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1、貝塞爾函數(shù)的性質(zhì)?貝塞爾函數(shù)的性質(zhì)n1x2n??J?(x)??(?1)()n!?(n???1)2一、遞推公式n?0dJx()J()x???1()??(1)??dxxx?2n證明:dJx?()dn1x()?[?(1)?]?2n??dxxdxn?0n!(????n1)2?2n?1n2nx??(1)?2n??n?1n!(????n1)2?2k?1k?12(k?1)x??(1)?2k??2?k?0(k?1)!(?k???2)2?2k?1?(1)?k1xJ??1()x??2k??1????.k?0k!(?k???2
2、)2x第四章-貝塞爾函數(shù)的性質(zhì)22?n1x2n???(n???1)?(n??)?(n??)J?(x)??(?1)()n!?(n???1)2n?0d??(xJx())?xJ()(2)x???1dx?2n?2?d?dn1x證明:(Jxx?())?[?(1)?2n??]dxdxn?0n!(?n???1)2?2n?2??1n2(n??)x??(1)?2n??n?0n!(?n???1)2?2n???1?n(n??)x?x?(1)?2n???1n?0n!(?n???1)2?2n???1?n1x??x?(1)?2n??
3、?1?xJ??1(x)n?0n!(?n??)2第四章-貝塞爾函數(shù)的性質(zhì)3dJx?()J??1()xd(xJx?())?xJ?()(2)x()??(1)?????1dxxxdx由（1）和（2）式可得(等式左邊展開(kāi))xJx?()??Jx()??xJ()(3)x????1?Jx()?xJx?()?xJ()(4)x????1由（3）和（4）式相加減分別可得2?J()x?J()x?Jx()(5)??1??1?xJ()x?J()x?2()(6)Jx???1??1?第四章-貝塞爾函數(shù)的性質(zhì)44注：從這些遞推關(guān)系可以得到
4、Jx0?()??Jx1()(把??0代入(3)即得)注：對(duì)所有正整數(shù)m,Jm(x)都可以用J0(x)和J1(x)(??J0?(x))線性表示。2?由(5)式得J()x?Jx()?J()x??1???1x因此只需對(duì)J2(x)證明該結(jié)論即可。2Jx()?Jx()?Jx()210xxJx?()??Jx()??xJ()(3)x????12?J()x?J()x?Jx()(5)??1??1?x第四章-貝塞爾函數(shù)的性質(zhì)55貝塞爾函數(shù)常用遞推公式：dJx?()J??1()x()??(1)??Jx?()?J()xdxxx0
5、?1d??(xJx())?xJ()(2)x(xJx())??xJx()???110dxxJx?()??Jx()??xJ()(3)x????1nn(xJx())??xJ()xnn?1?Jx()?xJx?()?xJ()(4)x????1?n?n(xJxn())???xJn?1()x2?J()x?J()x?Jx()(5)??1??1?J()x?J()x?2()Jx?xn?1n?1n???J()xJ()x2()(6)Jx??1??1?2nJ()x?J()x?Jx()n?1n?1nx貝塞爾函數(shù)遞推公式的應(yīng)用之一就是
6、計(jì)算貝塞爾函數(shù)的積分。主要用于被積函數(shù)為冪函數(shù)與貝塞爾函數(shù)的乘積的情形。第四章-貝塞爾函數(shù)的性質(zhì)6J()x?J()x?2()(6)Jx?xJxdx().??1??1?例求?2d??(xJx())?xJ()(2)x???1dx解：Jx()?Jx()2(),?Jx?d201(xJx())?xJx()10dxxJxdx()?xJxdx()?2xJxdx?()?2?0?1?xJx()2(?xJx()?Jxdx())11?1?xJx()2(?xJx()?Jxdx?())11?0??xJx()2?Jx()?c.10第
7、四章-貝塞爾函數(shù)的性質(zhì)7諾伊曼函數(shù)也有與第一類貝塞爾函數(shù)相同的遞推關(guān)系式，只不過(guò)將上述(1)—(6)中的Jx()v換成Nx()v第四章-貝塞爾函數(shù)的性質(zhì)88二.半整數(shù)階貝塞爾函數(shù)第一類和第二類貝塞爾函數(shù)都不是初等函數(shù)，但是半整數(shù)階貝塞爾函數(shù)是初等函數(shù)，即若m是整數(shù)則時(shí)，J1()x和N1()x都是初等函數(shù)。m?m?22?1n1x2n?證明：由于J()x??(1)?()21?122n?0n!(?n?)2?11x2n?n2??(1)?()(2n?1)!!2n?0n!?n21112n?12n?311(2n?1)!
8、!?(n?)?(n?)(?n?)???()??n22222222第四章-貝塞爾函數(shù)的性質(zhì)9?11x2n?J()xn21??(1)?()?(2n?1)!!22n?0n!?n2?2n2nx??(1)??xn?0(2)!n2cosx??x2cosxJ()x?(7)1??x22sinx同樣可得J()x?(8)12?x第四章-貝塞爾函數(shù)的性質(zhì)1010121J3(x)?J1(x)?J1(x)?(?cosx?sinx)?2x22?xx23?
