Incomplete Bessel functions

In mathematics, the incomplete Bessel functions are types of special functions which act as a type of extension from the complete-type of Bessel functions.

Definition

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The incomplete Bessel functions are defined as the same delay differential equations of the complete-type Bessel functions:

 
 
 
 
 
 

And the following suitable extension forms of delay differential equations from that of the complete-type Bessel functions:

 
 
 
 
 
 

Where the new parameter   defines the integral bound of the upper-incomplete form and lower-incomplete form of the modified Bessel function of the second kind:[1]

 
 

Properties

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  for integer  
 
 
 
 
  for non-integer  
 
 
 
 
  for non-integer  
  for non-integer  

Differential equations

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  satisfies the inhomogeneous Bessel's differential equation

 

Both   ,   ,   and   satisfy the partial differential equation

 

Both   and   satisfy the partial differential equation

 

Integral representations

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Base on the preliminary definitions above, one would derive directly the following integral forms of   ,  :

 
 

With the Mehler–Sonine integral expressions of   and   mentioned in Digital Library of Mathematical Functions,[2]

we can further simplify to   and   , but the issue is not quite good since the convergence range will reduce greatly to  .

References

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  1. ^ Jones, D. S. (February 2007). "Incomplete Bessel functions. I". Proceedings of the Edinburgh Mathematical Society. 50 (1): 173–183. doi:10.1017/S0013091505000490.
  2. ^ Paris, R. B. (2010), "Bessel Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
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