X o v 2d x 5 A0 1 2 Z 2 0 F d Am 1 Z 2 0 F cosm d Bm 1 Z 2 0 F sinm d 124 5 4 IX 28 F N O X 122 K 121 aq Laplace K 2u 1 u 1 2 2u 2 0 a 125a u a G 125b du 120 ln m3 k 5 AT B0 0 Am Bm 0 m N l u A0 X m 1 a m Cmcosm Dmsinm 126 125b A0 X m 1 Cmcosm Dmsinm G 127 u X A0 1 2 Z 2 0 G d Cm 1 Z 2 0 G cosm d Dm 1 Z 2 0 G sinm d 128 G N O X 126 K 125 XJ K k X a x k 5 K u u l 120 z u A0 B0ln X m 1 Am m Cm m cosm 129 AT 122 3 v Laplace 126 3 v Laplace 120 3 v Laplace 3 v Laplace K U u A0 n N c J e n K N a u E0 E0 R u m n A K Lda K N l n 2 d n K U e L A 1 n du N N S 5 4 IX 29 5du 3 3 K w k U 1 IX z q E0 x du k z Cz R u z xy L5 K K u G U 4 IX 1n K du vk u v Laplace K k u A K dd e K 2u 1 u 1 2 2u 2 0 a 130a u a 0 130b u E0 cos u0 130c u0 I L 5 5 u0 U 5 3O 4 L k 1o K dc K k x 5 K 129 130c K A0 B0ln X m 1 Am mcosm u0 E0 cos A0 u0 B0 0 A1 E0 Am 0 m 1 131 u u u0 E0 cos X m 1 Cm mcosm 132 2 130b u0 E0acos X m 1 Cma mcosm 0 u0 0 C1 E0a2 Cm 0 m 1 133 u E0 cos E0a2 cos 134 5 4 IX 30 1 J J 1 w 1 K d aA J 0 u n a 0 u a 2 0E0cos 135 w 2 2 2 3 2 K z o Q R 2 0 ad 2a 0E0 R 2 0 cos d 0 X 51 K a J 134 y 1 3 k 5 du K 3 w d 3 S 3k 3 v Laplace c dK L du 3 L K J g y3 2 2 2 d S ad A 4f du l 2acos 4f 4 ad 2acos 4a2 0E0cos2 d o 4 p R 2 24a 2 0E0 cos2 d 2 a2 0E0 x n 4f p 4f p ql l 0 q ql n 4f p 2 0 2 pcos 2 0 p J 134 1 3 NC E 2E0cos 3 0 N B g 1 XJ 5 z Q T K 2 XJ T KX J I k kg SN oPoisson n 0 e g Poisson n e S K 2u 1 u 1 2 2u 2 f a 136a u a F 136b K k n O0 Xe XJ f K L f 2 K m dE dt Z l 0 v t 2v t2 a2 v x x v t dx 6 5 533 1 1 f x t gS 1 dE dt Z l 0 v t 2v t2 a2 2v x2 dx a2 v x v t l 0 147a 147b 5 y3z 1 1 a dE dt 0 l E t E 0 d 147c E 0 0 u E t 0 E t K 7k v t 0 v x 0 u v x t 2d T l v x t 0 5 1 148 1 L U w PT U u U u X e 3gd e u 3x x x 3 v T v x x x v x x T 2v x2 x dv x t C v x t v x t Z l 0 T 2v x2 v dx T v v x l 0 T Z l 0 v x v x dx 149 g du v O k v v x 2 v x 2 2 v x v x 150 XJz 1 a 1 a K 149 1 150 149 Z l 0 T 2v x2 v dx 1 2T Z l 0 v x 2 dx 1 2T Z l 0 v v x 2 dx 151 V 1 2T Z l 0 v x 2 dx 1 2 a 2 Z l 0 v x 2 dx 152 K u V TNX X V U U O 147a 5 v k v v t t Z l 0 T 2v x2 v dx Z l 0 2v t2 v dx 1 2 Z l 0 v v t 2 dx 1 2 Z l 0 v t 2 dx U 153 U V 0 y TNX X V U 2 XJu 1na Xx l X k u x l vk u G u l x l x 0 x l T v x kv x l 0 T U 149 1 l Z l 0 T 2v x2 v dx 1 2