APPENDIX 
PROPERTIES OF BESSEL FUNCTIONS AND CYLINDER FUNCTIONS 
The Bessel differential equation may be written as 
=e) n? 
1d + 
es sel Nt pire ws -0 
Lee ? ae 
If the sign of ¢ is positive the solutions are the nth order Bessel functions of the first and 
second kinds J, (x) and Y (x). If the sign of ¢ is negative the solutions are the modified 
Bessel functions /, (x) and K, (x). If mis an integer the two sets of functions are related by 
the equations 
1, (iz) = Fi)” J, (@) 
K, (fiz) = Gar** ZU, (@) F 2 ¥, (@)] 
Some of the important properties used in this report are listed below.2"® As J, (a) and Y, (x) 
have the same properties, Z, (x) will be used here to represent either function or any linear 
combination of these functions. 
Z, (@) = Z,_,(@) hs Z, (t)=-Z,,,(@) + = Z,, (a) 
1, (a) = 1,_, (@) re I, (e) = 1,,,(@) + = I, (x) 
Ky (@)=-K,_,(@)+4 K, @)=-K,,,(2) + 2 K, () 
Zy (#) =-Z,(@) = Z_,(@) 
Iy(z)= 1,(@)= I, (2) 
Kf (@) = - K, (@) = - K_, () 
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