Mathematical Methods for Physicists: A concise introduction - Site Map

BESSEL'S EQUATION
b ˆ ln 2, where ˆ 0:577 215 664 90 ... is the so-called Euler constant, which
is de®ned as the limit of
1 ‡ 1 2 ‡‡1 s ln s
as s approaches in®nity. The standard particular solution thus obtained is known
as the Bessel function of the second kind of order zero or Neumann's function of
order zero and is denoted by Y 0 …x†:
Y 0 …x† ˆ2
J
0 …x† ln x ‡
2 ‡
X1
mˆ1
…1† m1
h m
2 2m …m!† 2 x2m : …7:84†
If ˆ 1; 2; ...; a second solution can be obtained by similar manipulations,
starting from Eq. (7.35). It turns out that in this case also the solution contains a
logarithmic term. So the second solution is unbounded near the origin and is
useful in applications only for x 6ˆ 0.
Note that the second solution is de®ned di€erently, depending on whether the
order is integral or not. To provide uniformity of formalism and numerical
tabulation, it is desirable to adopt a form of the second solution that is valid for
all values of the order. The common choice for the standard second solution
de®ned for all is given by the formula
cos J
Y …x† ˆJ…x† …x†
; Y
sin
n …x† ˆlim Y …x†:
!n
…7:85†
This function is known as the Bessel function of the second kind of order . Itis
also known as Neumann's function of order and is denoted by N …x† (Carl
Neumann 1832±1925, German mathematician and physicist). In G. N. Watson's
A Treatise on the Theory of Bessel Functions (2nd ed. Cambridge University Press,
Cambridge, 1944), it was called Weber's function and the notation Y …x† was
used. It can be shown that
Y n …x† ˆ…1† n Y n …x†:
We plot the ®rst three Y n …x† in Fig. 7.4.
A general solution of Bessel's equation for all values of can now be written:
y…x† ˆc 1 J …x†‡c 2 Y …x†:
In some applications it is convenient to use solutions of Bessel's equation that
are complex for all values of x, so the following solutions were introduced
9
H
…1† …x† ˆJ …x†‡iY …x†; =
…7:86†
…x† ˆJ …x†iY …x†:
;
H …2†
327