Bessel functions are all

solutions to the Bessel

equation

*x*^{2}*y*'' + *xy*' + (*x*^{2}-ν^{2})*y* = 0

which naturally appears in problems displaying cylindrical symmetries.

The **Bessel function of the first kind** is defined for parameter ν being a real number:

J_{ν}(*x*) = *x*^{ν} Sum_{m=0:inf} (-1)^{m}*x*^{2m} / 2^{2m+ν}*m*!Γ(ν+*m*+1)

Parameter ν is often an integer, so that ν = *n*; we then talk about the **Bessel function of the first kind of order ***n*:

J_{n}(*x*) = *x*^{n} Sum_{m=0:inf} (-1)^{m}*x*^{2m} / 2^{2m+n}*m*!(*n*+*m*)!

**Bessel functions of the second kind**, or Neumann's function, are also solutions to the Bessel equation:

Y_{ν}(*x*) = (1/sinνπ) (J_{ν}(*x*)cosνπ - J_{-ν}(*x*))

When ν = *n*, the definition for the Bessel function of the second kind of order *n* becomes:

Y_{n}(*x*) = lim_{ν->n} Y_{ν}(*x*)

When the solution seeked must be complex for real values of *x*, **Bessel functions of the third kind** or Hankel functions are used:

H_{ν}^{(1)} = J_{ν}(*x*) + *i*Y_{ν}(*x*)

H_{ν}^{(2)} = J_{ν}(*x*) - *i*Y_{ν}(*x*)

**Modified Bessel function** I_{ν}(*x*) = *i*^{-ν}J_{ν}(*ix*) is a solution to the modified Bessel equation

*x*^{2}*y*'' + *xy*' - (*x*^{2} + ν^{2})*y* = 0

Of course, there is also the **Modified Bessel function of the second kind** (also often called *of the third kind* for a reason unknown to me):

K_{ν}(*x*) = (π/2sinνπ)(I_{-ν}(*x*) - I_{ν}(*x*))

There are entire books devoted to the properties of Bessel functions; however, here's a short and useful list:

- J
_{-n}(*x*) = (-1)^{n}J_{n}(*x*);
- (d/d
*x*)(*x*^{ν}J_{ν}(*x*)) = *x*^{ν}J_{ν-1}(*x*);
- (d/d
*x*)(*x*^{-ν}J_{ν}(*x*)) = -*x*^{-ν}J_{ν+1}(*x*);
- J
_{ν-1}(*x*) + J_{ν+1}(*x*) = (2ν/*x*)J_{ν}(*x*);
- J
_{ν-1}(*x*) - J_{ν+1}(*x*) = 2J'_{ν}(*x*);
- J
_{1/2}(*x*) = sqrt(2/π*x*)sin*x*;
- J
_{-1/2}(*x*) = sqrt(2/π*x*)cos*x*

Primary source: *Advanced Engineering Mathematics*, Erwin Kreyszig