Mathematical Physics
Cylindrical Functions
Cylindrical Functions
?Fundamental Properties
?Eigenvalue Problem
?Symmetric Cylindrical Problems
?General Cylindrical Problems
?Conclusion
Fundamental Properties
Cylindrical Functions of order m
Definition：
Classification：
Bessel Function of
order m
Neumann Function
of order m
Hankel Function of
order m
0)('" 222 ???? ymxxyyx
? ?? ? ? ???? ?? 0 22)1(! )1()( k mkxkm mkkxJ
mx
xJmxxJxN mm
m s i n
)(c o s)()( ???
)()()( xNixJxH mmm ??
Special solution of
Fundamental Properties
?Graphs of Cylindrical Functions
– Bessel Functions
– Neumann Functions
?Properties of Cylindrical Functions
– Symmetry
? For m ? N,Zm(-x) =(-1)m Zm(x)
– Asymptotic Properties
– Null points
– Recurrence Formulas
Bessel Functions
Neumann Functions
Asymptotic Properties
As x → 0,we have,
m
x
m
m
x
mx
mm
xNxN
xJxJ
)()(,ln)(
)()(,1)(
2)!1(
02
2
0
2!
1
00
??
?
?
?
???
??
As x → ∞,we have,
)](e x p [)(
)];(e x p [)(
);s i n ()(
);c o s ()(
4
1
2
12
4
1
2
12
4
1
2
12
4
1
2
12
??
??
??
??
?
?
?
?
????
???
???
???
?
?
mxixH
mxixH
mxxN
mxxJ
xm
xm
xm
xm
Null Points of Bessel Functions
? From the asymptotic formula,one obtains
? According to the graph：
There are infinitely many positive zeros.
?? ?????? ? )( 1)()(3)(2)(10 mnmnmmm xxxxx
?
?????
)(
)(0)c o s (
4
1
2
1)(
2
1
4
1
2
1
4
1
2
1
???
????????
mnx
nmxmx
m
n
The positive zeros appear alternately.
?? ?????? ? )1(1)(1)2(1)1(1)0(10 mm xxxxx
The first positive zero increases with the order m.
??????? ?? )(3)1(2)(2)1(1)(10 mmmmm xxxxx
Recurrence Formulas
)() ] '([
)() ] '([
1
1
xZxxZx
xZxxZx
m
m
m
m
m
m
m
m
?
??
?
??
??
??
Basic recurrence formulas
Corollary 2
1
1
/'
/'
?
?
???
???
mmm
mmm
ZxmZZ
ZxmZZ
Corollary 1
11
11
/2
'2
??
??
??
??
mmm
mmm
ZZxmZ
ZZZ
Proof of the recurrence formula
? ?? ? ? ???? ?? 0 22)1(! )1()( k mkxkm mkkxJ
? ? )'()1(! )1(]'/)([ 20 221 kk mkkmm xmkkxxJ ? ? ? ???? ??
? ? 121 221)1(! )1(2 ?? ? ?? ??? ?? kk mkk xmkk k
? ? 121 12211 )1()!1( )1( ?? ? ???? ???? ??? kk mkk xmkk
1??kl
? ? mmll mll xxmll /)11(! )1( 120 1221 ???? ??? ???? ???
mm xxJ /)(1???
Applications of the recurrence formulas
]'[]'[
]'[]'[
1
1
01
1
11
1
xZxZZxZx
ZxxZZxZx
m
m
m
m
k
kk
km
m
m
m
?
?
??
?
??
?
??
????
?????
?? ????? dxJxxxdxJx mmmnmn ]'[ 111
? ??? ????? dxJxmnJx mnmn 111 )1(
?? ?? dxxJxxdxJx nn ]'[ 110
? ???? dxJxnJx nn 111 )1(
? ?? ????? dxJxnJxnJx nnn 022011 )1()1(
cJxdxJx mmmm ??? ? 1
?? ??? dxxJJxJxdxJx 0021303 42
?? ??? ????? dxJxmnJxdxJx mnmnmn 111 )1(
10 xJdxxJ ??
?? ?? ????? dxJxnJxnJxdxJx nnnn 0220110 )1()1(
Ex.1
Ex,2
01 JdxJ ???
Ex,4
Ex,3
?? ??? dxxJJxdxJx 00212 2
Ex,5 ?? ??? dxJxJdxxJ
112 2
cJxdxJx mmmm ??? ? 1
Applications of the recurrence formulas
Eigenvalue problems of Bessel’s Eq.
? The resolution of rotational symmetric
cylindrical problems
? General eigenvalue problems
– Eigenvalue problems
– Eigenvalues and eigenfunctions
– Orthogonality and completeness
? Typical eigenvalue problems
– Finiteness and boundary condition of the 1st kind
– Finiteness and boundary condition of the 2nd kind
– Boundary conditions of the 1st kind
uau t 22 ??
0' 22 ?? TkaT
?? imeRtTu )()(?
?? imt efu )(| 0 ??
)e x p ( 22 takAT nn ?? )()( ?? nmnnmn kNDkJCR ??
? ? ?? 1 )()(n imnn eRtTu ??? ? ?? 1 )()( n nn RAf ??
0)''( 22 ??? RkRR m ?? ?
The resolution of rotational
symmetric cylindrical problem
General eigenvalue problem
? Eigenvalue problems：
??
???
?
??????
co n d i t i o n sb o u n d ar yt y p eLS
0,0'" 222 kbxyxymxyyx
??
?
?
?
?
???
c o n d i t i o n sb o u n d a r yt y p eLi o u v i l l eS t u r m
0)''( 22 RkRR m ?? ?
Let x = k ρ,y(x) = R(ρ),then we have：
Eigenvalues and eigenfunctions
? The general solution of the universe eq,is
)()()( xBNxAJxy mm ??
? From boundary conditions,we get
? The eigenfunctions are
0
0/
)(
00
)(
0
??
???
m
m
nn
k
bk
?
?
?,3,2,10),/()/()()( )()(,???? nbNDbJCxyR mnmnmnmnnn ?????
Orthogonality and completeness
2,
0 )()()(
mnlnlnb NdxPR ???? ??
Norm
Orthogonality
??? dRN nbmn )()( 202 ??
Completeness
)()( 1 ?? nn n Rff ? ? ??
The generalized Fourier coefficients are
???? dRfNf nbm
n
n )()()(
1
02 ??
Finiteness and boundary condition of the 1st kind
? The eigenvalue problem is
? The eigenvalues and eigenfunctions are
?,3,2,1),/()()(
)(,/
)(
)()(
???
?
nbxJxyR
xJofz er op o s i t i ven t ht h eisxbxk
m
nmnn
m
m
n
m
nn
??；
??
???
?
????
? 0|
,0)''( 22
b
m
R
bRkRR
?
? ???
? The orthogonality and norm,
)()(
)()()(
)(2
1
2
2
12
2
,0
m
nm
m
n
m
nlnln
b
xJbN
NdxPR
??
?? ????
Finiteness and boundary condition of the 1st kind
???? dRfNf nbm
n
n )()()(
1
02 ??
)/()()( )(11 bxJfRff mnmn nnn n ??? ?? ? ?? ? ??
? Completeness,
? Fourier’s expansion coefficients：
x d xxJkxfNk mnbkm
nn
n )()/(
)(
1
022 ??
x d xxJxbxfNx b m
x m
nm
n
m
n
m
n )()/(
)()(
)(
0
)(
22)(
2 ?
?
Finiteness and boundary condition of the 1st kind
????? dRcNf nbm
nn
)()()( 1 02 ?? ?
)/()()( )(11 bxJfRfc mnmn nnn n ???? ?? ? ?? ? ???
)()( 1 2 ccRN nm
n
?
Ex,1,Expand f = δ(ρ-c) in [0,b] in a generalized Fourier
series of the Bessel functions Jm.
)(
)/(
)(2
1
2
21
)(
m
nm
m
nm
xJb
bcxcJ
?
?
Finiteness and boundary condition of the 1st kind
???? dRNf nmbm
nn
)()( 1 02 ??
)/()( )(11 bxJfRf mnmn nnn nm ??? ?? ? ?? ? ??
x d xxJxNx b mx mm
nmmn
m mn
)()()(
)(
022)(
2 ?
?
?
?
)(
2
)(1)( mnmmn
m
xJx
b
?
?
? ? )(01122)( 2 )()()( mnxmmm
nmmn
m
xJxNx b ???
?
?
bxx mn /)( ??
Ex,2,Expand f = ρm in [0,b] in a generalized Fourier series
of the Bessel functions Jm.
Finiteness and boundary condition of the 1st kind
)/()( )0(0112 bxJfRf nn nnn n ??? ?? ? ?? ? ??
???? dRNf nb
n
n )()(
1 2
020 ??
dxxJxNx b nx
nn
)()()( 0
0
3
204)0(
4 )0(?
?
? ? )0(010213)0(2
12214)0(
4
42)()( nx
nn
xJJxJxxJbx b ???
1021303 42 xJJxJxdxJx ????
)(]4)[()()( 2 )0(1)0(3)0()0(2
14)0(
2
nnn
nn
xJxxxJx b ??
Ex.3,Expand f = ρ2 in [0,b] in a generalized Fourier series
of the Bessel functions J0.
Finiteness and boundary condition of the 2nd kind
? The eigenvalue problem is
? The eigenvalues and eigenfunctions are
??
?
?
?
?????
????
??
0,3,2,1,0),/()()(
0;,3,2,1),/()()(
,0)(',/
)1(
0
)0(
0
)0(
0
)(
)()(
xnbJxyR
mnbJxyR
xJofr o o tp o s i t i v en t ht h eisw h e r ebk
nnn
m
nmnn
m
m
n
m
nn
????
???
??；?
?
??
???
?
????
? 0|'
,0)''( 22
b
m
R
bRkRR
?
? ???
? The orthogonality and norm,
)(/)(
)(/])/(1[)(
)()()(
)0(2
0
2
2
120
)(22)(2
2
12
2
,
0
nn
m
nm
m
n
m
n
m
nlnln
b
JbN
JmbN
NdPR
?
??
?????
?
??
??
Finiteness and boundary condition of the 2nd kind
????? dRcNf nbm
nn
)()()( 1 02 ?? ?
)/()()( )1(000 bxJfRfc nn nnn n ???? ?? ? ?? ? ???
)()( 1 2 ccRN nm
n
?
)(
)/(
)1(2
0
2
2
1
)1(
0
n
n
xJb
bcxcJ?
)1()0(
10 )()('
nn x
xJxJ
??
?
?
?
Ex,1,Expand f = δ(ρ-c) in [0,b] in a generalized Fourier
series of the Bessel functions J0.
Finiteness and boundary condition of the 2nd kind
??? dRNf nb
nn
)(1)( 1 020 ?? ?
)/()(1 )1(000 bxJfRf nn nnn n ?? ?? ? ?? ? ??
dxxJxNx b nx
nn
)()()( 0
0202)1(
2 )1(?
?
0?? ? )1(01202)1(
2
)()()( nx
nn
xxJNx b?
bxx n /)1( ??
1?
??? dRNf b )(1)( 1 0020
00
?? ?
?? dJb b 11)0(1 02
0221
?? ?
Ex.2,Expand f = 1 in [0,b] in a generalized Fourier
series of the Bessel functions J0.
Boundary conditions of the 1st kind
? The eigenvalue problem is
? By boundary conditions,one gets
0)()( )()( ??
?
?
??
?
??
?
??
?
B
A
kbNkbJ
kaNkaJ
mm
mm
??
???
??
?????
?? 0||
,0)''( 22
ba
m
RR
baRkRR
??
? ???
)()()( ??? kBNkAJR mm ??
? The general solution is
? The condition of non-zero solution is
0)()( )()( ?kbNkbJ kaNkaJ
mm
mm
Rotational symmetric problems
? Axial symmetric problems ( the case of m = 0)
– Heat problem
– Wave problem
– Steady problem
? Rotational symmetric problems
– Heat problem
– Wave problem
– Steady problem
Axial symmetric heat problems
)(0122 ?? nn n kJAb ? ? ???由初始条件得：
Example 1
??? ? ???? dkJbNA nb
n
n )()()(
1
022020根据完备性：
半径为 b的无限长圆柱体，柱面上温度为零，初始温度分布
为 f = b2 –ρ2，确定柱内温度 u 的变化。
??
???
????
???
??
22
0
2
2
|,0|
,
?
?
? bfuu
buau
tb
t定解问题为：
? ? ? ?? 1 0022 )]()()[e x p (n nnnnnn kNDkJCtkaAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ???
?
1
)0(
0
22 /),()e x p (
,0),(),0(
n nnnnn bxkkJtkaAu
tbutu
?
半通解化为有界，
Solution：以圆柱体的对称轴为 z 轴，建立柱坐标。
Axial symmetric heat problems
)(002 ?? nn n kJAA ? ? ??由初始条件得：
Example 2
?? ? ???? dkJANA nb
n
n )()(
1
02020根据完备性：
半径为 b的无限长圆柱体，柱面上绝热，初始温度分布为 f =
Aρ2,确定柱内温度 u 的变化。
??
???
???
???
?? 20
22
|,0|
,
?
?
?? Afuu
buau
tb
t定解问题为：
? ? ? ?? 0 0022 )]()()[e x p (n nnnnnn kNDkJCtkaAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ???
?
0
)1(
0
22 /),()e x p (
,0),(),0(
n nnnnn bxkkJtkaAu
tbutu
?
? 半通解化为有界，
Solution：以圆柱体的对称轴为 z 轴，建立柱坐标。
Axial symmetric wave problems
??
??
?
?
??
?
?
?
?
?
?
)(0
)(
01
01
22
?
??
nnn n
nn n
kJakB
kJAb
由初始条件得：
Example 3
???? ? ???? dkJbNAB nb
n
nn )()()(
1,0
022020根据完备性：
半径为 b的圆形膜，边缘固定，初始形状是旋转抛物面
f = b2 –ρ2，初始速度为零，求膜的振动情况。
??
???
????
???
??? 0|,|,0|
,
0220
22
tttb
tt
ubuu
buau
?
?
?
定解问题为：
? ? ? ??? 1 00 )]()()[s i nc o s(n nnnnnnnn kNDkJCtakBtakAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ???
?
1
)0(0 /),()s i nc o s(
,0),(),0(
n nnnnnnn bxkkJtakBtakAu
tbutu
?
半通解化为有界，
Solution：以圆形膜的中心为原点，建立极坐标。
Axial symmetric wave problems
??
??
?
??
?
?
?
?
?
?
?
)()(
)(0
01
01
???
?
nnn n
nn n
kJakBc
kJA
由初始条件得：
Example 4
???? ? ????? dkJcNakBA nb
nn
nn )()()(
1,0
0020根据完备性：
半径为 b的圆形膜，边缘固定，初始位移为零，初始速度为 f
= δ(ρ - c)，求膜的振动情况。
??
???
????
???
??? )(|,0|,0|
,
00
22
cuuu
buau
tttb
tt
??
?
?
定解问题为：
? ? ? ??? 1 00 )]()()[s i nc o s(n nnnnnnnn kNDkJCtakBtakAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ???
?
1
)0(0 /),()s i nc o s(
,0),(),0(
n nnnnnnn bxkkJtakBtakAu
tbutu
?
半通解化为有界，
Solution：以圆形膜的中心为原点，建立极坐标。
Axial symmetric steady problems
0)(0 01 ??? ? ? ? nnn n AkJA ?由下底条件得：
Example 5
?? ? ???? dkJNLkB nb
nnn
)()(s i n h 1 02020根据完备性：
半径为 b，高为 L的圆柱体，下底和侧面都保持零度，上底的温
度分布为 ρ2，求柱内的稳恒温度分布。
??
???
???
??????
??? 20
2
|,0|,0|
0,,0
?
?
? Lzzb
zz
uuu
Lzbuu定解问题为：
? ? ? ??? 1 00 )]()()[s i n hc o s h(n nnnnnnnn kNDkJCzkBzkAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ???
?
1
)0(0 /),()s i n hc o s h(
,0),(),0(
n nnnnnnn bxkkJzkBzkAu
zbuzu
?
半通解化为有界，
)()s i n h ( 012 ?? nnn n kJLkB? ? ??由上底条件得：
Solution：以圆柱体的轴为 z 轴，下底中心为原点，建立柱坐标。
Axial symmetric steady problems
??
??
?
?
?
?
?
?
?
?
?
)()s i n h (
)()s i n h (
01
2
01
??
?
nnn n
nnn n
kJLkBB
kJLkAA
由上下底条件得：
Example 6
??? ? nnb
nn
n BdkAJNLkA,)()(s i n h
1
0020 ???根据完备性：
半径为 b，高为 L的圆柱体，侧面电势保持为零，上底的电势为
A，下底的电势分布为 Bρ2，求柱内的电势分布。
??
???
???
??????
??? AuBuu
Lzbuu
Lzzb
zz
|,|,0|
0,,0
20
2
?
?
?
定解问题为：
? ? ? ???? 1 00 )]()()][(s i n hs i n h[n nnnnnnnn kNDkJCzLkBzkAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ????
?
1
)0(0 /),()](s i n hs i n h[
,0),(),0(
n nnnnnnn bxkkJzLkBzkAu
zbuzu
?
半通解化为有界，
Solution：以圆柱体的轴为 z 轴，下底中心为原点，建立柱坐标。
Rotational symmetric heat problems
)(11 ?? nn n kJAA ? ? ??由初始条件得：
Example 7
?? ? ???? dkJANA nb
n
n )()(
1
1021根据完备性：
半径为 b的无限长圆柱体，柱面上温度为零，初始温度分布
为 f = Aρ cos φ，确定柱内温度 u 的变化。
??
???
??
???
?? ??
?
? co s|,0|
,
0
22
Auu
buau
tb
t定解问题为：
??? c o s)]()()[e x p (1 1122? ? ? ?? n nnnnnn kNDkJCtkaAu
，相应的半通解为定解问题有转动对称性
? ? ? ???
?
1
)1(
1
22 /,c o s)()e x p (
,0),(),0(
n nnnnn bxkkJtkaAu
tbutu
??
半通解化为有界，
Solution：以圆柱体的对称轴为 z 轴，建立柱坐标。
Rotational symmetric wave problems
??
??
?
?
?
?
?
?
?
?
?
)(0
)(
21
21
2
?
??
nnn n
nn n
kJakB
kJA
由初始条件得：
Example 8
??? ? ???? dkJNAB nb
n
nn )()(
1,0
22022根据完备性：
半径为 b的圆形膜，边缘固定，初始形状是 ρ2sin2φ，初始
速度为零，求膜的振动情况。
??
???
????
???
??? 0|,2s i n|,0|
,
0220
22
tttb
tt
ubuu
buau
??
?
? ）（
定解问题为：
??? 2s i n)]()()[s i nc o s(1 22? ? ? ??? n nnnnnnnn kNDkJCtakBtakAu
，相应的半通解为定解问题有转动对称性
? ? ? ???
?
1
)2(2 /,2s i n)()s i nc o s(
,0),(),0(
n nnnnnnn bxkkJtakBtakAu
tbutu
??
半通解化为有界，
Solution：以圆形膜的中心为原点，建立极坐标。
Rotational symmetric steady problems
??
??
?
?
?
?
?
?
?
?
?
)()s i n h (
)()s i n h (0
11
11
??
?
nnn n
nnn n
kJLkBA
kJLkA
由上下底条件得：
Example 9
??? ? ???? dkJANLkBA nb
nn
nn )()(s i n h
10
1021，根据完备性：
半径为 b，高为 L的圆柱体，侧面和上底保持零度，下底的
温度分布为 Aρsinφ，求柱内的稳恒温度分布。
??
?
???
??????
??? 0|,s i n|,0|
0,,0
0
2
Lzzb
zz
uAuu
Lzbuu
??
?
?
定解问题为：
??? s i n)]()() ] [(s i n hs i n h[1 11? ? ? ???? n nnnnnnnn kNDkJCzLkBzkAu
，相应的半通解为定解问题有转动对称性
? ? ? ????
?
1
)1(1 /,s i n)()](s i n hs i n h[
,0),(),0(
n nnnnnnn bxkkJzLkBzkAu
zbuzu
??
半通解化为有界，
Solution：以圆柱体的轴为 z 轴，下底中心为原点，建立柱坐标。
General Cylindrical Problems
? Programer
– First,resolve the conditions into Fourier series
with respect to the variable ?,
– Then,find the symmetric cylindrical solutions for
each condition with trigonometric functions.
– Superpose the symmetric cylindrical solutions
obtained above,
? General heat problems
? General wave problems
? General steady problems
General heat problems
半径为 b的无限长圆柱体，柱面上温度为零，初始
温度分布为 f (ρ,φ)，确定柱内温度 u 的变化。
?
?
?
?
?
?
?
???
?
?
),(|
,0|
,
0
2
2
??
?
?
fu
u
buau
t
b
t
定解问题为：
bxk
imkJtkaAu
m
n
m
n
n
m
nm
m
nnmm
/
)ex p ()()ex p (
|)(|)(
1
)(2)(2
,
?
? ??
?
?
?
???
??
定解问题的半通解为
Hint：以圆柱体的对称轴为 z 轴，建立柱坐标。
General wave problems
半径为 b的圆形膜，边缘固定，初始形状是 f (ρ,
φ)，初始速度为零，求膜的振动情况。
?
?
?
?
?
?
?
?
?
?
???
?
?
?
0|
),|
,0|
,
0
0
2
2
tt
t
b
tt
u
fu
u
buau
??
?
?
（
定解问题为：
bxk
imkJtakBtakAu
m
n
m
n
n
m
nm
m
nnm
m
nnmm
/
)ex p ()()s i nco s(
|)(|)(
1
)()(
,
)(
,
?
?? ??
?
?
?
???
??
定解问题的半通解为
Hint：以圆形膜的中心为原点，建立极坐标。
General steady problems
Hint：以圆柱体的对称轴为 z 轴，下底中心为原点，
建立柱坐标。
?
?
?
?
?
?
?
?
?
?
??????
?
?
?
0|
),(|
0|
0,,0
0
2
Lz
z
b
zz
u
fu
u
Lzbuu
??
?
?
定解问题为：
bxk
imkJzLkBzkAu
m
n
m
n
n
m
nm
m
nnm
m
nnmm
/
)ex p ()()](s i n hs i n h[
|)(|)(
1
)()(
,
)(
,
?
??? ??
?
?
??
问题的半通解为
半径为 b，高为 L的圆柱体，侧面和上底保持零度，
下底的温度分布为 f ( ρ,φ)，求柱内的稳恒温度分布。
Conclusion
? The general cylindrical problem can be regarded as the
superposition of symmetric problems.
? By separation of variables,one obtains the eigenvalue
problem of Bessel’s equation from the symmetric
cylindrical problems.
? The eigenfunctions of Bessel’s equation are cylindrical
functions,and the eigenvalues are given by the boundary
conditions.
? Bessel functions and Neumann functions are typical
cylindrical functions,For the cylindrical problems,it is
important to be familiar with their properties such as the
symmetries,recurrence formulas,and so on.
Cylindrical Functions
Cylindrical Functions
?Fundamental Properties
?Eigenvalue Problem
?Symmetric Cylindrical Problems
?General Cylindrical Problems
?Conclusion
Fundamental Properties
Cylindrical Functions of order m
Definition：
Classification：
Bessel Function of
order m
Neumann Function
of order m
Hankel Function of
order m
0)('" 222 ???? ymxxyyx
? ?? ? ? ???? ?? 0 22)1(! )1()( k mkxkm mkkxJ
mx
xJmxxJxN mm
m s i n
)(c o s)()( ???
)()()( xNixJxH mmm ??
Special solution of
Fundamental Properties
?Graphs of Cylindrical Functions
– Bessel Functions
– Neumann Functions
?Properties of Cylindrical Functions
– Symmetry
? For m ? N,Zm(-x) =(-1)m Zm(x)
– Asymptotic Properties
– Null points
– Recurrence Formulas
Bessel Functions
Neumann Functions
Asymptotic Properties
As x → 0,we have,
m
x
m
m
x
mx
mm
xNxN
xJxJ
)()(,ln)(
)()(,1)(
2)!1(
02
2
0
2!
1
00
??
?
?
?
???
??
As x → ∞,we have,
)](e x p [)(
)];(e x p [)(
);s i n ()(
);c o s ()(
4
1
2
12
4
1
2
12
4
1
2
12
4
1
2
12
??
??
??
??
?
?
?
?
????
???
???
???
?
?
mxixH
mxixH
mxxN
mxxJ
xm
xm
xm
xm
Null Points of Bessel Functions
? From the asymptotic formula,one obtains
? According to the graph：
There are infinitely many positive zeros.
?? ?????? ? )( 1)()(3)(2)(10 mnmnmmm xxxxx
?
?????
)(
)(0)c o s (
4
1
2
1)(
2
1
4
1
2
1
4
1
2
1
???
????????
mnx
nmxmx
m
n
The positive zeros appear alternately.
?? ?????? ? )1(1)(1)2(1)1(1)0(10 mm xxxxx
The first positive zero increases with the order m.
??????? ?? )(3)1(2)(2)1(1)(10 mmmmm xxxxx
Recurrence Formulas
)() ] '([
)() ] '([
1
1
xZxxZx
xZxxZx
m
m
m
m
m
m
m
m
?
??
?
??
??
??
Basic recurrence formulas
Corollary 2
1
1
/'
/'
?
?
???
???
mmm
mmm
ZxmZZ
ZxmZZ
Corollary 1
11
11
/2
'2
??
??
??
??
mmm
mmm
ZZxmZ
ZZZ
Proof of the recurrence formula
? ?? ? ? ???? ?? 0 22)1(! )1()( k mkxkm mkkxJ
? ? )'()1(! )1(]'/)([ 20 221 kk mkkmm xmkkxxJ ? ? ? ???? ??
? ? 121 221)1(! )1(2 ?? ? ?? ??? ?? kk mkk xmkk k
? ? 121 12211 )1()!1( )1( ?? ? ???? ???? ??? kk mkk xmkk
1??kl
? ? mmll mll xxmll /)11(! )1( 120 1221 ???? ??? ???? ???
mm xxJ /)(1???
Applications of the recurrence formulas
]'[]'[
]'[]'[
1
1
01
1
11
1
xZxZZxZx
ZxxZZxZx
m
m
m
m
k
kk
km
m
m
m
?
?
??
?
??
?
??
????
?????
?? ????? dxJxxxdxJx mmmnmn ]'[ 111
? ??? ????? dxJxmnJx mnmn 111 )1(
?? ?? dxxJxxdxJx nn ]'[ 110
? ???? dxJxnJx nn 111 )1(
? ?? ????? dxJxnJxnJx nnn 022011 )1()1(
cJxdxJx mmmm ??? ? 1
?? ??? dxxJJxJxdxJx 0021303 42
?? ??? ????? dxJxmnJxdxJx mnmnmn 111 )1(
10 xJdxxJ ??
?? ?? ????? dxJxnJxnJxdxJx nnnn 0220110 )1()1(
Ex.1
Ex,2
01 JdxJ ???
Ex,4
Ex,3
?? ??? dxxJJxdxJx 00212 2
Ex,5 ?? ??? dxJxJdxxJ
112 2
cJxdxJx mmmm ??? ? 1
Applications of the recurrence formulas
Eigenvalue problems of Bessel’s Eq.
? The resolution of rotational symmetric
cylindrical problems
? General eigenvalue problems
– Eigenvalue problems
– Eigenvalues and eigenfunctions
– Orthogonality and completeness
? Typical eigenvalue problems
– Finiteness and boundary condition of the 1st kind
– Finiteness and boundary condition of the 2nd kind
– Boundary conditions of the 1st kind
uau t 22 ??
0' 22 ?? TkaT
?? imeRtTu )()(?
?? imt efu )(| 0 ??
)e x p ( 22 takAT nn ?? )()( ?? nmnnmn kNDkJCR ??
? ? ?? 1 )()(n imnn eRtTu ??? ? ?? 1 )()( n nn RAf ??
0)''( 22 ??? RkRR m ?? ?
The resolution of rotational
symmetric cylindrical problem
General eigenvalue problem
? Eigenvalue problems：
??
???
?
??????
co n d i t i o n sb o u n d ar yt y p eLS
0,0'" 222 kbxyxymxyyx
??
?
?
?
?
???
c o n d i t i o n sb o u n d a r yt y p eLi o u v i l l eS t u r m
0)''( 22 RkRR m ?? ?
Let x = k ρ,y(x) = R(ρ),then we have：
Eigenvalues and eigenfunctions
? The general solution of the universe eq,is
)()()( xBNxAJxy mm ??
? From boundary conditions,we get
? The eigenfunctions are
0
0/
)(
00
)(
0
??
???
m
m
nn
k
bk
?
?
?,3,2,10),/()/()()( )()(,???? nbNDbJCxyR mnmnmnmnnn ?????
Orthogonality and completeness
2,
0 )()()(
mnlnlnb NdxPR ???? ??
Norm
Orthogonality
??? dRN nbmn )()( 202 ??
Completeness
)()( 1 ?? nn n Rff ? ? ??
The generalized Fourier coefficients are
???? dRfNf nbm
n
n )()()(
1
02 ??
Finiteness and boundary condition of the 1st kind
? The eigenvalue problem is
? The eigenvalues and eigenfunctions are
?,3,2,1),/()()(
)(,/
)(
)()(
???
?
nbxJxyR
xJofz er op o s i t i ven t ht h eisxbxk
m
nmnn
m
m
n
m
nn
??；
??
???
?
????
? 0|
,0)''( 22
b
m
R
bRkRR
?
? ???
? The orthogonality and norm,
)()(
)()()(
)(2
1
2
2
12
2
,0
m
nm
m
n
m
nlnln
b
xJbN
NdxPR
??
?? ????
Finiteness and boundary condition of the 1st kind
???? dRfNf nbm
n
n )()()(
1
02 ??
)/()()( )(11 bxJfRff mnmn nnn n ??? ?? ? ?? ? ??
? Completeness,
? Fourier’s expansion coefficients：
x d xxJkxfNk mnbkm
nn
n )()/(
)(
1
022 ??
x d xxJxbxfNx b m
x m
nm
n
m
n
m
n )()/(
)()(
)(
0
)(
22)(
2 ?
?
Finiteness and boundary condition of the 1st kind
????? dRcNf nbm
nn
)()()( 1 02 ?? ?
)/()()( )(11 bxJfRfc mnmn nnn n ???? ?? ? ?? ? ???
)()( 1 2 ccRN nm
n
?
Ex,1,Expand f = δ(ρ-c) in [0,b] in a generalized Fourier
series of the Bessel functions Jm.
)(
)/(
)(2
1
2
21
)(
m
nm
m
nm
xJb
bcxcJ
?
?
Finiteness and boundary condition of the 1st kind
???? dRNf nmbm
nn
)()( 1 02 ??
)/()( )(11 bxJfRf mnmn nnn nm ??? ?? ? ?? ? ??
x d xxJxNx b mx mm
nmmn
m mn
)()()(
)(
022)(
2 ?
?
?
?
)(
2
)(1)( mnmmn
m
xJx
b
?
?
? ? )(01122)( 2 )()()( mnxmmm
nmmn
m
xJxNx b ???
?
?
bxx mn /)( ??
Ex,2,Expand f = ρm in [0,b] in a generalized Fourier series
of the Bessel functions Jm.
Finiteness and boundary condition of the 1st kind
)/()( )0(0112 bxJfRf nn nnn n ??? ?? ? ?? ? ??
???? dRNf nb
n
n )()(
1 2
020 ??
dxxJxNx b nx
nn
)()()( 0
0
3
204)0(
4 )0(?
?
? ? )0(010213)0(2
12214)0(
4
42)()( nx
nn
xJJxJxxJbx b ???
1021303 42 xJJxJxdxJx ????
)(]4)[()()( 2 )0(1)0(3)0()0(2
14)0(
2
nnn
nn
xJxxxJx b ??
Ex.3,Expand f = ρ2 in [0,b] in a generalized Fourier series
of the Bessel functions J0.
Finiteness and boundary condition of the 2nd kind
? The eigenvalue problem is
? The eigenvalues and eigenfunctions are
??
?
?
?
?????
????
??
0,3,2,1,0),/()()(
0;,3,2,1),/()()(
,0)(',/
)1(
0
)0(
0
)0(
0
)(
)()(
xnbJxyR
mnbJxyR
xJofr o o tp o s i t i v en t ht h eisw h e r ebk
nnn
m
nmnn
m
m
n
m
nn
????
???
??；?
?
??
???
?
????
? 0|'
,0)''( 22
b
m
R
bRkRR
?
? ???
? The orthogonality and norm,
)(/)(
)(/])/(1[)(
)()()(
)0(2
0
2
2
120
)(22)(2
2
12
2
,
0
nn
m
nm
m
n
m
n
m
nlnln
b
JbN
JmbN
NdPR
?
??
?????
?
??
??
Finiteness and boundary condition of the 2nd kind
????? dRcNf nbm
nn
)()()( 1 02 ?? ?
)/()()( )1(000 bxJfRfc nn nnn n ???? ?? ? ?? ? ???
)()( 1 2 ccRN nm
n
?
)(
)/(
)1(2
0
2
2
1
)1(
0
n
n
xJb
bcxcJ?
)1()0(
10 )()('
nn x
xJxJ
??
?
?
?
Ex,1,Expand f = δ(ρ-c) in [0,b] in a generalized Fourier
series of the Bessel functions J0.
Finiteness and boundary condition of the 2nd kind
??? dRNf nb
nn
)(1)( 1 020 ?? ?
)/()(1 )1(000 bxJfRf nn nnn n ?? ?? ? ?? ? ??
dxxJxNx b nx
nn
)()()( 0
0202)1(
2 )1(?
?
0?? ? )1(01202)1(
2
)()()( nx
nn
xxJNx b?
bxx n /)1( ??
1?
??? dRNf b )(1)( 1 0020
00
?? ?
?? dJb b 11)0(1 02
0221
?? ?
Ex.2,Expand f = 1 in [0,b] in a generalized Fourier
series of the Bessel functions J0.
Boundary conditions of the 1st kind
? The eigenvalue problem is
? By boundary conditions,one gets
0)()( )()( ??
?
?
??
?
??
?
??
?
B
A
kbNkbJ
kaNkaJ
mm
mm
??
???
??
?????
?? 0||
,0)''( 22
ba
m
RR
baRkRR
??
? ???
)()()( ??? kBNkAJR mm ??
? The general solution is
? The condition of non-zero solution is
0)()( )()( ?kbNkbJ kaNkaJ
mm
mm
Rotational symmetric problems
? Axial symmetric problems ( the case of m = 0)
– Heat problem
– Wave problem
– Steady problem
? Rotational symmetric problems
– Heat problem
– Wave problem
– Steady problem
Axial symmetric heat problems
)(0122 ?? nn n kJAb ? ? ???由初始条件得：
Example 1
??? ? ???? dkJbNA nb
n
n )()()(
1
022020根据完备性：
半径为 b的无限长圆柱体，柱面上温度为零，初始温度分布
为 f = b2 –ρ2，确定柱内温度 u 的变化。
??
???
????
???
??
22
0
2
2
|,0|
,
?
?
? bfuu
buau
tb
t定解问题为：
? ? ? ?? 1 0022 )]()()[e x p (n nnnnnn kNDkJCtkaAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ???
?
1
)0(
0
22 /),()e x p (
,0),(),0(
n nnnnn bxkkJtkaAu
tbutu
?
半通解化为有界，
Solution：以圆柱体的对称轴为 z 轴，建立柱坐标。
Axial symmetric heat problems
)(002 ?? nn n kJAA ? ? ??由初始条件得：
Example 2
?? ? ???? dkJANA nb
n
n )()(
1
02020根据完备性：
半径为 b的无限长圆柱体，柱面上绝热，初始温度分布为 f =
Aρ2,确定柱内温度 u 的变化。
??
???
???
???
?? 20
22
|,0|
,
?
?
?? Afuu
buau
tb
t定解问题为：
? ? ? ?? 0 0022 )]()()[e x p (n nnnnnn kNDkJCtkaAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ???
?
0
)1(
0
22 /),()e x p (
,0),(),0(
n nnnnn bxkkJtkaAu
tbutu
?
? 半通解化为有界，
Solution：以圆柱体的对称轴为 z 轴，建立柱坐标。
Axial symmetric wave problems
??
??
?
?
??
?
?
?
?
?
?
)(0
)(
01
01
22
?
??
nnn n
nn n
kJakB
kJAb
由初始条件得：
Example 3
???? ? ???? dkJbNAB nb
n
nn )()()(
1,0
022020根据完备性：
半径为 b的圆形膜，边缘固定，初始形状是旋转抛物面
f = b2 –ρ2，初始速度为零，求膜的振动情况。
??
???
????
???
??? 0|,|,0|
,
0220
22
tttb
tt
ubuu
buau
?
?
?
定解问题为：
? ? ? ??? 1 00 )]()()[s i nc o s(n nnnnnnnn kNDkJCtakBtakAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ???
?
1
)0(0 /),()s i nc o s(
,0),(),0(
n nnnnnnn bxkkJtakBtakAu
tbutu
?
半通解化为有界，
Solution：以圆形膜的中心为原点，建立极坐标。
Axial symmetric wave problems
??
??
?
??
?
?
?
?
?
?
?
)()(
)(0
01
01
???
?
nnn n
nn n
kJakBc
kJA
由初始条件得：
Example 4
???? ? ????? dkJcNakBA nb
nn
nn )()()(
1,0
0020根据完备性：
半径为 b的圆形膜，边缘固定，初始位移为零，初始速度为 f
= δ(ρ - c)，求膜的振动情况。
??
???
????
???
??? )(|,0|,0|
,
00
22
cuuu
buau
tttb
tt
??
?
?
定解问题为：
? ? ? ??? 1 00 )]()()[s i nc o s(n nnnnnnnn kNDkJCtakBtakAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ???
?
1
)0(0 /),()s i nc o s(
,0),(),0(
n nnnnnnn bxkkJtakBtakAu
tbutu
?
半通解化为有界，
Solution：以圆形膜的中心为原点，建立极坐标。
Axial symmetric steady problems
0)(0 01 ??? ? ? ? nnn n AkJA ?由下底条件得：
Example 5
?? ? ???? dkJNLkB nb
nnn
)()(s i n h 1 02020根据完备性：
半径为 b，高为 L的圆柱体，下底和侧面都保持零度，上底的温
度分布为 ρ2，求柱内的稳恒温度分布。
??
???
???
??????
??? 20
2
|,0|,0|
0,,0
?
?
? Lzzb
zz
uuu
Lzbuu定解问题为：
? ? ? ??? 1 00 )]()()[s i n hc o s h(n nnnnnnnn kNDkJCzkBzkAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ???
?
1
)0(0 /),()s i n hc o s h(
,0),(),0(
n nnnnnnn bxkkJzkBzkAu
zbuzu
?
半通解化为有界，
)()s i n h ( 012 ?? nnn n kJLkB? ? ??由上底条件得：
Solution：以圆柱体的轴为 z 轴，下底中心为原点，建立柱坐标。
Axial symmetric steady problems
??
??
?
?
?
?
?
?
?
?
?
)()s i n h (
)()s i n h (
01
2
01
??
?
nnn n
nnn n
kJLkBB
kJLkAA
由上下底条件得：
Example 6
??? ? nnb
nn
n BdkAJNLkA,)()(s i n h
1
0020 ???根据完备性：
半径为 b，高为 L的圆柱体，侧面电势保持为零，上底的电势为
A，下底的电势分布为 Bρ2，求柱内的电势分布。
??
???
???
??????
??? AuBuu
Lzbuu
Lzzb
zz
|,|,0|
0,,0
20
2
?
?
?
定解问题为：
? ? ? ???? 1 00 )]()()][(s i n hs i n h[n nnnnnnnn kNDkJCzLkBzkAu ??
相应的半通解为定解问题有轴对称性，
? ? ? ????
?
1
)0(0 /),()](s i n hs i n h[
,0),(),0(
n nnnnnnn bxkkJzLkBzkAu
zbuzu
?
半通解化为有界，
Solution：以圆柱体的轴为 z 轴，下底中心为原点，建立柱坐标。
Rotational symmetric heat problems
)(11 ?? nn n kJAA ? ? ??由初始条件得：
Example 7
?? ? ???? dkJANA nb
n
n )()(
1
1021根据完备性：
半径为 b的无限长圆柱体，柱面上温度为零，初始温度分布
为 f = Aρ cos φ，确定柱内温度 u 的变化。
??
???
??
???
?? ??
?
? co s|,0|
,
0
22
Auu
buau
tb
t定解问题为：
??? c o s)]()()[e x p (1 1122? ? ? ?? n nnnnnn kNDkJCtkaAu
，相应的半通解为定解问题有转动对称性
? ? ? ???
?
1
)1(
1
22 /,c o s)()e x p (
,0),(),0(
n nnnnn bxkkJtkaAu
tbutu
??
半通解化为有界，
Solution：以圆柱体的对称轴为 z 轴，建立柱坐标。
Rotational symmetric wave problems
??
??
?
?
?
?
?
?
?
?
?
)(0
)(
21
21
2
?
??
nnn n
nn n
kJakB
kJA
由初始条件得：
Example 8
??? ? ???? dkJNAB nb
n
nn )()(
1,0
22022根据完备性：
半径为 b的圆形膜，边缘固定，初始形状是 ρ2sin2φ，初始
速度为零，求膜的振动情况。
??
???
????
???
??? 0|,2s i n|,0|
,
0220
22
tttb
tt
ubuu
buau
??
?
? ）（
定解问题为：
??? 2s i n)]()()[s i nc o s(1 22? ? ? ??? n nnnnnnnn kNDkJCtakBtakAu
，相应的半通解为定解问题有转动对称性
? ? ? ???
?
1
)2(2 /,2s i n)()s i nc o s(
,0),(),0(
n nnnnnnn bxkkJtakBtakAu
tbutu
??
半通解化为有界，
Solution：以圆形膜的中心为原点，建立极坐标。
Rotational symmetric steady problems
??
??
?
?
?
?
?
?
?
?
?
)()s i n h (
)()s i n h (0
11
11
??
?
nnn n
nnn n
kJLkBA
kJLkA
由上下底条件得：
Example 9
??? ? ???? dkJANLkBA nb
nn
nn )()(s i n h
10
1021，根据完备性：
半径为 b，高为 L的圆柱体，侧面和上底保持零度，下底的
温度分布为 Aρsinφ，求柱内的稳恒温度分布。
??
?
???
??????
??? 0|,s i n|,0|
0,,0
0
2
Lzzb
zz
uAuu
Lzbuu
??
?
?
定解问题为：
??? s i n)]()() ] [(s i n hs i n h[1 11? ? ? ???? n nnnnnnnn kNDkJCzLkBzkAu
，相应的半通解为定解问题有转动对称性
? ? ? ????
?
1
)1(1 /,s i n)()](s i n hs i n h[
,0),(),0(
n nnnnnnn bxkkJzLkBzkAu
zbuzu
??
半通解化为有界，
Solution：以圆柱体的轴为 z 轴，下底中心为原点，建立柱坐标。
General Cylindrical Problems
? Programer
– First,resolve the conditions into Fourier series
with respect to the variable ?,
– Then,find the symmetric cylindrical solutions for
each condition with trigonometric functions.
– Superpose the symmetric cylindrical solutions
obtained above,
? General heat problems
? General wave problems
? General steady problems
General heat problems
半径为 b的无限长圆柱体，柱面上温度为零，初始
温度分布为 f (ρ,φ)，确定柱内温度 u 的变化。
?
?
?
?
?
?
?
???
?
?
),(|
,0|
,
0
2
2
??
?
?
fu
u
buau
t
b
t
定解问题为：
bxk
imkJtkaAu
m
n
m
n
n
m
nm
m
nnmm
/
)ex p ()()ex p (
|)(|)(
1
)(2)(2
,
?
? ??
?
?
?
???
??
定解问题的半通解为
Hint：以圆柱体的对称轴为 z 轴，建立柱坐标。
General wave problems
半径为 b的圆形膜，边缘固定，初始形状是 f (ρ,
φ)，初始速度为零，求膜的振动情况。
?
?
?
?
?
?
?
?
?
?
???
?
?
?
0|
),|
,0|
,
0
0
2
2
tt
t
b
tt
u
fu
u
buau
??
?
?
（
定解问题为：
bxk
imkJtakBtakAu
m
n
m
n
n
m
nm
m
nnm
m
nnmm
/
)ex p ()()s i nco s(
|)(|)(
1
)()(
,
)(
,
?
?? ??
?
?
?
???
??
定解问题的半通解为
Hint：以圆形膜的中心为原点，建立极坐标。
General steady problems
Hint：以圆柱体的对称轴为 z 轴，下底中心为原点，
建立柱坐标。
?
?
?
?
?
?
?
?
?
?
??????
?
?
?
0|
),(|
0|
0,,0
0
2
Lz
z
b
zz
u
fu
u
Lzbuu
??
?
?
定解问题为：
bxk
imkJzLkBzkAu
m
n
m
n
n
m
nm
m
nnm
m
nnmm
/
)ex p ()()](s i n hs i n h[
|)(|)(
1
)()(
,
)(
,
?
??? ??
?
?
??
问题的半通解为
半径为 b，高为 L的圆柱体，侧面和上底保持零度，
下底的温度分布为 f ( ρ,φ)，求柱内的稳恒温度分布。
Conclusion
? The general cylindrical problem can be regarded as the
superposition of symmetric problems.
? By separation of variables,one obtains the eigenvalue
problem of Bessel’s equation from the symmetric
cylindrical problems.
? The eigenfunctions of Bessel’s equation are cylindrical
functions,and the eigenvalues are given by the boundary
conditions.
? Bessel functions and Neumann functions are typical
cylindrical functions,For the cylindrical problems,it is
important to be familiar with their properties such as the
symmetries,recurrence formulas,and so on.
