# Bessel function

In mathematics, **Bessel functions**, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions *y*(*x*) of Bessel's differential equation:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0}**

for an arbitrary real number α (the *order*). The most common and important special case is where α is an integer, *n*.

Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g., so that the Bessel functions are mostly smooth functions of α).

## Applications

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates, and Bessel functions are therefore especially important for many problems of wave propagation, static potentials, and so on. (For cylindrical problems, one obtains Bessel functions of integer order α = *n*; for spherical problems, one obtains half integer orders α = *n*+½.) For example:

- electromagnetic waves in a cylindrical waveguide
- heat conduction in a cylindrical object.
- modes of vibration of a thin circular (or annular) membrane.

Bessel functions also have useful properties for other problems, such as signal processing (e.g., see FM synthesis or Kaiser window).

## Definitions

Since this is a second-order differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient, and the different variations are described below.

### Bessel functions of the first kind

Bessel functions of the first kind, denoted with *J*_{α}(*x*), are solutions of Bessel's differential equation which are finite at *x* = 0 for α an integer or α non-negative. The specific choice and normalization of *J*_{α} are defined by its properties below; another possibility is to define it by its Taylor series expansion around *x* = 0 (or a more general power series for non-integer α):

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \Gamma(m+\alpha+1)} {\left({\frac{x}{2}}\right)}^{2m+\alpha} }**

The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to 1/√*x* (see also their asymptotic forms, below), although their roots are not generally periodic except asymptotically for large *x*.

Here is the plot of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\alpha (x)}**
for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 0, 1, 2}**
:

If α is not an integer, the functions **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\alpha (x)}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_{-\alpha} (x)}**
are linearly independent and are therefore the two solutions of the differential equation. On the other hand, if the order **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha}**
is an integer, then the following relationship is valid:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_{-\alpha}(x) = (-1)^{\alpha} J_{\alpha}(x)\,}**

This means that they are no longer linearly independent. The second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.

#### Bessel's integrals

Another definition of the Bessel function is possible using an integral equation:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\alpha (x) = \frac{1}{2 \pi} \int_{0}^{2 \pi} \cos (\alpha \tau - x \sin \tau) d\tau.}**

This is the approach that Bessel used, and from this definition he derived several properties of the function. Another integral representation is:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\alpha (x) = \frac{1}{2 \pi} \int_{-\pi}^{\pi} e^{i(\alpha \tau - x \sin \tau)} d\tau}**

#### Relation to hypergeometric series

The Bessel functions can be expressed in terms of the hypergeometric series as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\alpha(z)=\frac{(z/2)^\alpha}{\Gamma(\alpha+1)} \;_0F_1 (\alpha+1; -z^2/4).}**

### Bessel functions of the second kind

These are perhaps the most commonly used forms of the Bessel functions.

The Bessel functions of the second kind, denoted by *Y*_{α}(*x*), are solutions of the Bessel differential equation. They are singular (infinite) at *x* = 0.

*Y*_{α}(*x*) is sometimes also called the **Neumann function**, and is occasionally denoted instead by *N*_{α}(*x*). It is related to *J*_{α}(*x*) by:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_\alpha(x) = \frac{J_\alpha(x) \cos(\alpha\pi) - J_{-\alpha}(x)}{\sin(\alpha\pi)},}**

where the case of integer α is handled by taking the limit.

When α is not an integer, the definition of *Y*_{α} is redundant (as is clear from its definition above). On the other hand, when α is an integer, *Y*_{α} is the second linearly independent solution of Bessel's equation; moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_{-n}(x) = (-1)^n Y_n(x)\,}**

Both *J*_{α}(*x*) and *Y*_{α}(*x*) are holomorphic functions of *x* on the complex plane cut along the negative real axis. When α is an integer, there is no branch point, and the Bessel functions are entire functions of *x*. If *x* is held fixed, then the Bessel functions are entire functions of α.

Here is the plot of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_\alpha (x)}**
for **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = 0, 1, 2}**
:

### Hankel functions

Another important formulation of the two linearly independent solutions to Bessel's equation are the **Hankel functions** *H*_{α}^{(1)}(*x*) and *H*_{α}^{(2)}(*x*), defined by:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_\alpha^{(1)}(x) = J_\alpha(x) + i Y_\alpha(x)}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_\alpha^{(2)}(x) = J_\alpha(x) - i Y_\alpha(x)}**

where *i* is the imaginary unit. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. The Hankel functions express inward- and outward-propagating cylindrical wave solutions of the cylindrical wave equation. They are named for Hermann Hankel.

Using the previous relationships they can be expressed as:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\alpha}^{(1)} (x) = \frac{J_{-\alpha} (x) - e^{-\alpha \pi i} J_\alpha (x)}{i \sin (\alpha \pi)}}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\alpha}^{(2)} (x) = \frac{J_{-\alpha} (x) - e^{\alpha \pi i} J_\alpha (x)}{- i \sin (\alpha \pi)}}**

if α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{-\alpha}^{(1)} (x)= e^{\alpha \pi i} H_{\alpha}^{(1)} (x) }**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{-\alpha}^{(2)} (x)= e^{-\alpha \pi i} H_{\alpha}^{(2)} (x) }**

### Modified Bessel functions

The Bessel functions are valid even for complex arguments *x*, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the **modified Bessel functions** of the first and second kind, and are defined by:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_\alpha(x) = i^{-\alpha} J_\alpha(ix)\,}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_\alpha(x) = \frac{\pi}{2} \frac{I_{-\alpha} (x) - I_\alpha (x)}{\sin (\alpha \pi)} = \frac{\pi}{2} i^{\alpha+1} H_\alpha^{(1)}(ix)}**

These are chosen to be real-valued for real arguments *x*. They are the two linearly independent solutions to the modified Bessel's equation:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - (x^2 + \alpha^2)y = 0.}**

Unlike the ordinary Bessel functions, which are oscillating, *I*_{α} and *K*_{α} are exponentially growing and decaying functions, respectively. Like the ordinary Bessel function *J*_{α}, the function *I*_{α} goes to zero at *x*=0 for α > 0 and is finite at *x*=0 for α=0. Analogously, *K*_{α} diverges at *x*=0.

### Spherical Bessel functions

When solving for separable solutions of Laplace's equation in spherical coordinates, the radial equation has the form:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 \frac{d^2 y}{dx^2} + 2x \frac{dy}{dx} + [x^2 - n(n+1)]y = 0.}**

The two linearly independent solutions to this equation are called the **spherical Bessel functions** *j*_{n} and *y*_{n} (also denoted *n*_{n}), and are related to the ordinary Bessel functions *J*_{n} and *Y*_{n} by:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_n(x) = \sqrt{\frac{\pi}{2x}} J_{n+1/2}(x),}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n(x) = \sqrt{\frac{\pi}{2x}} Y_{n+1/2}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-1/2}(x).}**

The spherical Bessel functions can also be written as:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_n(x) = (-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n\,\frac{\sin x}{x} ,}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n(x) = -(-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n\,\frac{\cos x}{x}.}**

The first spherical Bessel function **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_0(x)}**
is also known as the sinc function. The first few spherical Bessel functions are:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_0(x)=\frac{\sin x} {x}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_1(x)=\frac{\sin x} {x^2}- \frac{\cos x} {x}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j_2(x)=\left(\frac{3} {x^2} - 1 \right)\frac{\sin x}{x} - \frac{3\cos x} {x^2}}**

and

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_0(x)=-j_{-1}(x)=-\,\frac{\cos x} {x}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1(x)=j_{-2}(x)=-\,\frac{\cos x} {x^2}- \frac{\sin x} {x}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_2(x)=-j_{-3}(x)=\left(-\,\frac{3}{x^2}+1 \right)\frac{\cos x}{x}- \frac{3 \sin x} {x^2}.}**

There are also spherical analogues of the Hankel functions:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_n^{(1)}(x) = j_n(x) + i y_n(x)}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_n^{(2)}(x) = j_n(x) - i y_n(x).}**

In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers *n*:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_n^{(1)}(x) = (-i)^{n+1} \frac{e^{ix}}{x} \sum_{m=0}^n \frac{i^m}{m!(2x)^m} \frac{(n+m)!!}{(n-m)!!}}**

and *h*_{n}^{(2)} is the complex-conjugate of this (for real *x*). (!! is the double factorial.) It follows, for example, that *j*_{0}(*x*) = sin(*x*)/*x* and *y*_{0}(*x*) = -cos(*x*)/*x*, and so on.

### Riccati-Bessel functions

Riccati-Bessel functions only slightly differ from spherical Bessel functions:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n(x)=x j_n(x)=\sqrt{\pi x/2}J_{n+1/2}(x)}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_n(x)=-x y_n(x)=-\sqrt{\pi x/2}Y_{n+1/2}(x)}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta_n(x)=x h_n^{(2)}(x)=\sqrt{\pi x/2}H_{n+1/2}^{(2)}(x)=S_n(x)+iC_n(x)}**

They satisfy the differential equation:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 \frac{d^2 y}{dx^2} + [x^2 - n (n+1)] y = 0}**

This differential equation, and the Riccati-Bessel solutions, arises in the problem of scattering of electromagnetic waves by a sphere, known as Mie scattering after the first published solution by Mie (1908). See e.g. Du (2004) for recent developments and references.

Following Debye (1909), the notation **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_n,\chi_n}**
is sometimes used instead of
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_n,C_n}**
.

## Asymptotic forms

The Bessel functions have the following asymptotic forms. For small arguments 0 < *x* << 1, one obtains:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\alpha(x) \rightarrow \frac{1}{\Gamma(\alpha+1)} \left( \frac{x}{2} \right) ^\alpha }**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_\alpha(x) \rightarrow \left\{ \begin{matrix} \frac{2}{\pi} \ln (x/2) & \mbox{if } \alpha=0 \\ \\ -\frac{\Gamma(\alpha)}{\pi} \left( \frac{2}{x} \right) ^\alpha & \mbox{if } \alpha > 0 \end{matrix} \right.}**

where α is non-negative and Γ denotes the gamma function. For large arguments *x* >> 1, they become:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\alpha(x) \rightarrow \sqrt{\frac{2}{\pi x}} \cos \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right)}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_\alpha(x) \rightarrow \sqrt{\frac{2}{\pi x}} \sin \left( x-\frac{\alpha\pi}{2} - \frac{\pi}{4} \right).}**

Asymptotic forms for the other types of Bessel function follow straightforwardly from the above relations. For example, for large *x* >> 1, the modified Bessel functions become:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_\alpha(x) \rightarrow \frac{1}{\sqrt{2\pi x}} e^x,}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_\alpha(x) \rightarrow \sqrt{\frac{\pi}{2x}} e^{-x}.}**

## Properties

For integer order α = *n*, *J*_{n} is often defined via a Laurent series for a generating function:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{(x/2)(t-1/t)} = \sum_{n=-\infty}^\infty J_n(x) t^n,}**

an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.) Another important relation for integer orders is the **Jacobi-Anger identity**:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{iz \cos \phi} = \sum_{n=-\infty}^\infty i^n J_n(z) e^{in\phi},}**

which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone modulated FM signal.

The functions *J*_{α}, *Y*_{α}, *H*_{α}^{(1)}, and *H*_{α}^{(2)} all satisfy the recurrence relations:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{\alpha-1}(x) + Z_{\alpha+1}(x) = \frac{2\alpha}{x} Z_\alpha(x)}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{\alpha-1}(x) - Z_{\alpha+1}(x) = 2\frac{dZ_\alpha}{dx}}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \frac{d}{x dx} \right)^m \left[ x^\alpha Z_{\alpha} (x) \right] = x^{\alpha - m} Z_{\alpha - m} (x)}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \frac{d}{x dx} \right)^m \left[ \frac{Z_\alpha (x)}{x^\alpha} \right] = (-1)^m \frac{Z_{\alpha + m} (x)}{x^{\alpha + m}}}**

where *Z* denotes *J*, *Y*, *H*^{(1)}, or *H*^{(2)}. (These two identities are often combined, e.g. added or subtracted, to yield various other relations.) In this way, for example, one can compute Bessel functions of higher orders (or higher derivatives) given the values at lower orders (or lower derivatives).

Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by *x*, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^1 x J_\alpha(x u_{\alpha,m}) J_\alpha(x u_{\alpha,n}) dx = \frac{\delta_{m,n}}{2} J_{\alpha+1}(u_{\alpha,m})^2,}**

where α > -1, δ_{m,n} is the Kronecker delta, and *u*_{α,m} is the *m*-th zero of *J*_{α}(*x*). This orthogonality relation can then be used to extract the coefficients in the *Fourier-Bessel series*, where a function is expanded in the basis of the functions *J*_{α}(*x* *u*_{α,m}) for fixed α and varying *m*. (An analogous relationship for the spherical Bessel functions follows immediately.)

Another orthogonality relation is the *closure equation*:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^\infty x J_\alpha(ux) J_\alpha(vx) dx = \frac{1}{u} \delta(u - v)}**

for α > -1/2 and where δ is the Dirac delta function. For the spherical Bessel functions the orthogonality relation is:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) dx = \frac{\pi}{2u^2} \delta(u - v)}**

for α > 0.

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_\alpha(x) \frac{dB_\alpha}{dx} - \frac{dA_\alpha}{dx} B_\alpha(x) = \frac{C_\alpha}{x},}**

where *A*_{α} and *B*_{α} are any two solutions of Bessel's equation, and *C*_{α} is a constant independent of *x* (which depends on α and on the particular Bessel functions considered). For example, if *A*_{α} = *J*_{α} and *B*_{α} = *Y*_{α}, then *C*_{α} is 2/π. This also holds for the modified Bessel functions; for example, if *A*_{α} = *I*_{α} and *B*_{α} = *K*_{α}, then *C*_{α} is -1.

(There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.)

## References

- Milton Abramowitz and Irene A. Stegun, eds.,
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*(Dover: New York, 1972)- Chapter 9 Bessel Functions of integer order
- Section 9.1 J, Y (Weber) and H (Hankel)
- Section 9.6 Modified (I and K)
- Section 9.9 Kelvin functions

- Chapter 10 Bessel Functions of fractional order
- Section 10.1 Spherical Bessel Functions (j, y and h)
- Section 10.2 Modified Spherical Bessel functions (I and K)
- Section 10.3 Riccati-Bessel Functions
- Section 10.4 Airy functions

- Chapter 9 Bessel Functions of integer order
- George B. Arfken and Hans J. Weber,
*Mathematical Methods for Physicists*(Harcourt: San Diego, 2001). - Frank Bowman,
*Introduction to Bessel Functions*(Dover: New York, 1958) ISBN 0486604624. - G. N. Watson,
*A Treatise on the Theory of Bessel Functions, Second Edition*(Cambridge University Press, 1966). - G. Mie, \u201cBeiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,\u201d
*Ann. Phys. Leipzig***25**, 377\u2013445 (1908). - Hong Du, "Mie-scattering calculation,"
*Applied Optics***43**(9), 1951-1956 (2004).

de:Besselsche Differentialgleichung fr:Fonction de Bessel it:Funzioni di Bessel ja:ベッセル関数 pl:Funkcje Bessela sl:Besslova funkcija