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
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:
Differential equations
satisfies the inhomogeneous Bessel's differential equationBoth,, 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,
we can further simplify to and, but the issue is not quite good since the convergence range will reduce greatly to.