18.5 BESSEL FUNCTIONS
and hence thatJν(x)andJ−ν(x) are linearly dependent. So, in this case, we cannot
write the general solution to Bessel’s equation in the form (18.80). One therefore
defines the function
Yν(x)=Jν(x)cosνπ−J−ν(x)
sinνπ, (18.81)which is called a Bessel function of thesecond kindof orderν(or, occasionally,
aWeberorNeumannfunction). As Bessel’s equation is linear,Yν(x) is clearly a
solution, since it is just the weighted sum of Bessel functions of the first kind.
Furthermore, for non-integerνit is clear thatYν(x) is linearly independent of
Jν(x). It may also be shown that the Wronskian ofJν(x)andYν(x) is non-zero
forallvalues ofν. HenceJν(x)andYν(x) always constitute a pair of independent
solutions.
Ifnis an integer, show thatYn+1/ 2 (x)=(−1)n+1J−n− 1 / 2 (x).From (18.81), we have
Yn+1/ 2 (x)=Jn+1/ 2 (x)cos(n+^12 )π−J−n− 1 / 2 (x)
sin(n+^12 )π.
Ifnis an integer, cos(n+^12 )π= 0 and sin(n+^12 )π=(−1)n, and so we immediately obtain
Yn+1/ 2 (x)=(−1)n+1J−n− 1 / 2 (x), as required.
The expression (18.81) becomes an indeterminate form 0/0whenνis aninteger, however. This is so because for integerνwe have cosνπ=(−1)νand
J−ν(x)=(−1)νJν(x). Nevertheless, this indeterminate form can be evaluated using
l’Hopital’s rule (see chapter 4). Therefore, for integerˆ ν,weset
Yν(x) = lim
μ→ν[
Jμ(x)cosμπ−J−μ(x)
sinμπ]
, (18.82)which gives a linearly independent second solution for this case. Thus, we may
write the general solution of Bessel’s equation, valid forallν,as
y(x)=c 1 Jν(x)+c 2 Yν(x). (18.83)The functionsY 0 (x),Y 1 (x)andY 2 (x) are plotted in figure 18.6
Finally, we note that, in some applications, it is convenient to work withcomplex linear combinations of Bessel functions of the first and second kinds
given by
H(1)ν(x)=Jν(x)+iYν(x),Hν(2)(x)=Jν(x)−iYν(x);these are called, respectively,Hankel functionsof the first and second kind of
orderν.