Similar to the relations between the trigonometric functions and the hyperbolic trigonometric functions,

the modified Bessel functions of the first kind of order , , can be expressed by the Bessel function of the first kind

The modified Bessel's differential equation is defined in a similar manner by changing the variable to in Bessel's differential equation:

Its general solution is

where

are the modified Bessel functions of the first and second kind respectively.

Generating Function: The generating function of the Bessel Function of the first kind is

Asymptotic Approximations:

For large , i.e., fixed and ,