1、一、傅里葉積分及傅里葉變換, )sincos()(0kkkkktbtatg),2, 1 , 0(20kTkkk其中非連續(xù)參量.01kkk.sin)(2,cos)(2222TTkkTTkkkdfTbdfTa系數(shù), 0)(1limlim220TTTTdfTa則假定積分存在1221coscos)(2limcoslimkkTTkTkkkTtdfTta上式等于,coscos)(1lim122kkkTTkTtdf0.coscos)(1dtdf01.sinsin)(1sinlim,dtdftbkkkT同理稱為傅里葉積分.sin)(cos)()(lim)(00dtBdtAtgtfTdfcos)(1dfsin)

2、(1傅里葉積分還可以合并為如下形式：.)()()(,)(cos)()(2/1220BACdtCtf. 0)0(,sin)(2)(,sin)()(00fdfBdtBtf且.sin)(2)(,sin)(2)(00dfBdtBtf對(duì)稱形式. 0)0(,cos)(2)(,cos)()(00fdfAdtAtf且.cos)(2)(,cos)(2)(00dfAdtAtf對(duì)稱式2、復(fù)數(shù)形式的傅里葉積分cos,sin,22iiiiitttteeeett0000)()(21)()(21sin)(cos)()(deiBAdeiBAdtBdtAtftiti傅里葉積分復(fù)數(shù)形式傅里葉積分令兩項(xiàng)合并.)(-deFti (

3、)( )/ 2, (0)( ) ()()/ 2.(0)i -i-ABFAB其中dfBdfAsin)(1)(,cos)(1)(*1( )( ).i2Ffed.)(21)(,)(21)(-defFdeFtfiti對(duì)稱形式dttf)()()(limtgtfT2()021( )2limikTtTTkfed0()1( ).2iktddfed 令令1( )( ),2iFfed若令.)()(deFtfti則上述積分成立的條件是：（傅里葉積分定理傅里葉積分定理 ）.)(21)(defFi,)(21)(deFtfti.)(21)(dxexfkFxik,)(21)(dkekFxfikx傅里葉變換的對(duì)稱形式為：dx

4、exfxfxik)(21)(Fdxexfexfxikxik)( )(21)(21分部積分, 0)(limxfxdxeikxfxfxik)( )(210)(F故dxexfikxik)(21).(kikF).()()()(kFikxfnnF進(jìn)而可得).(1)(kFikdfxF),()(xfdfdxdx,)()()(xxdfikdfdxdxfFFF由導(dǎo)數(shù)定理知)(1)(xfikdfxFF故.1)(akFaaxfFdxeaxfaxfikx)(21)(Faydeyfyakiaxy)(21令dyeyfayaki)(211)(100kFaakk 令.1akFa).(1kFik).()(00kFexxfxik

5、Fdxexxfxxfikx)(21)(00F)()(2100 xxdeeyfikxikydyeyfeikyikx)(210).(0kFeikx).()(00kkFxfexikFdxexfeikxxik)(210左端dxexfxkki)(0)(21).(0kkFdxffxfxf)()()()(2121).(*)()()(2121kFkFxfxfFdxedxffikx)()(2121左端ddxexffikx)(21)(21交換積分次序dkFefik)()(21延遲定理)()(21221kFdefik).()(221kFkF0020121)()()(*)(dkkkFkFkFkF0201)(21)(0

6、dkdxexfekFikxikx位移定理dxexfdkekFikxikx)()(2120010交換積分次序.)()(2)()(2)()(*212*121dkkFkFdkkFkFdxxfxfdxdkekFxfikx)()(21左端dxexfxfikx)()(21201dxexfxfikxxx)()(21210取.)()(21xfxfFdkkFdxexfikx)()(21交換積分次序dkkFdxexfikx)()(2122*1dkkFdxexfikx)()(2122*1.)()(22*1dkkFkF.)(2)(22dkkFdxxf若f1(x)= f2(x)= f(x)，則可得能量積分111),()

7、,(dkezykFzyxfxik122112),(dkedkezkkFxikyik123321123),(dkedkedkekkkFxikyikzik ,),(321)(321321dkdkdkekkkFzkykxki,),(21),(11dxezyxfzykFxik其中,),(21),(2121dyezykFzkkFyikdzezkkFkkkFzik3),(21),(21321.),()(21)(3321dxdydzezyxfzkykxki,321i zi yi xr,332211jkjkjkk,)()(kdekFrfrk i則.)()2(1)(3rderfkFrk i,)()2(1)(23

8、kdekFrfrk i.)()2(1)(23rderfkFrk i,),(),(321)(ddkdkdkekFtrftrki.),()2(1),()(4dxdydzdtetrfkFtrki.),0()() 1 (222并推廣到三維高斯分布函數(shù)aaexfxa.1)()2(xxf02.1sinsin).( 0),(sin)() 3(dkkkxkxxxxf并求積分.2coscossin).1( 0),1(1)()4(032dkkkkkkxxxxf并求積分-ikxxadxeaekF22221)(dxkxikxeaxa)sin(cos22220cos220222kxdxeaxaaebxdxeabax40

9、2221cos注：222222122akeaa.222ake,)(2322222222222razayaxaeaaeaeaerf.)(22322212222akkkakeekF-ikxdxexkF121)(0sin220dxxkxi)0( 2)0(2kiki.2kki-ikxdxexfxfkF)(21)()(F0sin21-ikxdxex-dxkxikxx)sin(cossin210sinsin10kxdxxi/20) 1cos() 1cos(211dxxkxki01) 1sin(1) 1sin(21kxkkxki1) 1sin(1) 1sin(21kkkki1) 1sin() 1sin()

10、1sin() 1sin(212kkkkkki1cossin2sincos2212kkkki.)1 (sin2kik-ikx-dkekFkFxf)()()(1F-ikxdkekki21sin1-dkkxikxkki)sin(cos1sin12-dkkkxk21sinsin10,1sinsin202dkkkxk).( 0),(sin1sinsin202xxxdkkkxk021sinsindkkkxkI021)(cos)(cos21dkkxkxk,1)(2)(1kezfxik令,1)(2)(2kezfxik令12)(1)1 () 1 (Reszxizzef分式2)(xie2)sin()cos(xix

11、,2sincosxix.2sincos) 1(Res1xixf,2sincos) 1 (Res2xixf而.2sincos) 1(Res2xixf).(2121II 兩項(xiàng)即為含三角函數(shù)廣兩項(xiàng)即為含三角函數(shù)廣義積分中的系數(shù)義積分中的系數(shù)m)(2121III)1(Res) 1 (Res)1(Res) 1 (Res2212211ffffi)sin2(221xii.sin2x. 0)1(Res) 1 (Res)1(Res) 1 (Res2212211ffffiI-ikxdxexfxfkF)(21)()(F0)1 (21112-ikxdxex112)sin(cos)1 (21-dxkxikxx0cos)

12、1 (21112-kxdxxisinx-isinx102cos)1 (1kxdxx10102sin21sin)1 (1dxkkxxkkxx分部積分1010cos2cos20dxkkxkkkxxk分部積分).cos(sin23kkkk-ikx-dkekFkFxf)()()(1F-ikxdkekkkk3cossin2-dkkxikxkkkk)sin(coscossin230coscossin403kxdkkkkk).1( 0),1()1 (4coscossin203xxxkxdkkkkk)211 (42coscossin203dkkkkkk.163).(),(),( , 0002初始條件xuxux

13、uautttxxtt ).(),(),(00022kUkUUakUtt運(yùn)用導(dǎo)數(shù)定理,),(),(txutkUF,)()(xkF.)()(xkF.)()(),(ikatikatekBekAtkU),()()(0kBkAkUt).()()(0kBkAikakUt),(121)(21)(kikakkA).(121)(21)(kikakkB.)(121)(21)(121)(21),(ikatikatekikakekikaktkU故),(),(1tkUtxu-F.)(121)(21)(121)(21dkeekikakekikakikxikatikat,0 xat 令dkeekikxikx0)(21dkeekikxikx)(210)(210 xx延遲定理),(21atxdkeekikxikx0)(21),(21atxdkekikaxxik)(0)(121dkekikaikyxxy)(1210令yda)(21積分定理atxda)(21,)()(21atxatxatxddadkekikaxxik)(0)(121,)(21atxda.)(21)()(21),(atxatxdaatxatxtxu)().(),( , 002初始條件xuxuautxxt).(, 0022kUUakUt.)(),(22takekAtkU).(