Bessel function

File:Vibrating drum Bessel function.gif

Bessel functions describe the radial part of vibrations of a circular membrane.

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: $$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}+(x^2-{\alpha }^2)y=0,$$ where $\alpha$ 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 $\alpha$ and $-\alpha$, mathematicians define separate Bessel functions for each to ensure the functions behave smoothly as the order changes.

The most important cases are when $\alpha$ is an integer or a half-integer. When $\alpha$ 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 $\alpha$ 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.

Applications

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 (Template:Math); in spherical problems, one obtains half-integer orders (Template:Math). For example:

• Electromagnetic waves in a cylindrical waveguide
• Pressure amplitudes of inviscid rotational flows
• Heat conduction in a cylindrical object
• Modes of vibration of a thin circular or annular acoustic membrane (such as a drumhead or other membranophone) or thicker plates such as sheet metal (see Kirchhoff–Love plate theory, Mindlin–Reissner plate theory)
• Diffusion problems on a lattice
• Solutions to the Schrödinger equation in spherical and cylindrical coordinates for a free particle
• Position space representation of the Feynman propagator in quantum field theory
• Solving for patterns of acoustical radiation
• Frequency-dependent friction in circular pipelines
• Dynamics of floating bodies
• Angular resolution
• Diffraction from helical objects, including DNA
• Probability density function of product of two normally distributed random variables
• Analyzing of the surface waves generated by microtremors, in geophysics and seismology.

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

Definitions

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 $\alpha$ when $\alpha$ is known to be an integer.

Type	First kind	Second kind
Bessel functions	Template:Mvar	Template:Mvar
Modified Bessel functions	Template:Mvar	Template:Mvar
Hankel functions	Template:Math	Template:Math
Spherical Bessel functions	Template:Mvar	Template:Mvar
Modified spherical Bessel functions	Template:Mvar	Template:Mvar
Spherical Hankel functions	Template:Math	Template:Math

Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by Template:Mvar and Template:Mvar, respectively, rather than Template:Mvar and Template:Mvar.

Bessel functions of the first kind: J_α

File:BesselJ.png

Plot of Bessel function of the first kind, $J_{\alpha }(x)$, for integer orders $\alpha =0,1,2$.

File:Bessel half.png

Plot of Bessel function of the first kind $J_{\alpha }(z)$ with $\alpha =0.5$ in the plane from $-4-4i$ to $4+4i$.

Bessel functions of the first kind, denoted as Template:Math, are solutions of Bessel's differential equation. For integer or positive Template:Mvar, Bessel functions of the first kind are finite at the origin (Template:Math); while for negative non-integer Template:Mvar, Bessel functions of the first kind diverge as Template:Mvar approaches zero. It is possible to define the function by $x^{\alpha }$ times a Maclaurin series (note that Template:Mvar 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: $$J_{\alpha }(x)=\sum _{m=0}^{\mathrm{\infty }}\frac{(-1{)}^m}{m!\mathrm{\Gamma }(m+\alpha +1)}{\left(\frac{x}{2}\right)}^{2m+\alpha },$$ where Template:Math 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 $2$ in $x/2$; this definition is not used in this article. The Bessel function of the first kind is an entire function if Template:Mvar 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 $x^{-1/2}$ (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large Template:Mvar. (The series indicates that Template:Math is the derivative of Template:Math, much like Template:Math is the derivative of Template:Math; more generally, the derivative of Template:Math can be expressed in terms of Template:Math by the identities below.)

For non-integer Template:Mvar, the functions Template:Math and Template:Math are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order Template:Mvar, the following relationship is valid (the gamma function has simple poles at each of the non-positive integers): $$J_{-n}(x)=(-1{)}^n{J}_n(x).$$

This means that the two solutions are no longer linearly independent. In this case, 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, for integer values of Template:Mvar, is possible using an integral representation: $$J_n(x)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}\cos (n\tau -x\sin \tau )d\tau =\frac{1}{\pi }\mathrm{Re}\left(\underset{0}{\overset{\pi }{\int }}{e}^{i(n\tau -x\sin \tau )}d\tau \right),$$ which is also called Hansen-Bessel formula.

This was the approach that Bessel used, and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by one of Schläfli's integrals, for Template:Math: $$J_{\alpha }(x)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}\cos (\alpha \tau -x\sin \tau )d\tau -\frac{\sin (\alpha \pi )}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x\sinh t-\alpha t}dt.$$

Relation to hypergeometric series

The Bessel functions can be expressed in terms of the generalized hypergeometric series as $$J_{\alpha }(x)=\frac{{\left(\frac{x}{2}\right)}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}{}_0{F}_1(\alpha +1;-\frac{x^2}{4}).$$

This expression is related to the development of Bessel functions in terms of the Bessel–Clifford function.

Relation to Laguerre polynomials

In terms of the Laguerre polynomials Template:Mvar and arbitrarily chosen parameter Template:Mvar, the Bessel function can be expressed as $$\frac{J_{\alpha }(x)}{{\left(\frac{x}{2}\right)}^{\alpha }}=\frac{e^{-t}}{\mathrm{\Gamma }(\alpha +1)}\sum _{k=0}^{\mathrm{\infty }}\frac{L_k^{(\alpha )}\left(\frac{x^2}{4t}\right)}{(\genfrac{}{}{0ex}{}{k+\alpha }{k})}\frac{t^k}{k!}.$$

Bessel functions of the second kind: Y_α

File:Besselyn.png

Plot of Bessel function of the second kind, $Y_{\alpha }(x)$, for integer orders $\alpha =0,1,2$

The Bessel functions of the second kind, denoted by Template:Math, occasionally denoted instead by Template:Math, are solutions of the Bessel differential equation that have a singularity at the origin (Template:Math) and are multivalued. These are sometimes called Weber functions, as they were introduced by Template:Harvs, and also Neumann functions after Carl Neumann.

For non-integer Template:Mvar, Template:Math is related to Template:Math by $$Y_{\alpha }(x)=\frac{J_{\alpha }(x)\cos (\alpha \pi )-J_{-\alpha }(x)}{\sin (\alpha \pi )}.$$

In the case of integer order Template:Mvar, the function is defined by taking the limit as a non-integer Template:Mvar tends to Template:Mvar: $$Y_n(x)=\underset{\alpha \to n}{\lim }{Y}_{\alpha }(x).$$

If Template:Mvar is a nonnegative integer, we have the series $$Y_n(z)=-\frac{{\left(\frac{z}{2}\right)}^{-n}}{\pi }\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{\left(\frac{z^2}{4}\right)}^k+\frac{2}{\pi }{J}_n(z)\ln \frac{z}{2}-\frac{{\left(\frac{z}{2}\right)}^n}{\pi }\sum _{k=0}^{\mathrm{\infty }}(\psi (k+1)+\psi (n+k+1))\frac{{(-\frac{z^2}{4})}^k}{k!(n+k)!}$$ where $\psi (z)$ is the digamma function, the logarithmic derivative of the gamma function.

There is also a corresponding integral formula (for Template:Math): $$Y_n(x)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}\sin (x\sin \theta -n\theta )d\theta -\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}(e^{nt}+(-1{)}^n{e}^{-nt})e^{-x\sinh t}dt.$$

In the case where Template:Math: (with $\gamma$ being Euler's constant)$$Y_0\left(x\right)=\frac{4}{{\pi }^2}\underset{0}{\overset{\frac{1}{2}\pi }{\int }}\cos (x\cos \theta )(\gamma +\ln \left(2x{\sin }^{}\theta \right))d\theta .$$

File:Besselyhalf.png

Plot of the Bessel function of the second kind $Y_{\alpha }(z)$ with $\alpha =0.5$ in the complex plane from $-2-2i$ to $2+2i$.

Template:Math is necessary as the second linearly independent solution of the Bessel's equation when Template:Mvar is an integer. But Template:Math has more meaning than that. It can be considered as a "natural" partner of Template:Math. See also the subsection on Hankel functions below.

When Template:Mvar is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid: $$Y_{-n}(x)=(-1{)}^n{Y}_n(x).$$

Both Template:Math and Template:Math are holomorphic functions of Template:Mvar on the complex plane cut along the negative real axis. When Template:Mvar is an integer, the Bessel functions Template:Mvar are entire functions of Template:Mvar. If Template:Mvar is held fixed at a non-zero value, then the Bessel functions are entire functions of Template:Mvar.

The Bessel functions of the second kind when Template:Mvar is an integer is an example of the second kind of solution in Fuchs's theorem.

Hankel functions: HTemplate:Su, HTemplate:Su

File:Plot of the Hankel function of the first kind H n^(1)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg

Plot of the Hankel function of the first kind Template:Math with Template:Math in the complex plane from Template:Math to Template:Math

File:Plot of the Hankel function of the second kind H n^(2)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg

Plot of the Hankel function of the second kind Template:Math with Template:Math in the complex plane from Template:Math to Template:Math

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, Template:Math and Template:Math, defined as $$\begin{array}{rl}{H}_{\alpha }^{(1)}(x)& =J_{\alpha }(x)+i{Y}_{\alpha }(x),\\ H_{\alpha }^{(2)}(x)& =J_{\alpha }(x)-i{Y}_{\alpha }(x),\end{array}$$ where Template:Mvar 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. They are named after Hermann Hankel.

These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of a factor of the form Template:Math. For real $x>0$ where $J_{\alpha }(x)$, $Y_{\alpha }(x)$ are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of the first Hankel function and the real and negative imaginary parts of the second Hankel function. Thus, the above formulae are analogs of Euler's formula, substituting Template:Math, Template:Math for $e^{\pm ix}$ and $J_{\alpha }(x)$, $Y_{\alpha }(x)$ for $\cos (x)$, $\sin (x)$, as explicitly shown in the asymptotic expansion.

The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the sign convention for the frequency).

Using the previous relationships, they can be expressed as $$\begin{array}{rl}{H}_{\alpha }^{(1)}(x)& =\frac{J_{-\alpha }(x)-e^{-\alpha \pi i}{J}_{\alpha }(x)}{i\sin \alpha \pi },\\ H_{\alpha }^{(2)}(x)& =\frac{J_{-\alpha }(x)-e^{\alpha \pi i}{J}_{\alpha }(x)}{-i\sin \alpha \pi }.\end{array}$$

If Template:Mvar is an integer, the limit has to be calculated. The following relationships are valid, whether Template:Mvar is an integer or not: $$\begin{array}{rl}{H}_{-\alpha }^{(1)}(x)& =e^{\alpha \pi i}{H}_{\alpha }^{(1)}(x),\\ H_{-\alpha }^{(2)}(x)& =e^{-\alpha \pi i}{H}_{\alpha }^{(2)}(x).\end{array}$$

In particular, if Template:Math with Template:Mvar a nonnegative integer, the above relations imply directly that $$\begin{array}{rl}{J}_{-(m+\frac{1}{2})}(x)& =(-1{)}^{m+1}{Y}_{m+\frac{1}{2}}(x),\\ Y_{-(m+\frac{1}{2})}(x)& =(-1{)}^m{J}_{m+\frac{1}{2}}(x).\end{array}$$

These are useful in developing the spherical Bessel functions (see below).

The Hankel functions admit the following integral representations for Template:Math: $$\begin{array}{rl}{H}_{\alpha }^{(1)}(x)& =\frac{1}{\pi i}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }+\pi i}{\int }}{e}^{x\sinh t-\alpha t}dt,\\ H_{\alpha }^{(2)}(x)& =-\frac{1}{\pi i}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }-\pi i}{\int }}{e}^{x\sinh t-\alpha t}dt,\end{array}$$ where the integration limits indicate integration along a contour that can be chosen as follows: from Template:Math to 0 along the negative real axis, from 0 to Template:Math along the imaginary axis, and from Template:Math to Template:Math along a contour parallel to the real axis.

Modified Bessel functions: I_α, K_α

The Bessel functions are valid even for complex arguments Template:Mvar, 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 (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as $$\begin{array}{rl}{I}_{\alpha }(x)& =i^{-\alpha }{J}_{\alpha }(ix)=\sum _{m=0}^{\mathrm{\infty }}\frac{1}{m!\mathrm{\Gamma }(m+\alpha +1)}{\left(\frac{x}{2}\right)}^{2m+\alpha },\\ K_{\alpha }(x)& =\frac{\pi }{2}\frac{I_{-\alpha }(x)-I_{\alpha }(x)}{\sin \alpha \pi },\end{array}$$ when Template:Mvar is not an integer. When Template:Mvar is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments Template:Mvar. The series expansion for Template:Math is thus similar to that for Template:Math, but without the alternating Template:Math factor.

$K_{\alpha }$ can be expressed in terms of Hankel functions: $$K_{\alpha }(x)=\{\begin{array}{ll}\frac{\pi }{2}{i}^{\alpha +1}{H}_{\alpha }^{(1)}(ix)& -\pi <\arg x\le \frac{\pi }{2}\\ \frac{\pi }{2}(-i{)}^{\alpha +1}{H}_{\alpha }^{(2)}(-ix)& -\frac{\pi }{2}<\arg x\le \pi \end{array}$$

Using these two formulae the result to $J_{\alpha }^2(z)+Y_{\alpha }^2(z)$, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following $$J_{\alpha }^2(x)+Y_{\alpha }^2(x)=\frac{8}{{\pi }^2}\underset{0}{\overset{\mathrm{\infty }}{\int }}\cosh (2\alpha t)K_0(2x\sinh t)dt,$$

given that the condition Template:Math is met. It can also be shown that $$J_{\alpha }^2(x)+Y_{\alpha }^2(x)=\frac{8\cos (\alpha \pi )}{{\pi }^2}\underset{0}{\overset{\mathrm{\infty }}{\int }}{K}_{2\alpha }(2x\sinh t)dt,$$ only when Template:Math and Template:Math but not when Template:Math.

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if Template:Math): $$\begin{array}{rl}{J}_{\alpha }(iz)& =e^{\frac{\alpha \pi i}{2}}{I}_{\alpha }(z),\\ Y_{\alpha }(iz)& =e^{\frac{(\alpha +1)\pi i}{2}}{I}_{\alpha }(z)-\frac{2}{\pi }{e}^{-\frac{\alpha \pi i}{2}}{K}_{\alpha }(z).\end{array}$$

Template:Math and Template:Math are the two linearly independent solutions to the modified Bessel's equation: $$x^2\frac{d^{2}y}{d{x}^2}+x\frac{dy}{dx}-(x^2+{\alpha }^2)y=0.$$

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, Template:Mvar and Template:Mvar are exponentially growing and decaying functions respectively. Like the ordinary Bessel function Template:Mvar, the function Template:Mvar goes to zero at Template:Math for Template:Math and is finite at Template:Math for Template:Math. Analogously, Template:Mvar diverges at Template:Math with the singularity being of logarithmic type for Template:Mvar, and Template:Math otherwise.

File:Besseli.png Modified Bessel functions of the first kind, $I_{\alpha }(x)$, for $\alpha =0,1,2,3$.

File:Besselk.png Modified Bessel functions of the second kind, $K_{\alpha }(x)$, for $\alpha =0,1,2,3$.

Two integral formulas for the modified Bessel functions are (for Template:Math): $$\begin{array}{rl}{I}_{\alpha }(x)& =\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}{e}^{x\cos \theta }\cos \alpha \theta d\theta -\frac{\sin \alpha \pi }{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x\cosh t-\alpha t}dt,\\ K_{\alpha }(x)& =\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-x\cosh t}\cosh \alpha tdt.\end{array}$$

Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for Template:Math): $$2K_0(\omega )=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{e^{i\omega t}}{\sqrt{t^2+1}}dt.$$

It can be proven by showing equality to the above integral definition for Template:Math. This is done by integrating a closed curve in the first quadrant of the complex plane.

Modified Bessel functions of the second kind may be represented with Bassett's integral $$K_n(xz)=\frac{\mathrm{\Gamma }(n+\frac{1}{2})(2z{)}^n}{\sqrt{\pi }{x}^n}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\cos (xt)dt}{(t^2+z^2{)}^{n+\frac{1}{2}}}.$$

Modified Bessel functions Template:Math and Template:Math can be represented in terms of rapidly convergent integrals $$\begin{array}{rl}{K}_{\frac{1}{3}}(\xi )& =\sqrt{3}\underset{0}{\overset{\mathrm{\infty }}{\int }}\exp (-\xi (1+\frac{4x^2}{3})\sqrt{1+\frac{x^2}{3}})dx,\\ K_{\frac{2}{3}}(\xi )& =\frac{1}{\sqrt{3}}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{3+2x^2}{\sqrt{1+\frac{x^2}{3}}}\exp (-\xi (1+\frac{4x^2}{3})\sqrt{1+\frac{x^2}{3}})dx.\end{array}$$

The modified Bessel function $K_{\frac{1}{2}}(\xi )=(2\xi /\pi {)}^{-1/2}\exp (-\xi )$ is useful to represent the Laplace distribution as an Exponential-scale mixture of normal distributions.

The modified Bessel function of the second kind has also been called by the following names (now rare):

• Basset function after Alfred Barnard Basset
• Modified Bessel function of the third kind
• Modified Hankel function
• Macdonald function after Hector Munro Macdonald

Spherical Bessel functions: j_n, y_n

File:Plot of the spherical Bessel function of the first kind j n(z) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg

Plot of the spherical Bessel function of the first kind Template:Math with Template:Math in the complex plane from Template:Math to Template:Math

File:Plot of the spherical Bessel function of the second kind y n(z) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg

Plot of the spherical Bessel function of the second kind Template:Math with Template:Math in the complex plane from Template:Math to Template:Math

File:Sphericalbesselj.png

Spherical Bessel functions of the first kind $j_{\alpha }(x)$, for $\alpha =0,1,2$.

File:Sphericalbessely.png

Spherical Bessel functions of the second kind $y_{\alpha }(x)$, for $\alpha =0,1,2$.

When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form $$x^2\frac{d^{2}y}{d{x}^2}+2x\frac{dy}{dx}+(x^2-n(n+1))y=0.$$

The two linearly independent solutions to this equation are called the spherical Bessel functions Template:Mvar and Template:Mvar, and are related to the ordinary Bessel functions Template:Mvar and Template:Mvar by $$\begin{array}{rl}{j}_n(x)& =\sqrt{\frac{\pi }{2x}}{J}_{n+\frac{1}{2}}(x),\\ y_n(x)& =\sqrt{\frac{\pi }{2x}}{Y}_{n+\frac{1}{2}}(x)=(-1{)}^{n+1}\sqrt{\frac{\pi }{2x}}{J}_{-n-\frac{1}{2}}(x).\end{array}$$

Template:Mvar is also denoted Template:Mvar or Template:Mvar; some authors call these functions the spherical Neumann functions.

From the relations to the ordinary Bessel functions it is directly seen that: $$\begin{array}{rl}{j}_n(x)& =(-1{)}^n{y}_{-n-1}(x)\\ y_n(x)& =(-1{)}^{n+1}{j}_{-n-1}(x)\end{array}$$

The spherical Bessel functions can also be written as (Template:Va) $$\begin{array}{rl}{j}_n(x)& =(-x{)}^n{\left(\frac{1}{x}\frac{d}{dx}\right)}^n\frac{\sin x}{x},\\ y_n(x)& =-(-x{)}^n{\left(\frac{1}{x}\frac{d}{dx}\right)}^n\frac{\cos x}{x}.\end{array}$$

The zeroth spherical Bessel function Template:Math is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are: $$\begin{array}{rl}{j}_0(x)& =\frac{\sin x}{x}.\\ j_1(x)& =\frac{\sin x}{x^2}-\frac{\cos x}{x},\\ j_2(x)& =(\frac{3}{x^2}-1)\frac{\sin x}{x}-\frac{3\cos x}{x^2},\\ j_3(x)& =(\frac{15}{x^3}-\frac{6}{x})\frac{\sin x}{x}-(\frac{15}{x^2}-1)\frac{\cos x}{x}\end{array}$$ and $$\begin{array}{rl}{y}_0(x)& =-j_{-1}(x)=-\frac{\cos x}{x},\\ y_1(x)& =j_{-2}(x)=-\frac{\cos x}{x^2}-\frac{\sin x}{x},\\ y_2(x)& =-j_{-3}(x)=(-\frac{3}{x^2}+1)\frac{\cos x}{x}-\frac{3\sin x}{x^2},\\ y_3(x)& =j_{-4}(x)=(-\frac{15}{x^3}+\frac{6}{x})\frac{\cos x}{x}-(\frac{15}{x^2}-1)\frac{\sin x}{x}.\end{array}$$

The first few non-zero roots of the first few spherical Bessel functions are:

Non-zero Roots of the Spherical Bessel Function (first kind)

Order	Root 1	Root 2	Root 3	Root 4	Root 5
$j_0$	3.141593	6.283185	9.424778	12.566371	15.707963
$j_1$	4.493409	7.725252	10.904122	14.066194	17.220755
$j_2$	5.763459	9.095011	12.322941	15.514603	18.689036
$j_3$	6.987932	10.417119	13.698023	16.923621	20.121806
$j_4$	8.182561	11.704907	15.039665	18.301256	21.525418

Non-zero Roots of the Spherical Bessel Function (second kind)

Order	Root 1	Root 2	Root 3	Root 4	Root 5
$y_0$	1.570796	4.712389	7.853982	10.995574	14.137167
$y_1$	2.798386	6.121250	9.317866	12.486454	15.644128
$y_2$	3.959528	7.451610	10.715647	13.921686	17.103359
$y_3$	5.088498	8.733710	12.067544	15.315390	18.525210
$y_4$	6.197831	9.982466	13.385287	16.676625	19.916796

Generating function

The spherical Bessel functions have the generating functions $$\begin{array}{rl}\frac{1}{z}\cos \left(\sqrt{z^2-2zt}\right)& =\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{n!}{j}_{n-1}(z),\\ \frac{1}{z}\sin \left(\sqrt{z^2-2zt}\right)& =\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{n!}{y}_{n-1}(z).\end{array}$$

Finite series expansions

In contrast to the whole integer Bessel functions Template:Math, the spherical Bessel functions Template:Math have a finite series expression: $$\begin{array}{rl}{j}_n(x)& =\sqrt{\frac{\pi }{2x}}{J}_{n+\frac{1}{2}}(x)=\\ & =\frac{1}{2x}[e^{ix}\sum _{r=0}^n\frac{i^{r-n-1}(n+r)!}{r!(n-r)!(2x{)}^r}+e^{-ix}\sum _{r=0}^n\frac{(-i{)}^{r-n-1}(n+r)!}{r!(n-r)!(2x{)}^r}]\\ & =\frac{1}{x}[\sin (x-\frac{n\pi }{2})\sum _{r=0}^{\left[\frac{n}{2}\right]}\frac{(-1{)}^r(n+2r)!}{(2r)!(n-2r)!(2x{)}^{2r}}+\cos (x-\frac{n\pi }{2})\sum _{r=0}^{\left[\frac{n-1}{2}\right]}\frac{(-1{)}^r(n+2r+1)!}{(2r+1)!(n-2r-1)!(2x{)}^{2r+1}}]\\ y_n(x)& =(-1{)}^{n+1}{j}_{-n-1}(x)=(-1{)}^{n+1}\frac{\pi }{2x}{J}_{-(n+\frac{1}{2})}(x)=\\ & =\frac{(-1{)}^{n+1}}{2x}[e^{ix}\sum _{r=0}^n\frac{i^{r+n}(n+r)!}{r!(n-r)!(2x{)}^r}+e^{-ix}\sum _{r=0}^n\frac{(-i{)}^{r+n}(n+r)!}{r!(n-r)!(2x{)}^r}]=\\ & =\frac{(-1{)}^{n+1}}{x}[\cos (x+\frac{n\pi }{2})\sum _{r=0}^{\left[\frac{n}{2}\right]}\frac{(-1{)}^r(n+2r)!}{(2r)!(n-2r)!(2x{)}^{2r}}-\sin (x+\frac{n\pi }{2})\sum _{r=0}^{\left[\frac{n-1}{2}\right]}\frac{(-1{)}^r(n+2r+1)!}{(2r+1)!(n-2r-1)!(2x{)}^{2r+1}}]\end{array}$$

Differential relations

In the following, Template:Mvar is any of Template:Mvar, Template:Mvar, Template:Math, Template:Math for Template:Math $$\begin{array}{rl}{\left(\frac{1}{z}\frac{d}{dz}\right)}^m(z^{n+1}{f}_n(z))& =z^{n-m+1}{f}_{n-m}(z),\\ {\left(\frac{1}{z}\frac{d}{dz}\right)}^m(z^{-n}{f}_n(z))& =(-1{)}^m{z}^{-n-m}{f}_{n+m}(z).\end{array}$$

Spherical Hankel functions: hTemplate:Su, hTemplate:Su

File:Plot of the spherical Hankel function of the first kind h n^(1)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg

Plot of the spherical Hankel function of the first kind Template:Math with Template:Math in the complex plane from Template:Math to Template:Math

File:Plot of the spherical Hankel function of the second kind h n^(2)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg

Plot of the spherical Hankel function of the second kind Template:Math with Template:Math in the complex plane from Template:Math to Template:Math

There are also spherical analogues of the Hankel functions: $$\begin{array}{rl}{h}_n^{(1)}(x)& =j_n(x)+i{y}_n(x),\\ h_n^{(2)}(x)& =j_n(x)-i{y}_n(x).\end{array}$$

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 Template:Mvar: $$h_n^{(1)}(x)=(-i{)}^{n+1}\frac{e^{ix}}{x}\sum _{m=0}^n\frac{i^m}{m!(2x{)}^m}\frac{(n+m)!}{(n-m)!},$$ and Template:Math is the complex-conjugate of this (for real Template:Mvar). It follows, for example, that Template:Math and Template:Math, and so on.

The spherical Hankel functions appear in problems involving spherical wave propagation, for example in the multipole expansion of the electromagnetic field.

Riccati–Bessel functions: S_n, C_n, ξ_n, ζ_n

Riccati–Bessel functions only slightly differ from spherical Bessel functions: $$\begin{array}{rl}{S}_n(x)& =x{j}_n(x)=\sqrt{\frac{\pi x}{2}}{J}_{n+\frac{1}{2}}(x)\\ C_n(x)& =-x{y}_n(x)=-\sqrt{\frac{\pi x}{2}}{Y}_{n+\frac{1}{2}}(x)\\ {\xi }_n(x)& =x{h}_n^{(1)}(x)=\sqrt{\frac{\pi x}{2}}{H}_{n+\frac{1}{2}}^{(1)}(x)=S_n(x)-i{C}_n(x)\\ {\zeta }_n(x)& =x{h}_n^{(2)}(x)=\sqrt{\frac{\pi x}{2}}{H}_{n+\frac{1}{2}}^{(2)}(x)=S_n(x)+i{C}_n(x)\end{array}$$

Riccati–Bessel functions Sn complex plot from -2-2i to 2+2i

Riccati–Bessel functions Sn complex plot from −2 − 2i to 2 + 2i

They satisfy the differential equation $$x^2\frac{d^{2}y}{d{x}^2}+(x^2-n(n+1))y=0.$$

For example, this kind of differential equation appears in quantum mechanics while solving the radial component of the Schrödinger equation with hypothetical cylindrical infinite potential barrier. This differential equation, and the Riccati–Bessel solutions, also 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 Template:Mvar, Template:Mvar is sometimes used instead of Template:Mvar, Template:Mvar.

Asymptotic forms

The Bessel functions have the following asymptotic forms. For small arguments $0<z\ll \sqrt{\alpha +1}$, one obtains, when $\alpha$ is not a negative integer: $$J_{\alpha }(z)\sim \frac{1}{\mathrm{\Gamma }(\alpha +1)}{\left(\frac{z}{2}\right)}^{\alpha }.$$

When Template:Mvar is a negative integer, we have $$J_{\alpha }(z)\sim \frac{(-1{)}^{\alpha }}{(-\alpha )!}{\left(\frac{2}{z}\right)}^{\alpha }.$$

For the Bessel function of the second kind we have three cases: $$Y_{\alpha }(z)\sim \{\begin{array}{ll}\frac{2}{\pi }(\ln \left(\frac{z}{2}\right)+\gamma )& \text{if }\alpha =0\\ -\frac{\mathrm{\Gamma }(\alpha )}{\pi }{\left(\frac{2}{z}\right)}^{\alpha }+\frac{1}{\mathrm{\Gamma }(\alpha +1)}{\left(\frac{z}{2}\right)}^{\alpha }\cot (\alpha \pi )& \text{if }\alpha \text{ is a positive integer (one term dominates unless }\alpha \text{ is imaginary)},\\ -\frac{(-1{)}^{\alpha }\mathrm{\Gamma }(-\alpha )}{\pi }{\left(\frac{z}{2}\right)}^{\alpha }& \text{if }\alpha \text{ is a negative integer,}\end{array}$$ where Template:Mvar is the Euler–Mascheroni constant (0.5772...).

For large real arguments Template:Math, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless Template:Mvar is half-integer) because they have zeros all the way out to infinity, which would have to be matched exactly by any asymptotic expansion. However, for a given value of Template:Math one can write an equation containing a term of order Template:Math: $$\begin{array}{rlrl}{J}_{\alpha }(z)& =\sqrt{\frac{2}{\pi z}}(\cos (z-\frac{\alpha \pi }{2}-\frac{\pi }{4})+e^{|\mathrm{Im}(z)|}{𝒪}(|z{|}^{-1}))& & \text{for }|\arg z|<\pi ,\\ Y_{\alpha }(z)& =\sqrt{\frac{2}{\pi z}}(\sin (z-\frac{\alpha \pi }{2}-\frac{\pi }{4})+e^{|\mathrm{Im}(z)|}{𝒪}(|z{|}^{-1}))& & \text{for }|\arg z|<\pi .\end{array}$$

(For Template:Math, the last terms in these formulas drop out completely; see the spherical Bessel functions above.)

The asymptotic forms for the Hankel functions are: $$\begin{array}{rlrl}{H}_{\alpha }^{(1)}(z)& \sim \sqrt{\frac{2}{\pi z}}{e}^{i(z-\frac{\alpha \pi }{2}-\frac{\pi }{4})}& & \text{for }-\pi <\arg z<2\pi ,\\ H_{\alpha }^{(2)}(z)& \sim \sqrt{\frac{2}{\pi z}}{e}^{-i(z-\frac{\alpha \pi }{2}-\frac{\pi }{4})}& & \text{for }-2\pi <\arg z<\pi .\end{array}$$

These can be extended to other values of Template:Math using equations relating Template:Math and Template:Math to Template:Math and Template:Math.

It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, Template:Math is not asymptotic to the average of these two asymptotic forms when Template:Mvar is negative (because one or the other will not be correct there, depending on the Template:Math used). But the asymptotic forms for the Hankel functions permit us to write asymptotic forms for the Bessel functions of first and second kinds for complex (non-real) Template:Mvar so long as Template:Math goes to infinity at a constant phase angle Template:Math (using the square root having positive real part): $$\begin{array}{rlrl}{J}_{\alpha }(z)& \sim \frac{1}{\sqrt{2\pi z}}{e}^{i(z-\frac{\alpha \pi }{2}-\frac{\pi }{4})}& & \text{for }-\pi <\arg z<0,\\ J_{\alpha }(z)& \sim \frac{1}{\sqrt{2\pi z}}{e}^{-i(z-\frac{\alpha \pi }{2}-\frac{\pi }{4})}& & \text{for }0<\arg z<\pi ,\\ Y_{\alpha }(z)& \sim -i\frac{1}{\sqrt{2\pi z}}{e}^{i(z-\frac{\alpha \pi }{2}-\frac{\pi }{4})}& & \text{for }-\pi <\arg z<0,\\ Y_{\alpha }(z)& \sim i\frac{1}{\sqrt{2\pi z}}{e}^{-i(z-\frac{\alpha \pi }{2}-\frac{\pi }{4})}& & \text{for }0<\arg z<\pi .\end{array}$$

For the modified Bessel functions, Hankel developed asymptotic expansions as well: $$\begin{array}{rlrl}{I}_{\alpha }(z)& \sim \frac{e^z}{\sqrt{2\pi z}}(1-\frac{4{\alpha }^2-1}{8z}+\frac{(4{\alpha }^2-1)(4{\alpha }^2-9)}{2!(8z{)}^2}-\frac{(4{\alpha }^2-1)(4{\alpha }^2-9)(4{\alpha }^2-25)}{3!(8z{)}^3}+\cdots )& & \text{for }|\arg z|<\frac{\pi }{2},\\ K_{\alpha }(z)& \sim \sqrt{\frac{\pi }{2z}}{e}^{-z}(1+\frac{4{\alpha }^2-1}{8z}+\frac{(4{\alpha }^2-1)(4{\alpha }^2-9)}{2!(8z{)}^2}+\frac{(4{\alpha }^2-1)(4{\alpha }^2-9)(4{\alpha }^2-25)}{3!(8z{)}^3}+\cdots )& & \text{for }|\arg z|<\frac{3\pi }{2}.\end{array}$$

There is also the asymptotic form (for large real $z$) $$\begin{array}{r}{I}_{\alpha }(z)=\frac{1}{\sqrt{2\pi z}\sqrt[4]{1+\frac{{\alpha }^2}{z^2}}}\exp (-\alpha \mathrm{arcsinh}\left(\frac{\alpha }{z}\right)+z\sqrt{1+\frac{{\alpha }^2}{z^2}})(1+{𝒪}\left(\frac{1}{z\sqrt{1+\frac{{\alpha }^2}{z^2}}}\right)).\end{array}$$

When Template:Math, all the terms except the first vanish, and we have $$\begin{array}{rlrl}{I}_{1/2}(z)& =\sqrt{\frac{2}{\pi }}\frac{\sinh (z)}{\sqrt{z}}\sim \frac{e^z}{\sqrt{2\pi z}}& & \text{for }|\arg z|<\frac{\pi }{2},\\ K_{1/2}(z)& =\sqrt{\frac{\pi }{2}}\frac{e^{-z}}{\sqrt{z}}.\end{array}$$

For small arguments $0<|z|\ll \sqrt{\alpha +1}$, we have $$\begin{array}{rl}{I}_{\alpha }(z)& \sim \frac{1}{\mathrm{\Gamma }(\alpha +1)}{\left(\frac{z}{2}\right)}^{\alpha },\\ K_{\alpha }(z)& \sim \{\begin{array}{ll}-\ln \left(\frac{z}{2}\right)-\gamma & \text{if }\alpha =0\\ \frac{\mathrm{\Gamma }(\alpha )}{2}{\left(\frac{2}{z}\right)}^{\alpha }& \text{if }\alpha >0\end{array}\end{array}$$

Properties

For integer order Template:Math, Template:Mvar is often defined via a Laurent series for a generating function: $$e^{\frac{x}{2}(t-\frac{1}{t})}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_n(x)t^n$$ an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.)

Infinite series of Bessel functions in the form ${\sum }_{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{N\nu +p}(x)$ where $\nu ,p\in \mathbb{\mathbb{Z}},N\in {\mathbb{\mathbb{Z}}}^{+}$ arise in many physical systems and are defined in closed form by the Sung series. For example, when N = 3: ${\sum }_{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{3\nu +p}(x)=\frac{1}{3}[1+2\cos (x\sqrt{3}/2-2\pi p/3)]$. More generally, the Sung series and the alternating Sung series are written as: $$\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{N\nu +p}(x)=\frac{1}{N}\sum _{q=0}^{N-1}{e}^{ix\sin 2\pi q/N}{e}^{-i2\pi pq/N}$$ $$\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^{\nu }{J}_{N\nu +p}(x)=\frac{1}{N}\sum _{q=0}^{N-1}{e}^{ix\sin (2q+1)\pi /N}{e}^{-i(2q+1)\pi p/N}$$

A series expansion using Bessel functions (Kapteyn series) is $$\frac{1}{1-z}=1+2\sum _{n=1}^{\mathrm{\infty }}{J}_n(nz).$$

Another important relation for integer orders is the Jacobi–Anger expansion: $$e^{iz\cos \varphi }=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{i}^n{J}_n(z)e^{in\varphi }$$ and $$e^{\pm iz\sin \varphi }=J_0(z)+2\sum _{n=1}^{\mathrm{\infty }}{J}_{2n}(z)\cos (2n\varphi )\pm 2i\sum _{n=0}^{\mathrm{\infty }}{J}_{2n+1}(z)\sin ((2n+1)\varphi )$$ 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.

More generally, a series $$f(z)=a_0^{\nu }{J}_{\nu }(z)+2\cdot \sum _{k=1}^{\mathrm{\infty }}{a}_k^{\nu }{J}_{\nu +k}(z)$$ is called Neumann expansion of Template:Mvar. The coefficients for Template:Math have the explicit form $$a_k^0=\frac{1}{2\pi i}\underset{|z|=c}{\int }f(z)O_k(z)dz$$ where Template:Mvar is Neumann's polynomial.

Selected functions admit the special representation $$f(z)=\sum _{k=0}^{\mathrm{\infty }}{a}_k^{\nu }{J}_{\nu +2k}(z)$$ with $$a_k^{\nu }=2(\nu +2k)\underset{0}{\overset{\mathrm{\infty }}{\int }}f(z)\frac{J_{\nu +2k}(z)}{z}dz$$ due to the orthogonality relation $$\underset{0}{\overset{\mathrm{\infty }}{\int }}{J}_{\alpha }(z)J_{\beta }(z)\frac{dz}{z}=\frac{2}{\pi }\frac{\sin (\frac{\pi }{2}(\alpha -\beta ))}{{\alpha }^2-{\beta }^2}$$

More generally, if Template:Mvar has a branch-point near the origin of such a nature that $$f(z)=\sum _{k=0}{a}_k{J}_{\nu +k}(z)$$ then $${ℒ}\left\{\sum _{k=0}{a}_k{J}_{\nu +k}\right\}(s)=\frac{1}{\sqrt{1+s^2}}\sum _{k=0}\frac{a_k}{{(s+\sqrt{1+s^2})}^{\nu +k}}$$ or $$\sum _{k=0}{a}_k{\xi }^{\nu +k}=\frac{1+{\xi }^2}{2\xi }{ℒ}\{f\}\left(\frac{1-{\xi }^2}{2\xi }\right)$$ where ${ℒ}\{f\}$ is the Laplace transform of Template:Mvar.

Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula: $$\begin{array}{rl}{J}_{\nu }(z)& =\frac{{\left(\frac{z}{2}\right)}^{\nu }}{\mathrm{\Gamma }(\nu +\frac{1}{2})\sqrt{\pi }}\underset{-1}{\overset{1}{\int }}{e}^{izs}{(1-s^2)}^{\nu -\frac{1}{2}}ds\\ [5px]& =\frac{2}{{\left(\frac{z}{2}\right)}^{\nu }\cdot \sqrt{\pi }\cdot \mathrm{\Gamma }(\frac{1}{2}-\nu )}\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{\sin zu}{{(u^2-1)}^{\nu +\frac{1}{2}}}du\end{array}$$ where Template:Math and Template:Math. This formula is useful especially when working with Fourier transforms.

Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by Template:Mvar, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that: $$\underset{0}{\overset{1}{\int }}x{J}_{\alpha }\left(x{u}_{\alpha ,m}\right)J_{\alpha }\left(x{u}_{\alpha ,n}\right)dx=\frac{{\delta }_{m,n}}{2}{\left[J_{\alpha +1}\left(u_{\alpha ,m}\right)\right]}^2=\frac{{\delta }_{m,n}}{2}{\left[J_{{\alpha }^{\prime }}\left(u_{\alpha ,m}\right)\right]}^2$$ where Template:Math, Template:Math is the Kronecker delta, and Template:Math is the Template:Mvarth zero of Template:Math. 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 Template:Math for fixed Template:Mvar and varying Template:Mvar.

An analogous relationship for the spherical Bessel functions follows immediately: $$\underset{0}{\overset{1}{\int }}{x}^2{j}_{\alpha }\left(x{u}_{\alpha ,m}\right)j_{\alpha }\left(x{u}_{\alpha ,n}\right)dx=\frac{{\delta }_{m,n}}{2}{\left[j_{\alpha +1}\left(u_{\alpha ,m}\right)\right]}^2$$

If one defines a boxcar function of Template:Mvar that depends on a small parameter Template:Mvar as: $$f_{\epsilon }(x)=\frac{1}{\epsilon }\mathrm{rect}\left(\frac{x-1}{\epsilon }\right)$$ (where Template:Math is the rectangle function) then the Hankel transform of it (of any given order Template:Math), Template:Math, approaches Template:Math as Template:Mvar approaches zero, for any given Template:Mvar. Conversely, the Hankel transform (of the same order) of Template:Math is Template:Math: $$\underset{0}{\overset{\mathrm{\infty }}{\int }}k{J}_{\alpha }(kx)g_{\epsilon }(k)dk=f_{\epsilon }(x)$$ which is zero everywhere except near 1. As Template:Mvar approaches zero, the right-hand side approaches Template:Math, where Template:Mvar is the Dirac delta function. This admits the limit (in the distributional sense): $$\underset{0}{\overset{\mathrm{\infty }}{\int }}k{J}_{\alpha }(kx)J_{\alpha }(k)dk=\delta (x-1)$$

A change of variables then yields the closure equation: $$\underset{0}{\overset{\mathrm{\infty }}{\int }}x{J}_{\alpha }(ux)J_{\alpha }(vx)dx=\frac{1}{u}\delta (u-v)$$ for Template:Math. For the spherical Bessel functions the orthogonality relation is: $$\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^2{j}_{\alpha }(ux)j_{\alpha }(vx)dx=\frac{\pi }{2uv}\delta (u-v)$$ for Template:Math.

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions: $$A_{\alpha }(x)\frac{d{B}_{\alpha }}{dx}-\frac{d{A}_{\alpha }}{dx}{B}_{\alpha }(x)=\frac{C_{\alpha }}{x}$$ where Template:Mvar and Template:Mvar are any two solutions of Bessel's equation, and Template:Mvar is a constant independent of Template:Mvar (which depends on α and on the particular Bessel functions considered). In particular, $$J_{\alpha }(x)\frac{d{Y}_{\alpha }}{dx}-\frac{d{J}_{\alpha }}{dx}{Y}_{\alpha }(x)=\frac{2}{\pi x}$$ and $$I_{\alpha }(x)\frac{d{K}_{\alpha }}{dx}-\frac{d{I}_{\alpha }}{dx}{K}_{\alpha }(x)=-\frac{1}{x},$$ for Template:Math.

For Template:Math, the even entire function of genus 1, Template:Math, has only real zeros. Let $$0<j_{\alpha ,1}<j_{\alpha ,2}<\cdots <j_{\alpha ,n}<\cdots$$ be all its positive zeros, then $$J_{\alpha }(z)=\frac{{\left(\frac{z}{2}\right)}^{\alpha }}{\mathrm{\Gamma }(\alpha +1)}\prod _{n=1}^{\mathrm{\infty }}(1-\frac{z^2}{j_{\alpha ,n}^2})$$

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

Recurrence relations

The functions Template:Mvar, Template:Mvar, Template:Math, and Template:Math all satisfy the recurrence relations $$\frac{2\alpha }{x}{Z}_{\alpha }(x)=Z_{\alpha -1}(x)+Z_{\alpha +1}(x)$$ and $$2\frac{d{Z}_{\alpha }(x)}{dx}=Z_{\alpha -1}(x)-Z_{\alpha +1}(x),$$ where Template:Mvar denotes Template:Mvar, Template:Mvar, Template:Math, or Template:Math. 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). In particular, it follows that $$\begin{array}{rl}{\left(\frac{1}{x}\frac{d}{dx}\right)}^m[x^{\alpha }{Z}_{\alpha }(x)]& =x^{\alpha -m}{Z}_{\alpha -m}(x),\\ {\left(\frac{1}{x}\frac{d}{dx}\right)}^m\left[\frac{Z_{\alpha }(x)}{x^{\alpha }}\right]& =(-1{)}^m\frac{Z_{\alpha +m}(x)}{x^{\alpha +m}}.\end{array}$$

Using the previous relations one can arrive to similar relations for the Spherical Bessel functions:

$$\frac{2\alpha +1}{x}{j}_{\alpha }(x)=j_{\alpha -1}+j_{\alpha +1}$$

and

$$\frac{d{j}_{\alpha }(x)}{dx}=j_{\alpha -1}-\frac{\alpha +1}{x}{j}_{\alpha }$$

Modified Bessel functions follow similar relations: $$e^{\left(\frac{x}{2}\right)(t+\frac{1}{t})}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{I}_n(x)t^n$$ and $$e^{z\cos \theta }=I_0(z)+2\sum _{n=1}^{\mathrm{\infty }}{I}_n(z)\cos n\theta$$ and $$\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{e}^{z\cos (m\theta )+y\cos \theta }d\theta =I_0(z)I_0(y)+2\sum _{n=1}^{\mathrm{\infty }}{I}_n(z)I_{mn}(y).$$

The recurrence relation reads $$\begin{array}{rl}{C}_{\alpha -1}(x)-C_{\alpha +1}(x)& =\frac{2\alpha }{x}{C}_{\alpha }(x),\\ C_{\alpha -1}(x)+C_{\alpha +1}(x)& =2\frac{d}{dx}{C}_{\alpha }(x),\end{array}$$ where Template:Mvar denotes Template:Mvar or Template:Math. These recurrence relations are useful for discrete diffusion problems.

Transcendence

In 1929, Carl Ludwig Siegel proved that Template:Math, Template:Math, and the logarithmic derivative Template:Math are transcendental numbers when ν is rational and x is algebraic and nonzero. The same proof also implies that $\mathrm{\Gamma }(v+1)(2/x{)}^v{J}_v(x)$ is transcendental under the same assumptions.

Sums with Bessel functions

The product of two Bessel functions admits the following sum: $$\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{\nu }(x)J_{n-\nu }(y)=J_n(x+y),$$ $$\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{\nu }(x)J_{\nu +n}(y)=J_n(y-x).$$ From these equalities it follows that $$\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{\nu }(x)J_{\nu +n}(x)={\delta }_{n,0}$$ and as a consequence $$\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{\nu }^2(x)=1.$$

These sums can be extended to include a term multiplier that is a polynomial function of the index. For example, $$\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}\nu J_{\nu }(x)J_{\nu +n}(x)=\frac{x}{2}({\delta }_{n,1}+{\delta }_{n,-1}),$$ $$\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}\nu J_{\nu }^2(x)=0,$$ $$\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{\nu }^2{J}_{\nu }(x)J_{\nu +n}(x)=\frac{x}{2}({\delta }_{n,-1}-{\delta }_{n,1})+\frac{x^2}{4}({\delta }_{n,-2}+2{\delta }_{n,0}+{\delta }_{n,2}),$$ $$\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{\nu }^2{J}_{\nu }^2(x)=\frac{x^2}{2}.$$

Multiplication theorem

The Bessel functions obey a multiplication theorem $${\lambda }^{-\nu }{J}_{\nu }(\lambda z)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}{\left(\frac{(1-{\lambda }^2)z}{2}\right)}^n{J}_{\nu +n}(z),$$ where Template:Mvar and Template:Mvar may be taken as arbitrary complex numbers. For Template:Math, the above expression also holds if Template:Mvar is replaced by Template:Mvar. The analogous identities for modified Bessel functions and Template:Math are $${\lambda }^{-\nu }{I}_{\nu }(\lambda z)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}{\left(\frac{({\lambda }^2-1)z}{2}\right)}^n{I}_{\nu +n}(z)$$ and $${\lambda }^{-\nu }{K}_{\nu }(\lambda z)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{n!}{\left(\frac{({\lambda }^2-1)z}{2}\right)}^n{K}_{\nu +n}(z).$$

Zeros of the Bessel function

Bourget's hypothesis

Bessel himself originally proved that for nonnegative integers Template:Mvar, the equation Template:Math has an infinite number of solutions in Template:Mvar. When the functions Template:Math are plotted on the same graph, though, none of the zeros seem to coincide for different values of Template:Mvar except for the zero at Template:Math. This phenomenon is known as Bourget's hypothesis after the 19th-century French mathematician who studied Bessel functions. Specifically it states that for any integers Template:Math and Template:Math, the functions Template:Math and Template:Math have no common zeros other than the one at Template:Math. The hypothesis was proved by Carl Ludwig Siegel in 1929.

Transcendence

Siegel proved in 1929 that when ν is rational, all nonzero roots of Template:Math and Template:Math are transcendental, as are all the roots of Template:Math. It is also known that all roots of the higher derivatives $J_{\nu }^{(n)}(x)$ for Template:Math are transcendental, except for the special values $J_1^{(3)}(\pm \sqrt{3})=0$ and $J_0^{(4)}(\pm \sqrt{3})=0$.

Numerical approaches

For numerical studies about the zeros of the Bessel function, see Template:Harvtxt, Template:Harvtxt and Template:Harvtxt.

Numerical values

The first zeros in J₀ (i.e., j_0,1, j_0,2 and j_0,3) occur at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.

History

Waves and elasticity problems

The first appearance of a Bessel function appears in the work of Daniel Bernoulli in 1732, while working on the analysis of a vibrating string, a problem that was tackled before by his father Johann Bernoulli. Daniel considered a flexible chain suspended from a fixed point above and free at its lower end. The solution of the differential equation led to the introduction of a function that is now considered $J_0(x)$. Bernoulli also developed a method to find the zeros of the function.

Leonhard Euler in 1736, found a link between other functions (now known as Laguerre polynomials) and Bernoulli's solution. Euler also introduced a non-uniform chain that lead to the introduction of functions now related to modified Bessel functions $I_n(x)$.

In the middle of the eighteen century, Jean le Rond d'Alembert had found a formula to solve the wave equation. By 1771 there was dispute between Bernoulli, Euler, d'Alembert and Joseph-Louis Lagrange on the nature of the solutions of vibrating strings.

Euler worked in 1778 on buckling, introducing the concept of Euler's critical load. To solve the problem he introduced the series for $J_{\pm 1/3}(x)$. Euler also worked out the solutions of vibrating 2D membranes in cylindrical coordinates in 1780. In order to solve his differential equation he introduced a power series associated to $J_n(x)$, for integer n.

During the end of the 19th century Lagrange, Pierre-Simon Laplace and Marc-Antoine Parseval also found equivalents to the Bessel functions. Parseval for example found an integral representation of $J_0(x)$ using cosine.

At the beginning of the 1800s, Joseph Fourier used $J_0(x)$ to solve the heat equation in a problem with cylindrical symmetry. Fourier won a prize of the French Academy of Sciences for this work in 1811. But most of the details of his work, including the use of a Fourier series, remained unpublished until 1822. Poisson in rivalry with Fourier, extended Fourier's work in 1823, introducing new properties of Bessel functions including Bessel functions of half-integer order (now known as spherical Bessel functions).

Astronomical problems

In 1770, Lagrange introduced the series expansion of Bessel functions to solve Kepler's equation, a transcendental equation in astronomy. Friedrich Wilhelm Bessel had seen Lagrange's solution but found it difficult to handle. In 1813 in a letter to Carl Friedrich Gauss, Bessel simplified the calculation using trigonometric functions. Bessel published his work in 1819, independently introducing the method of Fourier series unaware of the work of Fourier which was published later. In 1824, Bessel carried out a systematic investigation of the functions, which earned the functions his name. In older literature the functions were called cylindrical functions or even Bessel–Fourier functions.

See also

• Anger function
• Bessel polynomials
• Bessel–Clifford function
• Bessel–Maitland function
• Fourier–Bessel series
• [[Hahn–Exton q-Bessel function|Hahn–Exton Template:Mvar-Bessel function]]
• Hankel transform
• Incomplete Bessel functions
• [[Jackson q-Bessel function|Jackson Template:Mvar-Bessel function]]
• Kelvin functions
• Kontorovich–Lebedev transform
• Lentz's algorithm
• Lerche–Newberger sum rule
• Lommel function
• Lommel polynomial
• Neumann polynomial
• Riccati-Bessel Functions
• Schlömilch's series
• Sonine formula
• Struve function
• Vibrations of a circular membrane
• Weber function (defined at Anger function)
• Gauss' circle problem

References

• Template:Abramowitz Stegun ref2
• Arfken, George B. and Hans J. Weber, Mathematical Methods for Physicists, 6th edition (Harcourt: San Diego, 2005). Template:ISBN.
• Reproduced as pages 84 to 109 in English translation of the text.
• Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). Template:ISBN.
• Template:Dlmf.
• .
• B Spain, M. G. Smith, Functions of mathematical physics, Van Nostrand Reinhold Company, London, 1970. Chapter 9 deals with Bessel functions.
• N. M. Temme, Special Functions. An Introduction to the Classical Functions of Mathematical Physics, John Wiley and Sons, Inc., New York, 1996. Template:ISBN. Chapter 9 deals with Bessel functions.
• Watson, G. N., A Treatise on the Theory of Bessel Functions, Second Edition, (1995) Cambridge University Press. Template:ISBN.
• .

External links

• Template:Main other.
• Template:Main other.
• Template:Main other.
• Wolfram function pages on Bessel J and Y functions, and modified Bessel I and K functions. Pages include formulas, function evaluators, and plotting calculators.
• Bessel functions J_ν, Y_ν, I_ν and K_ν in Librow Function handbook.
• F. W. J. Olver, L. C. Maximon, Bessel Functions (chapter 10 of the Digital Library of Mathematical Functions).