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

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

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

Differential equations

satisfies the inhomogeneous Bessel's differential equation

Both , , and satisfy the partial differential equation

Both and satisfy the partial differential equation

Integral representations

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

  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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.