The Modified Bessel's Differential Equation  
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. 
Important Properties  
Generating Function: The generating function of the Bessel Function of the first kind is
 
Asymptotic Approximations: For large , i.e., fixed and , 
Special Results  
