334
CHAPTER 38
38.68
Find the interval of convergence of
Use the ratio test.
convergence for all x. The function defined by this series is denoted J { (x) and is called a Bessel function of the
first kind of order one.
Therefore, we have
38.69
Show that
38.68.
J 0 (x) = — J^x), where J 0 (x) and /,(*) are the Bessel functions defined in Problems 38.67 and
Differentiate:
38.70
Find an ordinary differential equation of second order satisfied by J 0 (t).
Let x = -t
2 /4 in the series of Problem 38.66; the result is, by Problem 38.67, /„(/). Thus, the above
change of variable must take the differential equation
into a differential equation for
y = J 0 (t).
Explicitly,
and
and the desired differential equation (BesseVs equation of order zero) is
or
38.71
Show that z = /,(<) satisfies Bessel's equation of order 1:
By Problem 38.69, we obtain a differential equation for z = /,(f) by letting
in (j|) of Problem
Differentiating once more with respect to t:
38.70:
or
37.72
Find the power series expansion of
by division.
Let
Then
Hence. a 0 = 1,
a t =0, and, for
that is «* = -«*-2- Thus, 0 = a, = a, = • • -. For even subscripts, a 2 -—l, a 4 = l, a 6 = -l,
and, in general, a 2n = (—!)". Therefore,
1— x + x —x +x +•••. (Of course, this is obtained more easily by using a geometric series.)
38.73
Show that if f(x) =
a n x" for \x\ < r and f(x) is an even function [that is, /(—•*)=/(*)], then all
odd-order coefficients fl 2 *+i
= ^Equating coefficients, we see that, when n is odd, «„ = -«„,
and, therefore, a n = 0.
38.74
Show that if /(*) =
a n x" for |*|
and, therefore,
Equating coefficients, we see that, when n is even, a n = - a n ,
CHAPTER 38
38.68
Find the interval of convergence of
Use the ratio test.
convergence for all x. The function defined by this series is denoted J { (x) and is called a Bessel function of the
first kind of order one.
Therefore, we have
38.69
Show that
38.68.
J 0 (x) = — J^x), where J 0 (x) and /,(*) are the Bessel functions defined in Problems 38.67 and
Differentiate:
38.70
Find an ordinary differential equation of second order satisfied by J 0 (t).
Let x = -t
2 /4 in the series of Problem 38.66; the result is, by Problem 38.67, /„(/). Thus, the above
change of variable must take the differential equation
into a differential equation for
y = J 0 (t).
Explicitly,
and
and the desired differential equation (BesseVs equation of order zero) is
or
38.71
Show that z = /,(<) satisfies Bessel's equation of order 1:
By Problem 38.69, we obtain a differential equation for z = /,(f) by letting
in (j|) of Problem
Differentiating once more with respect to t:
38.70:
or
37.72
Find the power series expansion of
by division.
Let
Then
Hence. a 0 = 1,
a t =0, and, for
that is «* = -«*-2- Thus, 0 = a, = a, = • • -. For even subscripts, a 2 -—l, a 4 = l, a 6 = -l,
and, in general, a 2n = (—!)". Therefore,
1— x + x —x +x +•••. (Of course, this is obtained more easily by using a geometric series.)
38.73
Show that if f(x) =
a n x" for \x\ < r and f(x) is an even function [that is, /(—•*)=/(*)], then all
odd-order coefficients fl 2 *+i
= ^Equating coefficients, we see that, when n is odd, «„ = -«„,
and, therefore, a n = 0.
38.74
Show that if /(*) =
a n x" for |*|
and, therefore,
Equating coefficients, we see that, when n is even, a n = - a n ,