Page 313 - Schaum's Outline of Differential Equations

296 GAMMA AND BESSEL FUNCTIONS [CHAP. 30

The function J p(x) is a solution near the regular singular point x = 0 of Bessel's differential equation of order p:

In fact, (x) is that solution of Eq. (30.6) guaranteed by Theorem 28.1.
J p

ALGEBRAIC OPERATIONS ON INFINITE SERIES
Changing the dummy index. The dummy index in an infinite series can be changed at will without altering
the series. For example,

Change of variables. Consider the infinite series If we make the change of variables j = k + 1,
or k=j— 1, then

Note that a change of variables generally changes the limits on the summation. For instance, if j = k + 1, it follows
that 7' = 1 when k = 0,j = ^o when k=^o, and, as k runs from 0 to °°, j runs from 1 to °°.
The two operations given above are often used in concert. For example,

Here, the second series results from the change of variables j = k+2 in the first series, while the third
series is the result of simply changing the dummy index in the second series from7' to k. Note that all three series
equal

Solved Problems

30.1. Determine F(3.5).
It follows from Table 30-1 that T(1.5) = 0.8862, rounded to four decimal places. Using Eq. (30.2) withp = 2.5,
we obtain T(3.5) = (2.5)F(2.5). But also from Eq. (30.2), with p=1.5, we have T(2.5) = (1.5)r(1.5). Thus,
T(3.5) = (2.5)(1.5) T(1.5) = (3.75)(0.8862) = 3.3233.

30.2. Determine F(-0.5).
It follows from Table 30-1 that T(1.5) = 0.8862, rounded to four decimal places. Using Eq. (30.4) withp = 0.5,
we obtain T(0.5) = 2F(1.5). But also from Eq. (30.4), with p = -0.5, we have T(-0.5) = -2F(0.5). Thus, T(-0.5)
= (-2)(2) T(1.5) = -4(0.8862) = -3.5448.