Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

B.18 BESSEL FUNCTIONS

Definition: Bessel Functions Bessel functions of the first kind can be defined using a Maclaurin series (which is specified about x = 0):

(B.97)

where and is the gamma function. Modified Bessel functions of the first kind are generated from Jα(x) by imposing a purely imaginary argument with as follows:

(B.98)

where Iα(x) is a real-valued function of x (not to be confused with the indicator function). The series expansion of Iα(x) is identical to that of Jα(x) except for the (−1)n term in the numerator:

(B.99)

Modified Bessel functions of the second kind are given by

(B.100)

Examples of the various Bessel functions are shown in Figure B.15.

Figure B.15 Bessel functions. (a) Bessel functions of the first kind Jα(x). (b) Modified Bessel functions of the first kind ...