The Bessel Functions - Brown University

1 April 9, 201318:41 World Scientific Book - 9in x 6inMAINARDI BOOK-FINALA ppendix BThe Bessel FunctionsAs Rainville pointed out in his classic booklet[Rainville (1960)],no other special Functions have received such detailed treatment inreadily available treatises as the Bessel Functions . Consequently, wehere present only a brief introduction to the subject including therelated Laplace transform pairs used in this The standard Bessel functionsThe Bessel Functions of the first and second kind:J ,Y .The Bessel Functions of the first kindJ (z) are defined from theirpower series representation:J (z) := k=0( 1)k (k+ 1) (k+ + 1)(z2)2k+ ,( )wherezis a complex variable and is a parameter which can takearbitrary real or complex values. When is integer it turns out asan entire function; in this caseJ n(z) = ( 1)nJn(z), n= 1,2,..( )In factJn(z) = k=0( 1)kk!(k+n)!(z2)2k+n,J n(z) = k=n( 1)kk!

2 (k n)!(z2)2k n= s=0( 1)n+s(n+s)!s!(z2)2s+ 9, 201318:41 World Scientific Book - 9in x 6inMAINARDI BOOK-FINAL174 Fractional Calculus and Waves in Linear ViscoelasticyWhen is not integer the Bessel Functions exhibit a branch pointatz= 0 because of the factor (z/2) , sozis intended with|arg(z)|< that is in the complex plane cut along the negative real a suggestion by Tricomi, see[Gatteschi (1973)], we canextract from the series in ( ) that singular factor and set:JT (z) := (z/2) J (z) = k=0( 1)kk! (k+ + 1)(z2)2k.( )The entire functionJT (z) was referred to by Tricomi as theuniformBessel function. In some textbooks on special Functions , see [Kiryakova (1994)], p. 336, the related entire functionJC (z) :=z /2J (2z1/2) = k=0( 1)kzkk! (k+ + 1)( )is introduced and named theBessel-Clifford for fixedzin the cut plane the terms of the series ( )are analytic function of the variable , the fact that the series isuniformly convergent implies that the Bessel function of the firstkindJ (z) is an entire function of order.

3 The Bessel Functions are usually introduced in the framework ofthe Fucks Frobenius theory of the second order differential equationsof the formd2dz2u(z) +p(z)ddzu(z) +q(z)u(z) = 0,( )wherep(z) andq(z) are assigned analytic Functions . If we chose in( )p(z) =1z, q(z) = 1 2z2,( )and solve by power series, we would just obtain the series in ( ).As a consequence, we say that the Bessel function of the first kindsatisfies the equationu (z) +1zu (z) +(1 2z2)u(z) = 0,( )where, for shortness we have used the apices to denote differentiationwith respect toz. It is customary to refer to Eq. ( ) as theBesseldifferential 9, 201318:41 World Scientific Book - 9in x 6inMAINARDI BOOK-FINALA ppendix B: The Bessel Functions175 When is not integer the general integral of the Bessel equationisu(z) = 1J (z) + 2J (z), 1, 2 C,( )sinceJ (z) andJ (z) are in this case linearly independent withWronskianW{J (z),J (z)}= 2 zsin( ).

4 ( )We have used the notationW{f(z),g(z)}:=f(z)g (z) f (z)g(z).In order to get a solution of Eq. ( ) that is linearly independentfromJ also when =n(n= 0, 1, ) we introduce the Besselfunction of the second kindY (z) :=J (z) cos( ) J (z)sin( ).( )For integer the of ( ) becomes indeterminate so in thiscase we defineYn(z) as the limitYn(z) := lim nY (z) =1 [ J (z) =n ( 1)n J (z) =n].( )We also note that ( ) impliesY n(z) = ( 1)nYn(z).( )Then, when is an arbitrary real number, the general integral of Eq.( ) isu(z) = 1J (z) + 2Y (z), 1, 2 C,( )and the corresponding Wronskian turns out to beW{J (z),Y (z)}=2 z.( )The Bessel Functions of the third kind:H(1) ,H(2) .In ad-dition to the Bessel Functions of the first and second kind it is cus-tomary to consider the Bessel function of the third kind, or Hankelfunctions, defined asH(1) (z) :=J (z) +iY (z), H(2) (z) :=J (z) iY (z).( )These Functions turn to be linearly independent with WronskianW{H(1) (z),H(2) (z)}= 4i z.

5 ( )April 9, 201318:41 World Scientific Book - 9in x 6inMAINARDI BOOK-FINAL176 Fractional Calculus and Waves in Linear ViscoelasticyUsing ( ) to eliminateYn(z) from ( ), we obtain H(1) (z) :=J (z) e i J (z)isin( ),H(2) (z) :=e+i J (z) J (z)isin( ),( )which imply the important formulasH(1) (z) = e+i H(1) (z), H(2) (z) = e i H(2) (z).( )The recurrence relations for the Bessel func-tionsJ (z),Y (z),H(1) (z),H(2) (z) satisfy simplerecurrence rela-tions. Denoting any one of them byC (z) we have: C (z) =z2 [C 1(z) +C +1(z)],C (z) =12[C 1(z) C +1(z)].( )In particular we noteJ 0(z) = J1(z), Y 0(z) = Y1(z).We note thatC stands forcylinder function, as it is usual to call thedifferent kinds of Bessel Functions . The origin of the termcylinder isdue to the fact that these Functions are encountered in studying theboundary value problems of potential theory for cylindrical more general differential equation for the Bessel differential equation ( ) can be generalized by intro-ducing three additional complex parameters ,p,qin such a wayz2w (z) + (1 2p)zw (z) +( 2q2z2q+p2 2q2)w(z) = 0.

6 ( )A particular integral of this equation is provided byw(z) =zpC ( zq).( )We see that for = 1,p= 0,q= 1 we recover Eq. ( ).April 9, 201318:41 World Scientific Book - 9in x 6inMAINARDI BOOK-FINALA ppendix B: The Bessel Functions177 The asymptotic representations for the Bessel asymptotic representations of the standard Bessel Functions forz 0 andz are provided by the first term of the convergentseries expansion aroundz= 0 and by the first term of the asymptoticseries expansion forz , 0 (with|arg(z)|< if is not integer) we have: J n(z) ( 1)n(z/2)nn!, n= 0,1,..,J (z) (z/2) ( + 1), 6= 1, ( ) Y0(z) iH(1)0(z) iH(2)0(z) 2 log (z),Y (z) iH(1) (z) iH(2) (z) 1 ( )(z/2) , >0.( )Forz with|arg(z)|< and for any we have: J (z) 2 zcos(z 2 4),Y (z) 2 zsin(z 2 4),H(1) (z) 2 ze+i(z 2 4),H(2) (z) 2 ze i(z 2 4).( )The generating function of the Bessel Functions of Bessel Functions of the first kindJn(z) are simply re-lated to the coefficients of the Laurent expansion of the functionw(z,t) = ez(t 1/t)/2=+ n= cn(z)tn,0<|t|<.

7 ( )To this aim we multiply the power series of ezt/2, e z/(2t), and, aftersome manipulation, we getw(z,t) = ez(t 1/t)/2=+ n= Jn(z)tn,0<|t|< .( )The functionw(z,t) is called thegenerating functionof the Besselfunctions of integer order, and formula ( ) plays an importantrole in the theory of these 9, 201318:41 World Scientific Book - 9in x 6inMAINARDI BOOK-FINAL178 Fractional Calculus and Waves in Linear ViscoelasticyPlots of the Bessel Functions of integer of theBessel functionsJ (x) andY (x) for integer orders = 0,1,2,3,4are shown in Fig. and in Fig. , Plots ofJ (x) with = 0,1,2,3,4 for 0 x Plots ofY (x) with = 0,1,2,3,4 for 0 x Bessel Functions of semi-integer now con-sider the special cases when the order is a a semi-integer number =n+ 1/2 (n= 0, 1, 2, 3,..). In these cases the standardBessel function can be expressed in terms of elementary 9, 201318:41 World Scientific Book - 9in x 6inMAINARDI BOOK-FINALA ppendix B: The Bessel Functions179In particular we haveJ+1/2(z) =(2 z)1/2sinz , J 1/2(z) =(2 z)1/2cosz.

8 ( )The fact that any Bessel function of the first kind of half-integerorder can be expressed in terms of elementary Functions now followsfrom the first recurrence relation in ( ), 1+J +1=2 zJ (z),whose repeated applications gives J+3/2(z) =(2 z)1/2[sinzz cosz],J 3/2(z) = (2 z)1/2[sinz coszz],( )and so derive the corresponding formulas for Bessel Functions of thesecond and third kind we start from the expressions ( ) and ( )of these Functions in terms of the Bessel Functions of the first kind,and use ( ). For example, we have:Y1/2(z) = J 1/2(z) = (2 z)1/2cosz ,( )H(1)1/2(z) = i(2 z)1/2e+iz, H(2)1/2(z) = +i(2 z)1/2e iz.( )It has been shown by Liouville that the case of half-integer orderis the only case where the cylinder Functions reduce to is worth noting that when = 1/2 the asymptotic repre-sentations ( ) forz for all types of Bessel Functions re-duce to the exact expressions of the corresponding Functions providedabove.

9 This could be verified by using the saddle-point method forthe complex integral representation of the Bessel Functions , that wewill present in Subsection 9, 201318:41 World Scientific Book - 9in x 6inMAINARDI BOOK-FINAL180 Fractional Calculus and Waves in Linear The modified Bessel functionsThe modified Bessel Functions of the first and second kind:I , K .The modified Bessel Functions of the first kindJ (z) with IR andz Care defined by the power seriesI (z) := k=01 (k+ 1) (k+ + 1)(z2)2k+ .( )We also define the modified Bessel Functions of the second kindK (z):K (z) := 2I (z) I (z)sin( ).( )For integer the of ( ) becomes indeterminate so in thiscase we defineYn(z) as the limitKn(z) := lim nK (z).( )Repeating the consideration of Section , we find thatI (z)andK (z) are analytic Functions ofzin the cut plane and entirefunction of the order . We recall thatK (z) is sometimes referredto asMacdonald s function.

10 We note from the definitions ( ) and( ) the useful formulasI n(z) =In(z), n= 0, 1, 2,..( )K (z) =K (z), .( )The modified Bessel functionsI (z) andK (z) are simply relatedto the standard Bessel function of argumentzexp( i /2). If <arg(z)< /2, , /2<arg(zei /2)< /2,then ( ) impliesI (z) = e i /2J (zei /2).( )Similarly, according to ( ), for the same value ofzwe haveK (z) =i 2ei /2H1 (zei /2).( )On the other hand, if /2<arg(z)< , , <arg(ze i /2)< /2,then it is easily verified thatI (z) = e+i /2J (ze i /2),( )andK (z) = i 2e i /2H2 (ze i /2).( )April 9, 201318:41 World Scientific Book - 9in x 6inMAINARDI BOOK-FINALA ppendix B: The Bessel Functions181 The differential equation for the modified Bessel is an immediate consequence of their definitions thatI (z) andK (z) are linearly independent solutions of the differential equationv (z) +1zv (z) (1 + 2z2)v(z) = 0,( )which differs from the standard Bessel equation ( ) only by thesign of one term, and reduces to Eq.