a0 a1 star a2a3a4a5a6a7a8a9a10 2 star §19.3 a11a12a6a7 1 ? ¨'C A^ 11 1 ? ¨'C AA^^ 0 ?) ' §‰)flK , ? {§= ¨'C A^u3?) ' §‰)flK' ~^ ¨'C kLaplaceC FourierC ?' x19.1 LaplaceC 12 x19.1 LaplaceC LaplaceC ^u?)? m ' §‰)flK'C §gC ?’ 5~ '~X§ 5·x t gC ' §‰)flK§C I?)~ ' §(gC x) ‰)flK' ‘5§ o’ N·?)'? ? · ' ‰)flK ) …?§ 7L § U 'flK )' ~1 ??.\ 9D flK @u @t ?? @2u @x2 = f(x;t), ?1 < x < 1; t > 0; uflflt=0 = 0, ?1 < x < 1 )' ) LaplaceC '- u(x;t);U(x;p) = Z 1 0 u(x;t)e?ptdt; |^—'^ §k @u @t ;pU(x;p): rC …? w?·x …?§p·o?§?– @2u @x2 ; d2U(x;p) dx2 ; ?$ · …? ?'2? - f(x;t);F(x;p); ? §3?LLaplaceC §‰)flK C? pU(x;p)??d 2U(x;p) dx2 = F(x;p): |^~11.11 (J§ – U(x;p) = 12 1p?p Z 1 ?1 F(x0;p)exp ‰ ? rp ?jx?x 0j dx0: 2 LaplaceC ?“( ~10.7) 1p pe ?fipp: 1p …t exp ‰ ?fi 2 4t –9 ¨‰n( 10.3!)§ U u(x;t) = 12p?… Z 1 ?1 dx0 Z t 0 exp ‰ ?(x?x 0)2 4?(t??) f(x0;?) pt?? d?: 3???.?m ‰)flK￥§ ? ?( >.^ '￠S §?.?m§ · n ? § § ·L?3? ( m§ §￠￠￠)S§ K – 'ˇd§XJ / ‰)flK {§K A k>.^ uflflx!§1 ! 0: x19.1 LaplaceC 13 ^LaplaceC ?) ' §‰)flK§ –~ gC ?8– §, fi …? …?(~X § g §§ /“ U?E,)$ 7 N? §3? IAA^^ ¨‰n= ' ~2 ^LaplaceC ?)?.u ˉ?flK @2u @t2 ?a 2@ 2u @x2 = 0, ?1 < x < 1; t > 0; uflflt=0 = `(x); @u@t flfl fl t=0 = ?(x);?1 < x < 1: ) 3LaplaceC e§ u(x;t);U(x;p); u·§ 5 ‰)flK z p2U(x;p)?a2d 2U(x;p) dx2 = p`(x)+?(x): ~11.11 (J§ –? d § )(￠S ? U(x;p)3x !§1 1 ) U(x;p) = 12ap Z 1 ?1 h p`(x0)+?(x0) i exp n ?pa flfl flx?x0 flfl fl o dx0 = 12a Z 1 ?1 h `(x0)+ ?(x 0) p i exp n ?pa flfl flx?x0 flfl fl o dx0: ˇ e?fip;–(t?fi); 1pe?fip;·(t?fi); ?– u(x;t) = 12a Z 1 ?1 `(x0)– ? t? jx?x 0j a · dx0 + 12a Z 1 ?1 ?(x0)· ? t? jx?x 0j a · dx0 = 12 Z 1 ?1 `(x0)–(at?jx?x0j)dx0 + 12a Z 1 ?1 ?(x0)·(at?jx?x0j)dx0; 5? –(at?jx?x0j) = ( 0; jx?x0j6= at; 1; jx?x0j = at –9 ·(at?jx?x0j) = ( 0; jx?x0j > at; 1; jx?x0j < at; –? u(x;t) = 12 Z 1 ?1 `(x0)– ? t? jx?x 0j a · dx0 + 12a Z x+at x?at ?(x0)dx0 = 12 h `(x?at)+`(x+at) i + 12a Z x+at x?at ?(x0)dx0: x19.1 LaplaceC 14 ? ·1 n ￥^1ˉ{ (J' ^LaplaceC ?) ' §‰)flK k ‘:§? · 7 g >.^ gz(?+3? ~f￥§>.^ gz· 'N· )§ˇ ? k ' §‰)flK g>.^ =z ~ ' § g> .^ (ˇ 5 flK · gC flK)§??? ? 5 KK (J' x19.2 FourierC 15 x19.2 FourierC FourierC ? mC ?1' mC Cz?m§ – ^ F FourierC ?u?.?m(?1; 1) …?f(x)§XJ3??k ?m kk 4 4 k 1 am :§ ¨' Z 1 ?1 f(x)dx ?′?§K§ FourierC 3§ F(k) = 1p2… Z 1 ?1 f(x)e?ikxdx; _C ( )· f(x) = 1p2… Z 1 ?1 F(k)eikxdk: ?p FourierC _C /“ U G /“ k '/“ \? ?§ ??// n?[? ^' F uC {uC XJf(x)·‰′3 ?.?m[0; 1) §K x = 0 >.^ a.§ ^ uC F(k) = r 2 … Z 1 0 f(x)sinkxdx; f(x) = r 2 … Z 1 0 F(k)sinkxdk; ‰{uC F(k) = r 2 … Z 1 0 f(x)coskxdx; f(x) = r 2 … Z 1 0 F(k)coskxdk:: Fk u!{uC XJf(x)·‰′3k.?m §KA ^k u‰{uC ' !0 ?. m FourierC ' FourierC A^u?) ' §‰)flK§7, 9…? ! ? FourierC ' f(x) FourierC 3§? ?P F(k) = 1p2… Z 1 ?1 f(x)e?ikxdx = F[f(x)]; u·§ 1p 2… Z 1 ?1 f0(x)e?ikxdx = 1p2…f(x)e?ikx flfl fl 1 ?1 + ikp2… Z 1 ?1 f(x)e?ikxdx; x19.2 FourierC 16 du¨' Z 1 ?1 f(x)dx ?′?§ ‰k limx!§1f(x) = 0§?– F[f0(x)] = 1p2… Z 1 ?1 f0(x)e?ikxdx = ikp2… Z 1 ?1 f(x)e?ikxdx = ikF(k) = ikF[f(x)]: ? § , k F[f00(x)] = ?k2F[f(x)]: ^FourierC 5?) ! ~1 ~2' F?u~1§=?.\ 9D flK @u @t ?? @2u @x2 = f(x;t), ?1 < x < 1; t > 0; uflflt=0 = 0, ?1 < x < 1: –b u(x;t) FourierC 3§ U(k;t) = 1p2… Z 1 ?1 u(x;t)e?ikxdx; ? F(k;t) = 1p2… Z 1 ?1 f(x;t)e?ikxdx; ? §3 FourierC §‰)flK C dU(k;t) dt +?k 2U(k;t) = F(k;t); U(k;t)flflt=0 = 0; ^~?C·{?)? ~ ' § — flK§ U(k;t) = e??k2t Z t 0 F(k;?)e?k2?d?: 2? § u(x;t) = 1p2… Z 1 ?1 U(k;t)eikxdk = Z t 0 ? 1 p2… Z 1 ?1 F(k;?)e??k2(t??)eikxdk ? d?: |^1o ￥ (J§ Z 1 0 e?t2 cos2xtdt = 12p…e?x2; – 1p 2… Z 1 ?1 e??k2(t??)eikxdk = 1p2… Z 1 ?1 e??k2(t??) coskxdk = 1p2?(t??) exp ? ? x 2 4?(t??) ? ; x19.2 FourierC 17 2|^ f(x;t) = 1p2… Z 1 ?1 F(k;t)eikxdk; FourierC ¨?“§ F[f1(x)]F[f2(x)] = F ? 1 p2… Z 1 ?1 f1(?)f2(x??)d? ? ; U u(x;t) = Z t 0 ( 1p 2… Z 1 ?1 f(?;?)p 2?(t??) exp ? ? (x??) 2 4?(t??) ? d? ) d? = 12p?… Z t 0 ‰Z 1 ?1 f(?;?)exp ? ? (x??) 2 4?(t??) ? d? d? pt??: !￥ )“ ' l){ w§FourierC flKq ’LaplaceC { §ˇ I ^3?‰n5O ￥ y ‰¨'' ~ § ? { ^ ¨?“' F25)~2§?.u gd ?flK§ @2u @t2 ?a 2@ 2u @x2 = 0, ?1 < x < 1; t > 0; uflflt=0 = `(x); @u@t flfl fl t=0 = ?(x);?1 < x < 1: E u(x;t) FourierC 3§ U(k;t) = 1p2… Z 1 ?1 u(x;t)e?ikxdx; ? '(k) = 1p2… Z 1 ?1 `(x)e?ikxdx; “(k) = 1p2… Z 1 ?1 ?(x)e?ikxdx; u·§3 FourierC §‰)flK C d2U(k;t) dt2 +k 2a2U(k;t) = 0; U(k;t)flflt=0 = '(k); U(k;t)flflt=0 = “(k): ?· ~ ' § — flK§) = U(k;t) = '(k)coskat+“(k)sinkatka : x19.2 FourierC 18 FourierC ?“§ –? u(x;t) = 1p2… Z 1 ?1 ? '(k)coskat+“(k)sinkatka ? eikxdk: 5? 1p 2… Z 1 ?1 '(k)coskateikxdk = 1p2… 12 Z 1 ?1 '(k) h eik(x+at) +eik(x?at) i dk = 12￡`(x+at)+`(x?at)?; aq/§ k 1p 2… Z 1 ?1 “(k)sinkatka eikxdk = 1p2… Z 1 ?1 “(k) ?Z t 0 coska? d? ? eikxdk = Z t 0 ? 1 p2… Z 1 ?1 “(k)coska? eikxdk ? d? = 12 Z t 0 h ?(x+a?)+?(x?a?) i d? = 12a Z x+at x?at ?(?)d?; \ ? (J§ u(x;t) = 12 h `(x+at)+`(x?at) i + 12a Z x+at x?at ?(?)d?: ? , ^LaplaceC /“ ' ~3 ?)n ?. mˉ? § ‰)flK§ @2u @t2 ?c 2r2u = f(r;t);t > 0; uflflt=0 = `(r); @u@t flfl fl t=0 = ?(r): ) ?k FourierC '- U(k;t) = 1(2…)3=2 ZZZ u(r;t)expf?ik￠rgdr; F(k;t) = 1(2…)3=2 ZZZ f(r;t)expf?ik￠rgdr; '(k) = 1(2…)3=2 ZZZ `(r)expf?ik￠rgdr; “(k) = 1(2…)3=2 ZZZ ?(r)expf?ik￠rgdr; x19.2 FourierC 19 K‰)flKz ~ ' §— flK d2U dt2 +k 2c2U(k;t) = F(k;t); Uflflt=0 = '(k); dUdt flfl fl t=0 = “(k): 2 LaplaceC U(k;t) ;U(k;p); F(k;t) ;F(k;p): u·§‰)flK? C? ? § p2U(k;p)?p'(k)?“(k)+k2c2U(k;p) = F(k;p): ) = U(k;p) = 1p2 +k2c2￡F(k;p)+p'(k)+“(k)?: ? 'k LaplaceC §k U(k;t) = 1kcΨ(k)sinkct+Φ(k)coskct + 1kc Z t 0 sinkc? F(k;t??)d?: 2 FourierC § u(r;t) = 1(2…)3=2 ZZZ U(k;t) expfik￠rgdk = 1(2…)3=2 ZZZ Ψ(k) sinkctkc expfik￠rgdk + 1(2…)3=2 ZZZ Φ(k) coskctexpfik￠rgdk + 1(2…)3=2 ZZZ ? 1 kc Z t 0 sinkc? F(k;t??)d? ? expfik￠rgdk: |^FourierC ¨?“§ –? a §l “? ‰)flK )' ^k m ￥ I§ – F1 1 (2…)3=2 ZZZ sinkct kc expfik￠rgdk = 1(2…)3=2 ZZZ sinkct kc e ikrcos k2 sin dkd d` = 1p2…c Z 1 0 k sinkctdk Z … 0 eikrcos sin d = 1p2…c Z 1 0 sinkct 1?ir eikrcos flfl fl … 0 dk = 1p2…c 2r Z 1 0 sinkct sinkrdk = r… 2 1 cr –(r?ct): x19.2 FourierC 110 ?–§ FourierC ¨?“§ k 1 (2…)3=2 ZZZ Ψ(k) sinkctkc expfik￠rgdk = 1(2…)3=2 r… 2 1 c ZZZ 1 jr?r0j –(jr?r 0j?ct)?(r0)dr0 = 14…c ZZ Σ0 1 jr?r0j ?(r 0)dΣ0; ￥Σ0·–r ￥%!ct ? ￥?jr?r0j = ct' F1 1 (2…)3=2 ZZZ Φ(k) coskct expfik￠rgdk = 1(2…)3=2 @@t ZZZ Φ(k) sinkctkc expfik￠rgdk = 14…c @@t ZZ Σ0 1 jr?r0j `(r 0)dΣ0: F1n 1 (2…)3=2 ZZZ ? 1 kc Z t 0 sinkc? F(k;t??)d? ? expfik￠rgdk = Z t 0 ‰ 1 (2…)3=2 ZZZ ?sinkc? kc F(k;t??) ? expfik￠rgdk d? = Z t 0 ‰ 1 4…c ZZZ 1 jr?r0j –(jr?r 0j?c?)f(r0;t??)dr0 d? = 14…c ZZZ 1 jr?r0j ?Z t 0 –(jr?r0j?c?)f(r0;t??)d? ? dr0; w,§ Z t 0 –(jr?r0j?c?)f(r0;t??)d? = 8< : 1 cf(r 0;t?jr?r0j=c); jr?r0j < ct; 0; jr?r0j > ct: ?–§ 1 (2…)3=2 ZZZ ? 1 kc Z t 0 sinkc? F(k;t??)d? ? expfik￠rgdk = 14…c2 ZZZ jr?r0j<ct 1 jr?r0j f(r 0;t?jr?r0j=c)dr0: r ? (J8￥ 5§ ? u(r;t) = 14…c 2 4 ZZ Σ0 1 jr?r0j ?(r 0)dΣ0 + @ @t ZZ Σ0 1 jr?r0j `(r 0)dΣ0 3 5 + 14…c2 ZZZ jr?r0j<ct 1 jr?r0j f(r 0;t?jr?r0j=c)dr0: x19.3 ?. m / 111 x19.3 ?. m / ?u ?. m§ – ? ^ uC ‰{uC ' ^ uC ‰{uC K 3 uC e§ F(k) = r 2 … Z 1 0 f(x)sinkxdx; u· r 2 … Z 1 0 f0(x)sinkxdx = r 2 … ? f(x)sinkx flfl fl 1 0 ?k Z 1 0 f(x)coskxdx ? = ? r 2 …k Z 1 0 f(x)coskxdx; r 2 … Z 1 0 f00(x)sinkxdx = ? r 2 …k Z 1 0 f0(x)coskxdx = ? r 2 …k ? f(x)coskx flfl fl 1 0 +k Z 1 0 f(x)sinkxdx ? = r 2 …kf(0)?k 2F(k): dd §?u ' § ‰)flK§K k F‰)flK￥= y …?9 ?§ F 3 ?. m x = 0 ·1 a>.^ § – ^ uC ' §?u{uC G(k) = r 2 … Z 1 0 g(x)coskxdx; k r 2 … Z 1 0 g0(x)coskxdx = ? r 2 …g(0)+ r 2 …k Z 1 0 g(x)sinkxdx; r 2 … Z 1 0 g00(x)coskxdx = ? r 2 …g 0(0)?k2G(k): ?–§XJ u ? ‰§K k ‰)flK￥ F= y …?9 ?§ x19.3 ?. m / 112 F 3x = 0 ·1 a>.^ § – ^{uC ' ~4 ?) ?.\ 9D flK @u @t ?? @2u @x2 = f(x;t), 0 < x < 1; t > 0; uflflx=0 = u0, t > 0; uflflt=0 = 0, 0 < x < 1; ￥u0 ~?' ) AT^ uC ' U(k;t) · F[u(x;t)] = r 2 … Z 1 0 u(x;t)sinkxdx; Kk F ?@2u(x;t) @x2 ? = r 2 … Z 1 0 @2u(x;t) @x2 sinkxdx = r 2 …ku0 ?k 2U(k;t): u·§3?LFourierC §‰)flK =z ~ ' § — flK dU(k;t) dt +?k 2U(k;t) = r 2 …?ku0; U(k;t)flflt=0 = 0; ) U(k;t) = r 2 … u0 k h 1?e??k2t i : ?– u(x;t) = 2u0… Z 1 0 1 k h 1?e??k2t i sinkxdk: \1 ￥ (J§ Z 1 0 sinkx k dk = … 2; Z 1 0 sinkx k e ??k2tdk = p… Z x=2p?t 0 e??2d?; u(x;t) = u0 ? 2u0p… Z x=2p?t 0 e??2d? = u0 erfc x2p?t: ? ‰)flKw, u~19.3 l ! 1 /' T‰)flK ) 4 l ! 1§ˇ ??￥￥ k 0§?= – ? (J' x19.3 ?. m / 113 ~5 ?) ?. m ?‰flK @2u @x2 + @2u @y2 = 0; ?1 < x < 1; y > 0; uflfly=0 = f(x); ?1 < x < 1: ) KEAT^ uC ' U(x;k) · F[u(x;y)] = r 2 … Z 1 0 u(x;y)sinkydy; Kk F ?@2u(x;y) @y2 ? = r 2 … Z 1 0 @2u(x;y) @y2 sinkxdx = r 2 …kf(x)?k 2U(x;y): u·§3?LFourierC §‰)flK =z ??.?m?1 < x < 1 d2U(x;k) dx2 ?k 2U(k;t) = r 2 …kf(x) k.)'2|^~11.11 (J§k U(x;k) = 1p2… Z 1 ?1 e?kjx?x0jf(x0)dx0: § uC _C ?“§ –? u(x;y) = 1… Z 1 0 ?Z 1 ?1 e?kjx?x0jf(x0)dx0 ? sinkydk = 1… Z 1 ?1 f(x0) ?Z 1 0 e?kjx?x0jsinkydk ? dx0 = y… Z 1 ?1 f(x0) (x?x0)2 +y2dx 0: ? · ?? Poisson?“' x19.4 ’u¨'C ? 114 x19.4 ’u¨'C ? ¨'C ? F(k) = Z b a K(k;x)f(x)dx; §rgC x 2 [a; b](?p x – L m) …?f(x)C EC k …?F(k)§ ￥K(k;x)·¨'C § LaplaceC K(k;x) = e?kx; 0 ? x < 1; FourierC K(k;x) = e?ikx; ?1 < x < 1; uC K(k;x) = sinkx; 0 ? x < 1; {uC K(k;x) = coskx; 0 ? x < 1; ? k HankelC K(k;x) = xJn(kx); 0 ? x < 1; MellinC K(k;x) = xk?1; 0 ? x < 1: N ' §( E u ' §)‰)flK § .A ^= ?¨ 'C § ?–eA K F K1 ? 9 gC Cz?m TC ?·? ' ? XJgC Cz?m·(?1; 1)§ – ? ^FourierC ? ? XJgC Cz?m·[0; 1)§K ?A?¨'C – ?' F K2 ?u …?5 )§AO·…?3x ! (§)1 1 § T?¨ 'C ·? 3' ? ?uLaplaceC §C ·e?kx§ˇd?f(x) ? $§$ –#Nx ! 1 f(x)·u ? ? HankelC ·P~ …?§?f(x) ? p? ? FourierC ! uC {uC ·–ˇ …?§ˇd ?x ! (§)1 f(x) ! 0? ? MellinC ?…?f(x) ? p§ˇ § C · …?xk?1' F K3 ?…?f(x)9 ?f(n)(x)3TC ek{ ?’X' ? LaplaceC FourierC v? ?' ? ?u uC {uC § k…? ? ?(~Xf00(x)) “ UL? ?F(k) 5…?§…? ? ?K?'? § k‰)flK 9 …? 9 ?TgC ? ? § – ^ u‰{uC ' x19.4 ’u¨'C ? 115 ?u9D §§?+t Cz ·[0; 1)§ u‰{uC ? § dduu §￥￥ y …??t ?§?– ?{AA^^? ?C ' ?uˉ? §§ §￥ y?t ?( u3 n ??{ Z)§@o§33 KK §N ? mC t u‰{uC ' F K4 3 v K3 ?: §…?9 ?3TC e 3{ ?’X§@o§ 3?? ?’X￥§ ; /? y…?9 $ ? Aˇ ' ,§ U??/ ￠yTC §7L ?? Aˇ fi § ?(/‘§ ?? Aˇ —d‰)fl K￥ ‰‰)^ ' ?uLaplaceC § ?—'^ . ‰)^ §=XJ ? …? ? A ˇ {§ ‰·…?9 ?3 :: ? ' ?u ? ?¨'C § ?>.^ . ‰‰)^ ' ? K3 La ?^u~X? ' §'XJ? CX? ' §‰)fl K§? K I ?U'~X§?u ' § ￡L 1(x)+L2(y) ?u(x;y) = f(x;y); ￥L1(x) L2(y)'O·x y ( ) ' ?§b‰ ?L1(x) ￥ X? ·x ￠…?§ 2 u(x;y) ·￠…?§K¨'C Z b a K(k;x)u(x;y)dx = U(k;y) e§ §C Z b a K(k;x)f(x;y)dx = Z b a K(k;x)￡L1(x)u(x;y)?dx+L2(y) Z b a K(k;x)u(x;y)dx = Z b a ￡M 1(x)K(k;x) ?u(x;y)dx+L 2(y)U(k;y) = F(k;y); ￥M1(x)· ?L1(x) ?' y ’uU(k;y) ' §§M1(x)K(k;x)7L K(k;x)? ’§ M1(x)K(k;x) = ?K(k;x): ? § ‰ ?U J C K(k;x)'~X§3? IX?)Laplace §‰Poisson § §?u? C r§ U ^HankelC ' – ? ·?.‰ ?.?m ¨'C § A:·EC k ·oY §Ly _C ( ?“) ·?k ¨''3￠^￥ kk.?m ¨'C §? x19.4 ’u¨'C ? 116 k U l §?– –rC P Kn(x)'~X§~ k.?m ¨'C k k uC Kn(x) = sinnx; n = 1;2;3;￠￠￠ ; 0 ? x ? …; k {uC Kn(x) = cosnx; n = 0;1;2;￠￠￠ ; 0 ? x ? …; LegendreC Kn(x) = Pn(x); n = 0;1;2;￠￠￠ ; ?1 ? x ? 1: –w §? ¨'C T— ·‰′3 g?m …?'3A^? C ?) ' § ‰)flK § 'lC {?vk o K O§ )“ 'lC { ' x19.5 ˉC {0 117 x19.5 ˉC {0 / ˉ' 0§·Cc5u— 5 ’ # n K A^?? {' ˉ§ L?…? ?# ?§ mcurrency1“˙' ?E § fi?/? # ??? § § ?3 u— L§ ￥' ! Ul¨'C §? ˉC \ 5 0 'O(/‘§ ˉ' 8c ) '§=/¨' ˉC 0 / ˉ??0' k' eD FourierC ' [ §3(?1; 1) ‰′ …?f(t)§ f(t) v ‰^ §K§ FourierC F(!) = 1p2… Z 1 ?1 f(t)e?i!tdt –9_C f(t) = 1p2… Z 1 ?1 F(!)ei!td! 3'ˇ~§C t L m§! L“˙'F(!) & f(t) “ 'lFourierC ?“ –w § ? & “ A5§7Lu & 3(?1; 1) m S Cz ?§$ ) 5 Cz' Xn ? §3￠^ ?·?J ' , ?§XJ& 3, ( S)u) Cz§@o§ “ ? K ' 4 /§ y3t0 & –(t?t0)§ “ ·e?i!t0=p2…§ CX “˙ ' FourierC v§@3 ?Vc§ k<J L/\I0FourierC §ˇ L ? m z /I…?0gfi(t?b)§ J?? a(–t = b ￥% ‰ S) & §ˇL* & 3, “˙NC “ –… d& v ( &E' ?￡I…?(=UCb) –CX 'Gauss. …? · ? I…?'d uGauss.…? gfi(t) = 12p…fi exp n ? t 2 4fi o FourierC 1p 2… Z 1 ?1 1 2p…fi exp n ? t 2 4fi o e?i!tdt = 1p2…e?fi!2; E,·Gauss. …?§?–§ FourierC ¨?“ Z 1 ?1 f1(t)f2(t)e?i!tdt = Z 1 ?1 F1(?)F2(! ??)d?; 1p 2… Z 1 ?1 f1(t)e?i!tdt = F1(!); 1p 2… Z 1 ?1 f2(t)e?i!tdt = F2(!); ?u??…?f(t)§ k Z 1 ?1 f(t)gfi(t?b)e?i!tdt = 1p2… Z 1 ?1 F(?)e?i(!??)be?fi(!??)2d?; (#) ￥§F(!)·f(t) FourierC § F(!) = 1p2… Z 1 ?1 f(t)e?i!tdt: x19.5 ˉC {0 118 –r(#)“ n) …?gfi(t?b)ei!t f(t) S¨ ? gfi(t?b)ei!t; f(t) · = Z 1 ?1 h gfi(t?b)ei!t i? f(t)dt; m K·§ C S¨§ ? 1 p2…ei(!??)be?fi(!??)2; F(?) · = Z 1 ?1 h 1 p2…ei(!??)be?fi(!??)2 i? F(?)d?: ? (J L· ?H Parseval § ?f 1(t); f2(t) ￠ = ?F 1(?); F2(?) ￠ A~'dugfi(t?b)3t = b?k kb ?§ limfi!0gfi(t?b) = –(t?b); ?–§3(#)“￥§?u ¨' z 5gt = bNC§ ?um ¨' zK 5 g? = !NC' ?{‘§& f(t)3t = b &E –ˇL3“˙? = !NC* ? & “ ' –r(#)“ m U ? 1 p2fie?i!b ·Z 1 ?1 F(?)g1=(4fi)(? ?!)ei?bd?; (#)“‘?§e gfi(t) & mI…?§Kg1=(4fi)(?) A “ “˙I…?' ‰′ ˉtgfi = ?gfi(t); tgfi(t)￠ = Z 1 ?1 tflflgfi(t)flfl2dt gfi(t) ￥%'dugfi(t) …?§?–ˉtgfi = 0'3d?: § –? ‰′gfi(t) ￠gfi = s? (t?ˉt)gfi(t); (t?ˉt)gfi(t)￠? gfi(t); gfi(t)￠ w,k￠gfi = pfi' § –? g1=(4fi)(?) ￥% ￥ % ˉ!g1=(4fi) = 0; ￠g1=(4fi) = 12pfi: ?–§ mI ￡“˙I ~?§ ? 2￠gfi · ￡ ? 2￠g1=(4fi) · = 2: –3t ? ??? –(b; !): ￥% /b ? ￠gfi ? t ? b + ￠gfi; ! ? ￠g1=(4fi) ? ? ? ! +￠g1=(4fi)5/ z/L? m–“˙ z( a19.1)'? / ￡b?￠ gfi; b+￠gfi ?￡￡! ?￠ g1=(4fi); ! +￠g1=(4fi) ? ? a\IFourierC m–“˙I' mI 2￠gfi ? m–“˙I §“ ˙I 2￠g1=(4fi)? m–“˙I p ' m–“˙I !p –9?¨ · ‰ ' x19.5 ˉC {0 119 a19.1 GaborC m-“˙I – 0 · ?Aˇ/“ \IFourierC §=I…? Gauss.…? \IFourierC §? GaborC ' – O /“ I…?w(t)§X ?w(t)9tw(t) ? ¨(ˇ d ￠w k )§ §§ FourierC W(?)9?W(?) ? ¨(ˇ ￠W k )§? \IFourierC ? ? FourierC 'GaborC ,·? FourierC ?' GaborC ‰ ?? FourierC ":· § m–“˙I· ‰ § U “˙ p$ N 'ˇ “˙ mS –ˇ?? ’§?–§n ?· ( ?p “y § A ? mI? ?$“y §K mI'GaborC ‰ ?? FourierC ? u?n“ !Cz- & ' ¨' ˉC K· ??aflK u— 5 ' 0 ¨' ˉC §?k ?? ˉ9 ˉ Vg' XJh(t)9 FourierC H(!) ? ¨…?§ k § v N5^ Ch = Z 1 ?1 jH(!)j2 j!j d! < 1; K?h(t) ? ˉ' N5^ ? X ‰kH(0) = 0§= Z 1 ?1 h(t)dt = 0: ? ·? ˉ ˇ'3 ?¨' ˉC §I ^ N5^ ' ? ˉh(t) §ˇL?￡ § – x…? hb;a(t) = 1pjajh t?b a ? ; a 6= 0; ? ˉ' & f(t) ¨' ˉC K‰′ (Whf)(b;a) = 1pjaj Z 1 ?1 f(t)h? t?b a ? dt = ?hb;a; f￠: (z) …?h(t) ￥% 'O ˉt ￠h,K…?hb;a(t)·￥%3b+aˉt jaj￠h I…?'ˇd§(z)“L? ¨' ˉC & f(t)3 mI ￡b+aˉt?jaj￠ h; b+aˉt+jaj￠h ? S &E'jajC mIC??jajC mIC ' x19.5 ˉC {0 120 N·? hb;a(t) C Hb;a(?) = 1p2…a Z 1 ?1 h t?b a ? e?i?tdt = r a 2…e ?ib?H(a?): H(?) ￥% 'O ˉ! ￠H§- ·(? ? ˉ!) · H(?); K·(?)·￥% 'O 0 ￠H I…?' Parseval §§ – (Whf)(b;a) = r a 2… Z 1 ?1 ·?(a? ? ˉ!)F(?)d?: du·(a? ? ˉ!) ￠H=jaj§?–§ “ q·H(?)3“˙I ?ˉ! a ? ￠H jaj ; ˉ! a + ￠H jaj ? S &E'? m–“˙I · ￡b+aˉt?jaj￠ h; b+aˉt+jaj￠h ?￡? ˉ! jaj ? 1 jaj￠H; ˉ! jaj + 1 jaj￠H ? ; 2jaj￠h'ˇd§¨' ˉC k/C 0A5 3u p“y (=jaj ) §I?g ?C?? u $“y (=jaj ) §I?g?C ( a19.2)' ·??C A5§? ¨' ˉC ? N?n ?9 §E A^ k ' a19.2 ˉC m-“˙I 2? §d& f(t) ¨' ˉC (Whf)(b;a) –? f(t)(= )§ f(t) = 1C h Z 1 ?1 ?Z 1 ?1 (Whf)(b;a)hb;a(t)db ? da a2 : ? –n) o?? ˉ \ N5^ ' 3￠SA^￥§ k¨' ˉC l /“'~X§ˇLHaar…? h(t) = 8> >< >>: 1; 0 ? t < 1=2; ?1; 1=2 ? t < 1; 0; t < 0‰t ? 1 x19.5 ˉC {0 121 ? ?￡§ hj;k(t) = 2?j=2h?2?jt?k￠; j;k ?? ?; ? | 8 ?§ ? hj;k; hl;m￠ = –jl–km; ?? ? ¨ …?f(t) –(3? ′??′e)—m ˉ??§ f(t) = 1X j;k=?1 cj;khj;k(t); —mX?cj;k cj;k = ?hj;k;f￠ = (Whf) k 2j; 1 2j ? :