The Bessel functions of order λ (Cylindrical functions of the first kind) are defined by the following relationships:

J_{λ}(x) is an analytic function of a complex variable for all values of x (except maybe for the point x = 0) and an analytic function of λ for all values of λ. It is represented in the form x^{λ}f_{λ}(x^{2}), where f_{λ}(x^{2}) is an integer function.

Bessel functions are the partial solution of the Bessel differential equation:

which appears in problems of heat transfer when solving the Laplace Equation, written in cylindrical coordinates by the method of separation of variables.

For Bessel functions, the following recurrent relationships are valid:

The equation J_{λ}(x), for λ > –1 has infinitely many real roots. All these roots are simple (except for 0, possibly) and are limited by the roots of the function J_{λ+1}(x). They are arranged symmetrically about the point 0 and have no finite limit points.

The following asymptotic expression is valid for a Bessel function with large x:

Hence, an approximate formula for the roots J_{λ}(x) is:

For real positive values of x and λ, a Bessel function is real, with its curve in the form of decaying oscillations.

For an integer λ = n, the J_{n}(x) is the coefficient for t^{n} in expanding its generating function to:

In particular,

For a Bessel function J_{n}(x), an integral presentation is valid:

For even n, Bessel functions are even; for odd n, they are odd.

Bessel functions of order obtained (λ = m + ½) are expressed in terms of elementary functions:

In terms of Bessel functions are expressed.

Other types of cylindrical functions, which are of great importance for solving problems of heat transfer, include *Neumann functions* (cylindrical functions of the second kind) and *Hankel functions* (cylindrical functions of the third kind).

Heat & Mass Transfer, and Fluids Engineering