講者： 許永昌 老師
1
Contents
 Contour Integral Representation of the Hankel
Functions
 Steepest descents  Asymptotic series
 Modified Bessel function
2
Contour Integral Representation of
the Hankel Functions (請預讀P707~P710)
 Since
x
g  x , t   exp 
2

1 

t


 
t 


Jn xt ,
n
n  
 g(x,t) will have essential singularities at t=0, .
 Residue theorem?
 It is worked for n is an integer number case; otherwise, it will have a
branch point at t=0.
 Derived by yourself:

 B essel equation:  x 2  2x  x  x  x 2  

If
 x 2  2x  x  x   x 2  

 
w e get f
 x, t  
2
 
g  x, t 
t
 1
2
  J  x   0,



t
 g  x , t  f
 x , t   ,
x 
1
t
2 
t 
.

t
2 2
2
T herefore,  x  x  x  x   x  

2
   a
b
g  x, t 
t
 1
dt  gf
b
a
.
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Contour Integral Representation of
the Hankel Functions (continue)
 You will find both t=0+ and t=- will let fg=0 Re{x}>0.
 Bessel J :
J
z 
1
2 i
 Check:

z 1
J  z     
 2  2 i


z 1
  
 2  2 i



s0
z
 
2
 2s

e
 i 2

e
e
 i
 i

exp   y

2

 z   
1  
 
 2 y   

 y 

e
1 
z 
exp   t   
t 
2 
dt
 1
t
 i 2
exp   y 
 y 

 1
,


s
!
s
!


s
 1


s0



 1
z
 
2

e
C
2s
z
dy
 1
 y 
z

s
s
dy
s!
2 i 
dz  2 i    1  sin   

    
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Contour Integral Representation of
the Hankel Functions (continue)
 Hankel function:


 H 1   z  




 2
H  z  

1
i
1
i

e
 i
0

0
e
 i
1 
z 
exp   t   
t 
2 
dt
 1
t
1 
z 
exp   t   
t 
2 
dt
 1
t
1

1 
2
J  z    H   z   H   z  ,


2
 
 N  z   1  H 1   z   H  2   z  



 
2i 
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Contour Integral Representation of
the Hankel Functions (continue)
 Prove that N   z   1  H 1   z   H  2   z  


2i
 Prove that
 H 1   x   e  i  H 1  x 
 2
2
i 
 H   x   e H    x 
at first.
 Use these equations to find out J-(x).
 Finally, we get the formula of N shown in Ch11.3.
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Asymptotic series of the Hankel
Functions (請預讀P719~P723，此page只大略講)
 When z and Re{t-1/t}<0, Exp(z/2[t-1/t])0
 i.e. (|t|-1/|t|)cos(ang(t)) <0
0
 The main contribution is at the
saddle point i.

 Steepest descent:

z
2

t 1
t

z
  zi
2
  iz  y
1 

H
z 

1
i

2
z
e
i
0
iz
i t
e e
2

t

z
 if y   i 2
1  1
1

2
z
exp 
2
i   zO


 t      1 dt 
t  t
i

1 
 
 i     
2 
 2
 2 *
 H
z,
z
i


1
2


.
e
3
g

0

t

i  , dy   i z
e
iz
i 
 1
y
2
i z 2 
2

1
2

dt 

dy
1
2
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Modified Bessel Functions, I & K
(請預讀P713~P716，另一本無)
 Helmholtz eq. :
[2+k2]y=0
 Bessel eq. :
x2y’’+xy’+[x2-n2]y=0
 Modified Helmholtz eq. :
[2k2]y=0
 Modified Bessel eq. :
x2y’’+xy’-[x2+n2]y=0

----(1)
----(2)
Eq. (2) can be transformed from Eq. (1) by the transformation
x ix.
 I(x)iJ(ix)=ei/2J(xei/2). [i is used to make sure
I(x) if x ].
 K(x)/2 i1H(1)(ix). [i1 is used to make sure K(x)
if x . Besides, it will tend to zero when x.]
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Homework
 11.6.3(a~c) (12.3.2e)
 11.6.5 (12.3.3e)
 11.4.7
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