贝塞尔函数详细介绍(全面)

y AJn (x) BYn (x)
A、B为任意常数， n为任意实数
三 贝塞尔函数的性质
J
n
(
x)
m0
(1) m!(n
m
m
1)
x 2
n2m
Yn
(x)
lim
n
J
(x)
cos sin
J
(x)
性质1 有界性
Jn (x)
性质2 奇偶性 当n为正整数时
Yn (0)
x 0 Yn (x)
(0) j
)
1 2
J 0 (i(0) x)
i 1
(0) i
J
1
(i(0)
)
d
dx
xnJn (x)
xn Jn1(x)
d
dx
xn J n (x)
x n J n1 (x)
J n1 (x) J n1 (x) 2J n (x) 2n
J n1 (x) J n12区间内展成
第五章 贝塞尔函数(bessel)
一 贝塞尔函数的引出
u(ut,a,02) 2u(a,2 ),2u2
1
u
1
2
2u
2
,
R,0 2 ,t 0 R,0 2
u(R, ,t) 0,
令： u(, ,t) V (, )T (t)
0 2 ,t 0
令： V (, ) ()( )
VT a22V T
J (n1) (x) 2
2
x
n
1 2
1
d
n cosx
x dx x
J
n
(x)
m0
(1) m m!(n m
1)
x 2
n2m
Yn
(x)
lim
n
J
(x)
cos sin
J
(x)
性质7 大宗量近似
Jn (x)
2 cos x 1 n x 4 2
Yn (x)
R 0
xJ1
(
x)
c
os
x
xJ1
(
x)
sin
xdx
RJ0 (R) cos R
R 0
xJ1
(
x)
s
in
x
dx
RJ0 (R) cos R RJ1(R) sin R
(7)
xn1Jn ( x)dx
t α
n1
J
n
(t
)d
t α
1
n2
t n1Jn (t)dt
1
n2
dt n1J n1(t)
xJ1(x) 2 dJ0 (x) xJ1(x) 2J0 (x) C
d
dx
xnJn (x)
xn Jn1(x)
d
dx
xn J n (x)
x n J n1 (x)
(5)
x3J0 (x)dx x3J1(x) 2
x2dxJ1(x) x3J1(x) 2 x2J1(x)dx dx2J2 (x) x3J1(x) 2x2 J2 (x) C
2 2 n2 0, R
(R) 0, (0)
x x/
d( ) d
d( x
dx
)
dx
d
dy ( x) dx
y
y
x2 y xy x2 n2 y 0, x R
y( R) 0, y(0)
n阶贝塞尔方程
二 贝塞尔方程的求解
n阶贝塞尔方程
J
n
(
x)
m0
(1)m m!(n m)!
x 2
n2m
n 0,1, 2,L
c n 时
J
n
(
x)
m0
(1)m m!(n m
1)
x 2
n2m
n 1, 2,L
J
n
(
x)
m0
(1)m m!(n m
1)
x 2
n2m
1 n不为整数时，贝塞尔方程的通解
n阶第一类贝塞尔函数
Jn (x) 和 Jn (x) 线性无关
1 2
J0 (0)
J2
(0)
1 2
n 1
J n
(0)
1 2
J n1 (0)
J n1 (0)
0
性质5 零点
有无穷多个对称分布的零点
Jn (m(n) ) 0
J n (x)和 J n1 (x) 的零点相间分布 J n (x) 的零点趋于周期分布，
lim
m
(n) m1
(n) m
J
n
(
x)
m0
J0
(x)
1 2
J
2
(x)
J2
(
x)
(3) 3J0(x) 4J0(x) 3J1(x) 4J1(x) 3J1(x) 2J0 (x) 2J2 (x)
3J1(x) 2J1(x) J1(x) J3(x) J3 (x)
(4) xJ2(x)dx x2x1J2 (x)dx x2dx1J1(x) xJ1(x) x1J1(x)dx2 xJ1(x) 2 J1(x)dx
2
x
sin
x
1
4
n
2
x , Jn (x) 0,Yn (x) 0
性质8 正交性
R
0 rJn
(n) m R
r
J
n
(n) k R
r dr
R2
2
J
2 n1
(m(n)
)
0, R2
2
J
2 n1
(m(n)
),
mk mk
xJn (x) nJn (x) xJn1(x)
xJn (x) nJn (x) xJn1(x)
(1)m m!(n m
1)
x 2
n2m
性质6 半奇数阶的贝塞尔函数
J
1 2
(
x)
m0
(1)m m!(3
m)
x 2
12m 2
2
(1)m
m0 m! 1 1 1 2 1
2 2 2
x
1 2m
2
m 1 (1 ) 2
2 2
m0
m!1
3
(1)m 2m1
5 2m 1
(
1
J n1 (x) J n1 (x) 2J n (x)
J n1 (x)
J n1 (x)
2n x
Jn
(x)
(6)
R 0
J0 (x) cos
xdx
xJ0 (x) cos
x
|0R
R 0
xdJ0 (x) cos
x
RJ0 (R) cos R
R 0
xJ
0
(
x)
cos
x
J
0
(
x)
s
in
xdx
RJ0(R) cosR
dx
xnYn (x)
x
Y n n1
(
x)
Yn1 ( x)
Yn1 ( x)
2n x
Yn
(x)
Yn1(x) Yn1(x) 2Yn(x)
例1 求下列微积分
(1)
d dx
J0
(
x)
J 0
(x)
J1(x)
(2)
J0(x)
1 x
J0(x)
J1(x)
1 x
J1(x)
1 2
J
0
(x)
1 2
J2
(x)
1 2
y AJn (x) BJn (x)
A cot n B csc n
Yn (x)
Jn (x) cos n sin n
Jn (x)
n阶第二类贝塞尔函数（牛曼函数）
y AJ n (x) BYn (x)
2 n为整数时，贝塞尔方程的通解
n为整数时
1
0
(n m 1)
m 0,1, 2L (N 1)
(1)m 2n 2m x2n2m1
m0 2n2m m!(n m 1)
x
n
m0
(1) 2 n 2 m 1
x m 2n2m1 m!(n m)
xn J n1(x)
d dx
xn
Jn
( x)
xn
J n1 ( x)
d dx
xn
Jn
(x)
xn
J
n1
(
x)
d dx
xJ1 ( x)
xJ0 (x)
J n (x) (1)n J n (x)
Yn (x) (1)n Yn (x)
J
n
(x)
m0
(1) m m!(n m
1)
x 2
n2m
Yn
(x)
lim
n
J
(x)
cos sin
J
(x)
性质3 递推性
d
dx
x
n
J
n
(
x)
d
(1)m x2n2m
dx m0 2n2m m!(n m 1)
n任意实数或复数
x2 y xy x2 n2 y 0
假设 n 0
令： y xc (a0 a1x a2 x 2 ak x k ) ak xck k 0 (c k)(c k 1) (c k) (x2 n2 ) ak xck 0 k 0
(c 2 n2 )a0 xc (c 1)2 n2 a1xc1 (c k )2 n 2 ) ak ak2 xck 0
2n x
Jn
(x)
Jn1(x) Jn1(x) 2Jn (x)
d dx
xn Jn (x)
xn Jn1(x)
d dx
x
n
J
n
(
x)
x
n
J
n
1
(
x)
J n1 ( x)
J n1 ( x)
2n x
Jn
(x)
Jn1(x) Jn1(x) 2Jn (x)
d
dx
xnYn (x)
xnYn1(x)
d
2V T
V a2T
T ' a2T 0 T (t) Aea2t
2V V 0
( 0)
1 1 0
2
2 2
0
2 2 0
0
2 2 0
n2
n 0,1,2,3,
n An cos n Bn sin n
Ci
i 1
1 0
xJ
0
(
(0 j
)
x)
J
0
(
(0) i
x)dx
1
(0) 2 j
0
( j
0
)
tJ
0
(t
)dt
C j
1 0
xJ
2 0
(
(0) j
x)dx
1
(0) 2 j
0
(0) j
dtJ1
(t
)
Cj
1 2
J12
(
(0) j
)
1
(0) j
J1
(
(0) j
)
Cj
2
(0) j
J
1
(
t n1
n2
Jn1(t) C
J
n
(x)
m0
(1) m m!(n m
1)
x 2
n2m
Yn
(x)
lim
n
J
(x)
cos sin
J
(x)
性质4 初值
J 0 (0) 1
J n (0) 0 (n 0)
Yn (0)
J n1 (x) J n1 (x) 2J n (x)
J1(0)
x2 t 2Jm(t) tJm (t) t 2 m2 Jm (t)
0
例3：将1在 0 x 1 区间内展成
1
Ci
J
0
(
( i
0)
x)
i 1
J
0
(
(0) i
x)
的级数形式
1 0
xJ
0
(
(0) j
x)dx
1 0
xJ
0
(
(0) j
x)
Ci J0 (i(0) x)dx
i 1
J n (x) (1)n J n (x)
Yn
(x)
lim
n
J
(x)
cos sin
J
(x)
y AJn (x) BYn (x)
x2 y xy x2 n2 y 0
J
n
(
x)
m0
(1)m m!(n m
1)
x 2
n2m
Yn
(
x)
lim
n
J
(x)
cos sin
J
(
x)
)
x 2
1 2
2m
2
(1)m 22m1
x
1 2
2m
m0 2m 1 ! 2
(1)m 2 x2m1
m0 2m 1! x
2
x
(1)m x2m1
m0 2m 1 !
2 sin x
x
J 1 (x) 2
2 cosx
x
J n1 (x) (1)n 2
2
x
n
1 2
1
d
n sin x
x dx x
J 'n (m(n) ) Jn1(m(n) ) Jn1(m(n) )
Jn (m(n) ) 0
R 0
rJ n 2
(n) m R
r dr
贝塞尔函数
J
n
的m(n模) r
R
f
(r)
m1
Am Jn
(n) m R
r
Am
R2 2
1
J
2 n1
(
(n m
)
)
R 0
rf
(r)Jn
(n) m R
r
dr
例2：证明
y 1 2
x
y 2
2 m2
x2
y
0 的解为
y x Jm (x)
y x 1J m (x) x J m (x)
y 1x 2 Jm (x) x 1Jm (x) x 1Jm (x) x 2 Jm(x)
x 2 Jm(x) 2x 1Jm (x) 1 x 2 Jm (x)
d dx J0 (x) J1(x)
xn Jn (x) nxn1Jn (x) xn Jn1(x) xn Jn (x) nxn1Jn (x) xn Jn1(x)
xJn (x) nJn (x) xJn1(x) xJn (x) nJn (x) xJn1(x)
J n1 ( x)
J n1 ( x)
x 2 Jm(x) 2x 1Jm (x) 1x 2 Jm (x)
1 2 x
x 1Jm (x) x Jm (x)
2
2
m2 x2
x
J
m
(x)
x 2 Jm(x) x 1Jm (x) x2 2 m2 x 2 Jm (x)
x 2 x2 2 Jm(x) xJm (x) x2 2 m2 Jm (x)
J1
(
(1) i 2