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April 9, 2013 18:41 World Scientic Book - 9in x 6in MAINARDI_BOOK-FINAL

Appendix B

The Bessel Functions

As Rainville pointed out in his classic booklet

[Rainville (1960)], no other special functions have received such detailed treatment in readily available treatises as the Bessel functions. Consequently, we here present only a brief introduction to the subject including the related Laplace transform pairs used in this book.

B.1 The standard Bessel functions

The Bessel functions of the rst and second kind:J;Y.

The Bessel functions of the rst kindJ

(z) are dened from their power series representation: J (z) :=1X k=0(1)k(k+ 1)(k++ 1) z2

2k+;(B:1)

wherezis a complex variable andis a parameter which can take arbitrary real or complex values. Whenis integer it turns out as an entire function; in this case

Jn(z) = (1)nJn(z); n= 1;2;:::(B:2)

In fact

J n(z) =1X k=0(1)kk!(k+n)! z2 2k+n; J n(z )=1X k=n(1)kk!(kn)! z2 2kn=1 X s=0(1)n+s(n+s)!s! z2 2s+n: 173
April 9, 2013 18:41 World Scientic Book - 9in x 6in MAINARDI_BOOK-FINAL

174Fractional Calculus and Waves in Linear Viscoelasticy

Whenis not integer the Bessel functions exhibit a branch point atz= 0 because of the factor (z=2), sozis intended withjarg(z)j< that is in the complex plane cut along the negative real semi-axis.

Following a suggestion by Tricomi, see

[Gatteschi (1973)], we can extract from the series in (B.1) that singular factor and set: J

T(z) := (z=2)J(z) =1X

k=0(1)kk!(k++ 1) z2

2k:(B:3)

The entire functionJT(z) was referred to by Tricomi as theuniform Bessel function. In some textbooks on special functions, see e.g. Kiryakova (1994)], p. 336, the related entire function J

C(z) :=z=2J(2z1=2) =1X

k=0(1)kzkk!(k++ 1)(B:4) is introduced and named theBessel-Cliord function. Since for xedzin the cut plane the terms of the series (B.1) are analytic function of the variable, the fact that the series is uniformly convergent implies that the Bessel function of the rst kindJ(z) is an entire function of order. The Bessel functions are usually introduced in the framework of the Fucks{Frobenius theory of the second order dierential equations of the form d 2dz

2u(z) +p(z)ddz

u(z) +q(z)u(z) = 0;(B:5) wherep(z) andq(z) are assigned analytic functions. If we chose in (B.5) p(z) =1z ; q(z) = 12z

2;(B:6)

and solve by power series, we would just obtain the series in (B.1). As a consequence, we say that the Bessel function of the rst kind satises the equation u

00(z) +1z

u0(z) + 12z 2 u(z) = 0;(B:7) where, for shortness we have used the apices to denote dierentiation with respect toz. It is customary to refer to Eq. (B.7) as theBessel dierential equation. April 9, 2013 18:41 World Scientic Book - 9in x 6in MAINARDI_BOOK-FINAL

Appendix B: The Bessel Functions175

Whenis not integer the general integral of the Bessel equation is u(z) =

1J(z) +

2J(z);

1;

22C;(B:8)

sinceJ(z) andJ(z) are in this case linearly independent with

Wronskian

WfJ(z);J(z)g=2z

sin():(B:9) We have used the notationWff(z);g(z)g:=f(z)g0(z)f0(z)g(z). In order to get a solution of Eq. (B.7) that is linearly independent fromJalso when=n(n= 0;1;2:::) we introduce the Bessel function of the second kind Y (z) :=J(z) cos()J(z)sin():(B:10) For integerthe R.H.S of (B.10) becomes indeterminate so in this case we deneYn(z) as the limit Y n(z):= lim!nY(z)=1 @J(z)@ =n(1)n@J(z)@ =n :(B:11)

We also note that (B.11) implies

Y n(z) = (1)nYn(z):(B:12) Then, whenis an arbitrary real number, the general integral of Eq. (B.7) is u(z) =

1J(z) +

2Y(z);

1;

22C;(B:13)

and the corresponding Wronskian turns out to be

WfJ(z);Y(z)g=2z

:(B:14) The Bessel functions of the third kind:H(1);H(2).In ad- dition to the Bessel functions of the rst and second kind it is cus- tomary to consider the Bessel function of the third kind, or Hankel functions, dened as H (1)(z) :=J(z) +iY(z); H(2)(z) :=J(z)iY(z):(B:15) These functions turn to be linearly independent with Wronskian

WfH(1)(z);H(2)(z)g=4iz

:(B:16) April 9, 2013 18:41 World Scientic Book - 9in x 6in MAINARDI_BOOK-FINAL

176Fractional Calculus and Waves in Linear Viscoelasticy

Using (B.10) to eliminateYn(z) from (B.15), we obtain 8>>< >:H (1)(z) :=J(z)eiJ(z)isin(); H (2)(z) :=e+iJ(z)J(z)isin();(B:17) which imply the important formulas H (1) (z) = e+iH(1)(z); H(2) (z) = eiH(2)(z):(B:18) The recurrence relations for the Bessel functions.The func- tionsJ(z),Y(z),H(1)(z),H(2)(z) satisfy simplerecurrence rela- tions. Denoting any one of them byC(z) we have: 8>>>< >>:C (z) =z2[C1(z) +C+1(z)]; C

0(z) =12

[C1(z) C+1(z)]:(B:19)

In particular we note

J

00(z) =J1(z); Y00(z) =Y1(z):

We note thatCstands forcylinder function, as it is usual to call the dierent kinds of Bessel functions. The origin of the termcylinder is due to the fact that these functions are encountered in studying the boundary{value problems of potential theory for cylindrical coordi- nates. A more general dierential equation for the Bessel func- tions.The dierential equation (B.7) can be generalized by intro- ducing three additional complex parameters,p,qin such a way z

2w00(z)+(12p)zw0(z)+2q2z2q+p22q2w(z) = 0:(B:20)

A particular integral of this equation is provided by w(z) =zpC(zq):(B:21) We see that for= 1,p= 0,q= 1 we recover Eq. (B.7). April 9, 2013 18:41 World Scientic Book - 9in x 6in MAINARDI_BOOK-FINAL

Appendix B: The Bessel Functions177

The asymptotic representations for the Bessel functions. The asymptotic representations of the standard Bessel functions for z!0 andz! 1are provided by the rst term of the convergent series expansion aroundz= 0 and by the rst term of the asymptotic series expansion forz! 1, respectively. Forz!0 (withjarg(z)j< ifis not integer) we have:8>< :J n(z)(1)n(z=2)nn!; n= 0;1;:::; J (z)(z=2)(+ 1); 6=1;2::::(B:22) 8 >>:Y

0(z) iH(1)

0(z)iH(2)

0(z)2 log(z); Y (z)iH(1)(z)iH(2)(z)1 ()(z=2); >0:(B:23) Forz! 1withjarg(z)j< and for anywe have:8>>>>>>>>>>>< >>>>>>>>>>:J (z)r2 z cos z2 4 Y (z)r2 z sin z2 4 H (1)(z)r2 z e+i z2 4 H (2)(z)r2 z ei z2 4 :(B:24) The generating function of the Bessel functions of integer order.The Bessel functions of the rst kindJn(z) are simply re- lated to the coecients of the Laurent expansion of the function w(z;t) = ez(t1=t)=2=+1X n=1c n(z)tn;0To this aim we multiply the power series of e zt=2, ez=(2t), and, after some manipulation, we get w(z;t) = ez(t1=t)=2=+1X n=1J n(z)tn;0178Fractional Calculus and Waves in Linear Viscoelasticy Plots of the Bessel functions of integer order.Plots of the Bessel functionsJ(x) andY(x) for integer orders= 0;1;2;3;4

are shown in Fig. B.1 and in Fig. B.2, respectively.Fig. B.1 Plots ofJ(x) with= 0;1;2;3;4 for 0x10.Fig. B.2 Plots ofY(x) with= 0;1;2;3;4 for 0x10.

The Bessel functions of semi-integer order.We 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 standard Bessel function can be expressed in terms of elementary functions. April 9, 2013 18:41 World Scientic Book - 9in x 6in MAINARDI_BOOK-FINAL

Appendix B: The Bessel Functions179

In particular we have

J +1=2(z) =2z 1=2 sinz ; J1=2(z) =2z 1=2 cosz :(B:27) The fact that any Bessel function of the rst kind of half-integer order can be expressed in terms of elementary functions now follows from the rst recurrence relation in (B.19), i.e. J

1+J+1=2z

J(z); whose repeated applications gives 8>>>< >>:J +3=2(z) =2z

1=2sinzz

cosz J

3=2(z) =2z

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