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.
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Generating Function: The generating function of the Bessel Function of the first kind is
Asymptotic Approximations:
For large , i.e., fixed and ,
, 
, can be expressed by the 
to 
in 
,
is complex infinity.
