Generalized Functions in Minkowski Space

Transcription

1 Geelize Ftio i Miowi Spe Chiw Ch Agt t, Mthemti Deptmet, The Uiveit of Aizo Cl DeVito Mthemti Deptmet, The Uiveit of Aizo PDF ete with FiePit pffto til veio

2 . Itoio Peioi ftio e i to m e of mthemti phi. The fmetl emple of h ftio e the fmili ie oie of tigoomet. O ppoe hee i to geelize the tigoomet ftio. I follow Dvi Shelp ie [] to itoe the lph et ftio. Thee ftio e the loge of ie oie i Miowi pe. i.e. We eple the Pthgoe theoem with fo ome itege. I m iteete i tfomig the Lple etio ito pol ooite of Miowi Spe. The lltio le to ptil iffeetil etio tht, whe, ee to Lple etio. We e eptio of vile to fi ptil oltio to thi etio. PDF ete with FiePit pffto til veio

3 . Geelize Tigoometi Ftio The ie oie ftio e peioi ftio with peio p. O t i thi etio i to ete the l of tigoometi ftio. I will fit give ief itoio to peioi ftio [6] i the iffeetil tem tht eie ie oie. Afte tht, I will follow Shelp metho [] to geete the lph et ftio... Peioi Ftio Peioi ftio o feetl. Thi of lo, o, the ight, et. B peioi we me omethig tht epet it motio i ott legth of time. Hee, ftio f i peioi if ( t T ) f $ T ' f. (..) To me moe foml mthemtil efiitio, fo ftio f : Æ, let ( f ) { T Œ f ( t T ) f } P :. (..) Clel P ( f ) oti zeo fo ftio. Now we efie f e peioi ftio if thee eit ozeo elemet i P ( f ). O the othe h, ozeo elemet T i P ( f ) i lle peioi of f. If thee eit mllet ozeo elemet T i P ( f ), the T i ofte lle the fmetl peio of.. The Diffeetil Stem Deiptio of Sie Coie f. Lie m othe peil ftio, the ie oie e efie iffeetil tem. Let oie two ftio. The the iffeetil tem Ï Ó (..) with the iitil oitio ( ),, h ie oltio i o. PDF ete with FiePit pffto til veio 3

4 .3. The Alph Bet Ftio The ie of the geelizig the ftio ito Miowi pe tt fom geelizig the iffeetil tem (..). Coie tl me, we wite Ï Ó, (.3.) Whe, (.3.) ee to (..) o we hve it ot Aw, we will ee oie.. hve m iteetig popetie imil to ie Mltiplig the two etio i (.3.) Ï Ó The left h ie of (.3.) e el, hee, we oti. (.3.). (.3.3). (.3.4) Itegtig (.3.4), we ole fo ome ott C, C. (.3.5) Set t ppl the iitil oitio i (.3.), the ott i el to oe. Hee, (.3.6) Note tht whe, thi i the fmili i o t. PDF ete with FiePit pffto til veio 4

5 .4. The Ivee Alph Bet Ftio The ehvio of e ot totll etoo. Howeve, thei ivee ftio hve ve ie epeio. We will e the ottio g to epeet the ivee ftio. Let ( ). With (.3.) (.3.6) we oti. (.4.) Solve eptig vile, we get ( ) C t Ú. (.4.) B the fmetl lw of ll the efiitio of ftio of i Ú ( ' ) 5, we ole the ivee g '. (.4.3) Simill, let o tht ( ). The we hve. (.4.4) ( ) C t Ú. (.4.5) Agi we ppl the fmetl lw of ll the efiitio of Ú ( ' ), g '. (.4.6) A iteetig popet of eve. The the iteg of limit i o ftio. Theefoe, e otie hee. Sppoe i g (.4.3) i eve o tht the itegl of the ppe i o ftio ie it ivee i o. Moeove, ie i o ee i eve We ole tht i eve ftio. PDF ete with FiePit pffto til veio i o.

6 3. Geelize Pi Miowi Spe Eveo ow wht i p, t el le to epe wh p h it vle. I ft, it i well ow tht we get eie epeio of p fom the ivee tget. I thi etio, we efie the geelize Pi the ivee lph ftio o ivee et ftio; lze the popetie of thi me. 3.. The Geelize Pi Rell i the tigoometi ftio, we hve gi g o p. We follow thi efie me p the eltio p g g Ú ( ). (3..) The p i ow the geelize Pi. Fom the eltio (3..) it i le tht Ï p p Ó. (3..) Now we hve how tht i o ftio omi of 3.. The Peioi Alph Bet efie i the itevl t p. Howeve, we ve i eve ftio whe i eve, o the e efie i the itevl p t p fo eve. I thi etio we wt to how whe i eve, e peioi ftio with peio p. We will tt howig Ï p p Ó t t. (3..) To pove (3..) i te, we wite the ight h ie of (3..) PDF ete with FiePit pffto til veio 6

7 Ú ( ) Ú ( ' ) ' Ú ( ' ) Whih i the me '. (3..) p g g. (3..3) We te g to the let pt oth ie ito the lph ftio, the Let t g p g (. (3..4) ), we hve p t. (3..5) Whih i the fit pt of (3..). Simill, fom etio (3..3) we te left pt oth ie ito the et ftio, the Let t g p g (. (3..6) ), the p t Whih i the eo pt of (3..). We oie the ftio. (3..7) t p, fom (3...) we oti p p p ( t p ) t t. (3..8) g to the Sppoe i eve, the the lph ftio i o et ftio i eve, hee p p t ( t) ( t). (3..9) PDF ete with FiePit pffto til veio 7

8 We ppl the imil metho to the et ftio. The we ole Ï Ó ( t p ) ( t p ) If we ow eple p Ï Ó ( t p ) ( t p ), whe i eve. (3..) t p i (3..) we get, whe i eve. (3..) Theefoe, lph et ftio of eve itege oe e peioi ftio with the peio p Miowi Spe Sppoe we hve two iepeet vile. We wt to e the lph et ftio to geete pol ooitee. Let Ï Ó ( ) ( ) B (.3.5), we oti. (3.3.). (3.3.) We wt to how the tfomtio (3..) ove the whole pe. We ete o etig of tigoometi ftio ~ [,p ) [, ) thi tfomtio. Hee, (3..) we e i the Miowi Spe. PDF ete with FiePit pffto til veio 8

9 9 4. Geelize Lple, Etio Lple etio i well ow thi pplitio i phi. It i t polem to hge the Lple etio to pol ooite; thi ie i oetio with the Diihlet polem, fo emple. Moeove, whe we oie o pol ooite i Miowi pe, fthemoe iteetig elt ppe. I thi etio I will fit epoe the t poee to olvig the Diihlet Polem, mel, oti the pol fom of Lple etio. The I will ete it the imil metho ito the Miowi pe peet how to get the geelize Lple etio. 4.. The Diihlet Polem Rell the Lple etio. (4..) We hve o i. O gol i to wite () i tem of. The ptil iffeetil of e Ó Ï i o, Ó Ï o i. (4..) Theefoe, we oti i o (4..3) i o i o i o (4..4) PDF ete with FiePit pffto til veio

10 o i (4..5) i o o i i o o i i o o o i i. (4..6) Some tem i ppee i. I ft, if we loo efll, we fi tht. (4..7) Hee, we get the pol fom of Lple etio. (4..8) 4.. Diihlet Polem i Miowi Spe Geelize Lple, Etio We wt to wite the etio ito the pol ooite i Miowi Spe. A we i i etio 3.3, let. The we hve Ó Ï, Ó Ï (4..) Theefoe, (4..) PDF ete with FiePit pffto til veio

11 (4..3) (4..4) (4..5) Thee e ome tem i etio (5..5) ppee i (5..) (5..3). Althogh thee PDF ete with FiePit pffto til veio

12 etio e log, we till implifie them witig ow 3 4. (4..6) Set the ove etio e zeo, we hve. (4..7) Defie ppoe, (5..7) eome (4..8) Whih i o geelize Lple Etio 4.3. The Soltio of Geelize Lple, Etio Powe Seie A the t w to olve ptil iffeetil etio, I e eptig vile to oti two oi iffeetil etio. Let (4.3.) i tivil oltio. So we e ow oette fo the e h tht. Divie oth ie, we oti (4.3.) We ge tht o oltio ee to tif the etio lthogh we v t eep ott, o v t eep ott. Theefoe, eh tem ee to e ott. A thee ott p to e zeo. Let it ott we e p with two oi iffeetil etio: PDF ete with FiePit pffto til veio

13 3 Ó Ï (4.3.3) We fit olve the e pt of the epte etio. Let  (4.3.4) Pt o powe eie fom of ito the fit etio of (5.3.3), we oti   (4.3.5)    3 (4.3.6) Compig the oeffiiet, we ole tht e it ott, Moeove, fo, we hve (4.3.7) (4.3.8) I tem of, o eie eome... (4.3.9) We gop thi ito two oe i tem of the othe ito, whih i jtifie lte, P P   m m m m m m (4.3.) PDF ete with FiePit pffto til veio

14 I thi eie ovegee? I e the tio tet to he oth tem of. m ( ) ( ) lim Æ P m ( m ) P m m ( m ) lim Æ ( ) ( ( ) ) (4.3.) lim Æ P m ( ) ( ) ( m ) m P m ( m ) m lim Æ (( ) )( ) (4.3.) Hee oth of them e oltel oveget fo. Thi jtifie o egig i (4.3.). Uig the imil metho, we fi the oltio of e pt the imil metho ( ) '  '  (4.3.3) P ( ) m m P ( m ) m m m It i le tht the two tem o the ight of (4.3.3) e ovegee fo. The eo I e to e o ott i we gop tht with o iepeet vile. Nmel, let A, B, etio (4.3.3) ee to ( ) A  B Â. (4.3.4) P ( ) m m P ( m ) m m m Let C ', D ', etio (4.3.3) ee to () ( ) C  D  (4.3.5) P ( ) m m P ( m ) m m m Coie the ftio  m ( ). (4.3.6) P m ( m ) 4 PDF ete with FiePit pffto til veio

15 Â t t t. (4.3.7) P m m m ( m ) () Â t t (4.3.8) P m () m ( m ) Â t t t (4.3.9) P m ( m ) The (4.3.4) (4.3.5) e witte Ï Ó A ( ) B ( ) C ( ) D ( ) (4.3.) Hee, ptil oltio to the geelize Lple etio h the fom ( ) ( A ( ) B ( ) )( C ( ) D ( ) ), (4.3.) Sie the geelize Lple etio i lie, we p the oltio fo oti ew oltio. Theefoe, the mot geel oltio i ( ) ( A ( ) B ( ) )( C ( ) D ( ) ), Â (4.3.) 5 PDF ete with FiePit pffto til veio

16 5. Refeee [] Dvi Shelp, A Geeliztio of the Tigoometi Ftio, Amei Mth. Mothl, pp , 959 [] Jme Stewt, Cll, foth eitio, Boo/Cole Plihig Comp, 999 [3] Iv Geogievih Petovi, Ptil Diffeetil Etio, Loo:Iliffe, 967 [4] Gett Bioff Gi Clo Rot, Oi Diffeetil Etio, Wlthm, M., Bliell P. Co.,969 [5] Fi B. Hile, Ave Cll fo Applitio, eo eitio, PetieHll, I. Eglewoo Cliff, New Jee, 976 [6] Ewi Kezig, Ave Egieeig Mthemti, eighth eitio, Joh Wile & So, I PDF ete with FiePit pffto til veio