Bessel function for complex variable

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1 Besse fuctio for compex variabe Kauhito Miuyama May 4, 7 Besse fuctio The Besse fuctio Z ν () is the fuctio wich satisfies + ) ( + ν Z ν () =. () Three kids of the soutios of this equatio are give by { ( J ν () P ν () cos ν + π { ( N ν () P ν () si ν + π H (±) ( π < arg < π) ) π ) π ν () J ν () ± in ν () = π e±i (ν+ ) π Pν () ± iq ν () ( Q ν () si ν + ( + Q ν () cos ν + ) } π ) } π { ( π < arg < π) ( π < arg < π) () (3) (4) where P ν () ad Q ν () are poyomias give by P ν () + Q ν () Note that J ν, N ν ad H (±) ν ( C ν,k () (5) ( + C ν,k () (6) C ν,m () (4ν )(4ν 3 ) (4ν (m ) ) m!(8) m, (m ) (7) Spherica Besse fuctio are so-caed Besse, Neuma ad Hake fuctios as the soutios of Eq.(). By repacig ν by + ad defiig f () π Z + (), the Eq.() ca be rewritte as + ( + ( + ) ) f () =. (8)

2 The spherica Besse j (), Neuma (), ad Hake h (±) () fuctios are give by π j () J + () = { P + () cos ( + ) π Q + () si ( + ) π } () h (±) () π N + () = { P + () si ( + ) π + Q + () cos ( + ) π } ( π < arg < π) π H(±) () = j + () ± i () = e±i (+) π P + () ± iq + () { ( π < arg < π) ( π < arg < π) (9) () () where P + () + ( C +,k () () Q + + () ( + C +,k () (3) C +,m () (( + ) )(( + ) 3 ) (( + ) (m ) ) m!(8) m, = ( + m)! m!() m ( m)!. ( m < + ) (4) Therefore we, fiay, obtai Spherica besse fuctio h (±) + e±i () = ( i) ( ±i ( + k)! k!( k)! (5) Note that the symmetric properties of h (±) () ca be obtaied from Eq.(5) as ( ) = ( ) + e i (+i) ( + e i ) = (+i) ( i k ( + k)! k!( ( k)! = ( ) h ( ) () (6) ( i ( + k)! k!( ( k)! = h( ) () (7) Aso Eqs.(9) ad () ca be rewritte as j () = () = i () + h ( ) () () h ( ) () (8) (9) Usig Eq.(6), the symmetric properties of j ad ca be obtaied as j ( ) = ( ) = i ( ) + h ( ) ( ) ( ) h ( ) ( ) = ( ) h ( ) () + () = ( ) j () () = ( ) h ( ) () () = ( ) + () () i

3 3 Asymtotic behaviour & Wroskia From Eq.(5), we ca get the asymtotic behabiour of the spherica Hake fuctio at the imit of as im h(±) e±i () = im ( i)+ (±i ( + k)! e±i k!( ( i)+ ( k)! π e±i = ( i). () Usig Eq.(), the asymtotic behaviour of j ad ca be aso obtaied as j () si π, (3) () cos π. (4) Now et us itroduce oe of the very importat quatity, the so-caed Wroskia. The Wroskia is defied by usig two kids of the ieary idepedet fuctios as W (f (), f () ) f () () () f () f () () () f () (5) Usig the fact f () obey + ( ) ( + ) f () =, (6) the derivative of the Wroskia is give as W (f (), f () ) = f () = f () () () f () f () () () f () ( ) ( + ) () f () () ( ) ( + ) f () () =. (7) +f () () The, we ca fid that the Wroskia is a costat for. Therefore, the Wroskia ca be cacuated by usig the asymtotic property of f () at the imit of as W (f (), f () ) = im f () () () f () f () () () f () = cost. (8) Wroskias for Besse fuctios Thus we ca obtai the foowig resuts. W (j, ) = im j () () () j () =, (9) W (j, h (±) ) = im W (, h (±) ) = im W (, h ( ) ) = im j () h(±) () h(±) () h( ) () h (±) () h (±) () h ( ) () j () = ±i (3) () () = (3) () h(+) () = i (3) 3

4 4 Recurrece formua Usig Eq.(5), we ca obtai h (±) + e±i () = ( i) + h (±) () = ( i) + e ±i h (±) () = ( i) + e ±i + h (±) h (±) () () = ( i) e ±i ( + k)! k!( k)! k ( + k)! k!( k)! (+k+) ( + k)! k!( k)! k ( + k)! k!( k)! ( ±i +( i) + e ±i k ( k)( + k)! k!( k)! = ( i) e ±i +( i) + e ±i + k ( + k + )! (k + )!( k )! + ( i) e ±i ( ±i k ( k)( + k)! k!( k)! ( + ±i = ( i) e ±i k ( + k)! ( k )! (k + )! + ( i) e ±i ( ±i = ( i) e ±i k ( + k )! + ( i) e ±i ( k )! k! ( ±i = ( i) e ±i k ( + k )! = + h (±) () (36) ( k )! k! ( ±i = ( i) e ±i (+k+) ( + k)! k!( k)! ( ±i ( i) + e ±i (+k+) ( + k + )! k!( k)! = ( i) e ±i + (+k+) ( + k + )! (k + )!( k )! + ( i) e ±i (+) ( ( ) ±i ( i) + e ±i (+k+) ( + k + )! ±i ( i) + e ±i (+) ( + )! k!( k)!! = ( i) e ±i + (+k+) ( + k + )! (k + )!( k)! +( i) e ±i (+) ( i) + e ±i ( ±i = ( i) e ±i (+k+) ( + k + )! k!( k + )! +( i) e ±i (+) + ( i) e ±i + = ( i) e ±i ) (+) ( + )!! ) + (+) ( + )! ( + )! (33) (34) (35) ( ±i (+k+) ( + k + )! k!( k + )! = h (±) + () (37) 4

5 Recurrece formuas We ca derive the foowig recurrece formuas usig Eqs.(36) ad (37) as + h (±) () = h (±) () + h(±) + () (38) ( + ) h(±) () = h (±) () ( + )h(±) + () (39) 5 Wave fuctio of free partice The Schrodiger equatio for the free partice ca be expressed as m χ(k, k; r) = E(k)χ(k, k; r), (4) i χ(k, k; r) = kχ(k, k; r), (4) where k is a uit vector which gives the directio of the mometum vector, i.e., k = k k. Aso E(k) = k m with compex k. χ(k, k; r) is kow to be χ(k, k; r) = e +ik r = i j (π) 3 (π) 3 (kr)y ( r)y ( k). (4) The compex cojugate of this fuctio is χ (k, k; r) = e ik r (π) 3 ( i) j (k r)y ( r)y ( k) = χ( k, k; r) = (π) 3 (π) 3 i j ( kr)y ( r)y ( k) χ(k, k; r) = (π) 3 i j ( kr)y ( r)y ( k) = (Usig j (k r) = j (kr), j ( kr) = ( ) j (kr) ad Y ( k) = ( ) Y ( k)) ( i) j (π) 3 (kr)y( r)y ( k) (43) (π) 3 (π) 3 (Note that k = k if k is rea. I this study, we cosider the compex k i most of the cases.) From the defiitio of the Deta fuctio, we get δ(r r ) Y ( r) π δ(k k ) dke +ik (r r ) = dre ) r +i(k k = Y ( k) π dkk j (kr)j (kr ) Y( r ) (44) drr j (kr)j (k r) Y( k ). (45) the orthogoa & competeess reatio for the spherica Besse fuctio Thus we obtai the orthogoa & competeess reatio for the spherica Besse fuctio as δ(r r ) = π δ(k k ) = π dk(kr) j (kr)j (kr ) (46) dr(kr) j (kr)j (k r) (47) 5

6 the orthogoa ad competeess reatio for χ I terms of the free partice wave fuctio χ, the orthogoa ad competeess reatio is give by drχ (k, k; r)χ(k, k ; r) = δ(k k ) (48) dkχ(k, k; r)χ (k, k; r ) = δ(r r ) (49) 6 Free partice Gree s fuctio The free partice Gree s fuctio is defied by E(k) + m G F (rr ; k) = δ(r r ). (5) It is very easy to prove G F (rr ; k) ca be represeted as G F (rr ; k) = m dq χ(q, q; r)χ (q, q; r ) q k. (5) I the expressio of the partia wave expasio, this ca be rewritte as G F (rr ; k) = m π = m π = m Y ( r)y ( r ) Y ( r)y ( r ) Y ( r)y ( r ) dq q j (qr)j (qr ) q k dq dq q k + q + k q k + q + k qj (qr)j (qr ) qj (qr < ) + h ( ). (5) If we suppose k = k e +iθ with < θ < π, the Eq.(5) ca be cacuated as Eq.(5) = m Y ( r)y ( r C + q k qj (qr < ) m Y ( r)y ( r C q + k qj (qr < )h ( ) = m i Y ( r)y ( r )kj (kr < ) (kr > ) m i Y ( r)y ( r )kj ( kr < )h ( ) ( kr > ) = m ik Y ( r)y ( r )j (kr < ) (kr > ) = m e +ik r r r r (53) where C + (C ) is the coutour itegra path o the upper (ower) regio of the compex mometum q pae (see Fig.). We cosider that (qr)(h ( ) (qr)) is coverged at the imit of q o the upper(ower) regio of the compex q-pae. 6

7 Figure : Cotour path C ± o the compex mometum q-pae. Therefore G F (rr ; k ) = m Y ( r)y ( r C + q + k qj (qr < ) m Y ( r)y ( r C q k qj (qr < )h ( ) i Y ( r)y ( r )k j ( k r < ) ( k r > ) + m i Y ( r)y ( r )k j (k r < )h ( ) (k r > ) = m = m ik Y ( r)y ( r )j (k r < )h ( ) (k r > ) = m e ik r r r r (54) 7

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