Bessel functions are mathematical special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena with circular symmetry or cylindrical symmetry. They are named after the German astronomer and mathematician Friedrich Bessel, who studied them systematically in 1824.

Bessel functions are solutions to a particular type of ordinary differential equation: $${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0,}$$ where ${\displaystyle \alpha }$ is a number that determines the shape of the solution. This number is called the order of the Bessel function and can be any complex number. Although the same equation arises for both ${\displaystyle \alpha }$ and ${\displaystyle -\alpha }$, mathematicians define separate Bessel functions for each to ensure the functions behave smoothly as the order changes.

The most important cases are when ${\displaystyle \alpha }$ is an integer or a half-integer. When ${\displaystyle \alpha }$ is an integer, the resulting Bessel functions are often called cylinder functions or cylindrical harmonics because they naturally arise when solving problems (like Laplace's equation) in cylindrical coordinates. When ${\displaystyle \alpha }$ is a half-integer, the solutions are called spherical Bessel functions and are used in spherical systems, such as in solving the Helmholtz equation in spherical coordinates.

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (α = n); in spherical problems, one obtains half-integer orders (α = n + 1/2). For example:

Bessel functions also appear in other problems, such as signal processing (e.g., see FM audio synthesis, Kaiser window, or Bessel filter).

Because this is a linear differential equation, solutions can be scaled to any amplitude. The amplitudes chosen for the functions originate from the early work in which the functions appeared as solutions to definite integrals rather than solutions to differential equations. Because the differential equation is second-order, there must be two linearly independent solutions: one of the first kind and one of the second kind. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections.The subscript n is typically used in place of ${\displaystyle \alpha }$ when ${\displaystyle \alpha }$ is known to be an integer.

Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by N_n and n_n, respectively, rather than Y_n and y_n.

Bessel functions of the first kind, denoted as J_α(x), are solutions of Bessel's differential equation. For integer or positive α, Bessel functions of the first kind are finite at the origin (x = 0); while for negative non-integer α, Bessel functions of the first kind diverge as x approaches zero. It is possible to define the function by ${\displaystyle x^{\alpha }}$ times a Maclaurin series (note that α need not be an integer, and non-integer powers are not permitted in a Taylor series), which can be found by applying the Frobenius method to Bessel's equation: $${\displaystyle J_{\alpha }(x)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{m!\,\Gamma (m+\alpha +1)}}{\left({\frac {x}{2}}\right)}^{2m+\alpha },}$$ where Γ(z) is the gamma function, a shifted generalization of the factorial function to non-integer values. Some earlier authors define the Bessel function of the first kind differently, essentially without the division by ${\displaystyle 2}$ in ${\displaystyle x/2}$; this definition is not used in this article. The Bessel function of the first kind is an entire function if α is an integer, otherwise it is a multivalued function with singularity at zero. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to ${\displaystyle x^{-{1}/{2}}}$ (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large x. (The series indicates that −J₁(x) is the derivative of J₀(x), much like −sin x is the derivative of cos x; more generally, the derivative of J_n(x) can be expressed in terms of J_{n ± 1}(x) by the identities below.)