1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

314 Chapter 5 Higher Dimensions and Other Coordinates

Figure 7 Graphs of Bessel functions of the first kind. Also see the CD. (a)J 0 and
J 1 ,(b)Y 0 andY 1.

n

m 12 3 4

0 2.405 5.520 8.654 11.792

1 3.832 7.016 10.173 13.324

2 5.136 8.417 11.620 14.796

3 6.380 9.761 13.015 16.223

Table 1 Zeros of Bessel functions. The valuesαmn

satisfy the equationJm(αmn)=0.

Also, asr→∞,bothJμ(λr)andYμ(λr)tend to zero. Figure 7 gives graphs
of several Bessel functions, and Table 1 provides values of their zeros. Further
information can be found in most books of tables.
The modified Bessel equation differs from the Bessel equation only in the
sign of one term. It is

(

rR′

)′

−μ

2

rR−λ

(^2) rR= 0. (7)