Page 285 - Schaum's Outline of Theory and Problems of Advanced Calculus

276 INFINITE SERIES [CHAP. 11

Then
ð 1 1 1 1 1
1:4618
3
P 4 ðxÞ dx ¼ 1 þ þ þ þ
0 5ð2!Þ 7ð3!Þ 9ð4!Þ

1 1 1
ð ð ð 10
e 10 x e
x dx e <:0021

R 4 ðxÞ dx dx ¼
0 5! 0 5! 11:5
0
Thus, the maximum error is less than .0021 and the value of the integral is accurate to two decimal places.
SPECIAL TOPICS
1. Functions defined by series are often useful in applications and frequently arise as solutions of
differential equations. For example, the function defined by
( 2 4 )
p
x x x
2 p! 2ð2p þ 2Þ 2 4ð2p þ 2Þð2p þ 4Þ
J p ðxÞ¼ p 1 þ
n pþ2n
1
X
ð 1Þ ðx=2Þ
n!ðn þ pÞ!
¼ ð16Þ
n¼0
2
2
is a solution of Bessel’s differential equation x y þ xy þðx p Þy ¼ 0 and is thus called a
2 00
0
Bessel function of order p. See Problems 11.46, 11.110 through 11.113.
Similarly, the hypergeometric function
a B aða þ 1Þbðb þ 1Þ 2
Fða; b; c; xÞ¼ 1 þ x þ x þ ð17Þ
1 c 1 2 cðc þ 1Þ
is a solution of Gauss’ differential equation xð1 xÞy þfc ða þ b þ 1Þxgy aby ¼ 0.
00
0
These functions have many important properties.
1
X
n
2. Infinite series of complex terms,in particular power series of the form a n z , where z ¼ x þ iy
n¼0
and a n may be complex, can be handled in a manner similar to real series.
2
2
2
Such power series converge for jzj < R, i.e., interior to a circle of convergence x þ y ¼ R ,
where R is the radius of convergence (if the series converges only for z ¼ 0, we say that the radius
of convergence R is zero; if it converges for all z,we say that the radius of convergence is
infinite). On the boundary of this circle, i.e., jzj¼ R, the series may or may not converge,
depending on the particular z.
Note that for y ¼ 0 the circle of convergence reduces to the interval of convergence for real
power series. Greater insight into the behavior of power series is obtained by use of the theory
of functions of a complex variable (see Chapter 16).
1
X
3. Infinite series of functions of two (or more) variables, such as u n ðx; yÞ can be treated in a
n¼1
manner analogous to series in one variable. In particular, we can discuss power series in x and y
having the form
2 2
a 00 þða 10 x þ a 01 yÞþ ða 20 x þ a 11 xy þ a 02 y Þþ ð18Þ
using double subscripts for the constants. As for one variable, we can expand suitable functions
of x and y in such power series. In particular, the Taylor theroem may be extended as follows.
TAYLOR’S THEOREM (FOR TWO VARIABLES)
Let f be a function of two variables x and y.If all partial derivatives of order n are continuous in a
closed region and if all the ðn þ 1Þ partial derivatives exist in the open region, then