Bessel function



Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions $y(x)$ of Bessel's differential equation $$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0$$ for an arbitrary complex number $$\alpha$$, which represents the order of the Bessel function. Although $$\alpha$$ and $$-\alpha$$ produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of $$\alpha$$.

The most important cases are when $$\alpha$$ is an integer or half-integer. Bessel functions for integer $$\alpha$$ are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer $$\alpha$$ are obtained when solving the Helmholtz equation in spherical coordinates.

Applications of Bessel functions
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:


 * 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 radial 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. 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.

Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by $J_{α}$ and $Y_{α}$, respectively, rather than $I_{α}$ and $K_{α}$.

Bessel functions of the first kind: $H(1) α = J_{α} + iY_{α}$


Bessel functions of the first kind, denoted as $H(2) α = J_{α} − iY_{α}$, are solutions of Bessel's differential equation. For integer or positive $j_{n}$, Bessel functions of the first kind are finite at the origin ($h(1) n = j_{n} + iy_{n}$); while for negative non-integer $y_{n}$, Bessel functions of the first kind diverge as $N_{n}$ approaches zero. It is possible to define the function by $$x^\alpha$$ times a Maclaurin series (note that $n_{n}$ 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: $$ J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!\, \Gamma(m+\alpha+1)} {\left(\frac{x}{2}\right)}^{2m + \alpha},$$ where $h(2) n = j_{n} − iy_{n}$ is the gamma function, a shifted generalization of the factorial function to non-integer values. The Bessel function of the first kind is an entire function if $Y_{n}$ 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 $y_{n}$. (The series indicates that $J_{α}$ is the derivative of $J_{n}(z)$, much like $n = 0.5$ is the derivative of $−2 − 2i$; more generally, the derivative of $2 + 2i$ can be expressed in terms of $J_{α}(x)$ by the identities below.)

For non-integer $α$, the functions $α = 0, 1, 2$ and $J_{α}(x)$ are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order $α$, 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 $x$, is possible using an integral representation: $$J_n(x) = \frac{1}{\pi} \int_0^\pi \cos (n \tau - x \sin \tau) \,d\tau = \frac{1}{2\pi} \int_{-\pi}^\pi e^{i(n \tau-x \sin \tau )} \,d\tau,$$ 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 $x = 0$: $$J_\alpha(x) = \frac{1}{\pi} \int_0^\pi \cos(\alpha\tau - x \sin\tau)\,d\tau - \frac{\sin(\alpha\pi)}{\pi} \int_0^\infty e^{-x \sinh t - \alpha t} \, dt. $$

Relation to hypergeometric series
The Bessel functions can be expressed in terms of the generalized hypergeometric series as $$J_\alpha(x) = \frac{\left(\frac{x}{2}\right)^\alpha}{\Gamma(\alpha+1)} \;_0F_1 \left(\alpha+1; -\frac{x^2}{4}\right).$$

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 $α$ and arbitrarily chosen parameter $α$, the Bessel function can be expressed as $$\frac{J_\alpha(x)}{\left( \frac{x}{2}\right)^\alpha} = \frac{e^{-t}}{\Gamma(\alpha+1)} \sum_{k=0}^\infty \frac{L_k^{(\alpha)}\left( \frac{x^2}{4 t}\right)}{\binom{k+\alpha}{k}} \frac{t^k}{k!}.$$

Bessel functions of the second kind: $Γ(z)$
The Bessel functions of the second kind, denoted by $−J_{1}(x)$, occasionally denoted instead by $J_{0}(x)$, are solutions of the Bessel differential equation that have a singularity at the origin ($−sin x$) and are multivalued. These are sometimes called Weber functions, as they were introduced by, and also Neumann functions after Carl Neumann.

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

In the case of integer order $α$, the function is defined by taking the limit as a non-integer $n$ tends to $n$: $$Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x).$$

If $L_{k}$ is a nonnegative integer, we have the series $$Y_n(z) =-\frac{\left(\frac{z}{2}\right)^{-n}}{\pi}\sum_{k=0}^{n-1} \frac{(n-k-1)!}{k!}\left(\frac{z^2}{4}\right)^k +\frac{2}{\pi} J_n(z) \ln \frac{z}{2} -\frac{\left(\frac{z}{2}\right)^n}{\pi}\sum_{k=0}^\infty (\psi(k+1)+\psi(n+k+1)) \frac{\left(-\frac{z^2}{4}\right)^k}{k!(n+k)!}$$

where $$\psi(z)$$ is the digamma function, the logarithmic derivative of the gamma function.

There is also a corresponding integral formula (for $J_{n ± 1}(x)$): $$Y_n(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - n\theta) \, d\theta -\frac{1}{\pi} \int_0^\infty \left(e^{nt} + (-1)^n e^{-nt} \right) e^{-x \sinh t} \, dt.$$

In the case where $J_{α}(x)$, $$Y_{0}\left(x\right)=\frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos\left(x\cos\theta\right)\left(e+\ln\left(2x\sin^2\theta\right)\right)\, d\theta.$$

$J_{−α}(x)$ is necessary as the second linearly independent solution of the Bessel's equation when $t$ is an integer. But $Re(x) > 0$ has more meaning than that. It can be considered as a "natural" partner of $Y_{α}$. See also the subsection on Hankel functions below.

When $α$ 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 $Y_{n}(z)$ and $n = 0.5$ are holomorphic functions of $n$ on the complex plane cut along the negative real axis. When $α$ is an integer, the Bessel functions $n$ are entire functions of $n$. If $α$ is held fixed at a non-zero value, then the Bessel functions are entire functions of $α$.

The Bessel functions of the second kind when $x$ is an integer is an example of the second kind of solution in Fuchs's theorem.

Hankel functions: $−2 − 2i$, $2 + 2i$


Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, $Y_{α}(x)$ and $N_{α}(x)$, defined as $$\begin{align} H_\alpha^{(1)}(x) &= J_\alpha(x) + iY_\alpha(x), \\[5pt] H_\alpha^{(2)}(x) &= J_\alpha(x) - iY_\alpha(x), \end{align}$$

where $α$ 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 $x = 0$. 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 $Y_{α}(x)$, $J_{α}(x)$ for $$e^{\pm i x}$$ 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 $$\begin{align} H_\alpha^{(1)}(x) &= \frac{J_{-\alpha}(x) - e^{-\alpha \pi i} J_\alpha(x)}{i \sin \alpha\pi}, \\[5pt] H_\alpha^{(2)}(x) &= \frac{J_{-\alpha}(x) - e^{\alpha \pi i} J_\alpha(x)}{- i \sin \alpha\pi}. \end{align}$$

If $J$ is an integer, the limit has to be calculated. The following relationships are valid, whether $x$ is an integer or not: $$\begin{align} H_{-\alpha}^{(1)}(x) &= e^{\alpha \pi i} H_\alpha^{(1)} (x), \\[6mu] H_{-\alpha}^{(2)}(x) &= e^{-\alpha \pi i} H_\alpha^{(2)} (x). \end{align}$$

In particular, if $Y_{α}(x)$ with $x$ a nonnegative integer, the above relations imply directly that $$\begin{align} J_{-(m+\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\frac{1}{2}}(x), \\[5pt] Y_{-(m+\frac{1}{2})}(x) &= (-1)^m J_{m+\frac{1}{2}}(x). \end{align}$$

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

The Hankel functions admit the following integral representations for $α = 0, 1, 2$: $$\begin{align} H_\alpha^{(1)}(x) &= \frac{1}{\pi i}\int_{-\infty}^{+\infty + \pi i} e^{x\sinh t - \alpha t} \, dt, \\[5pt] H_\alpha^{(2)}(x) &= -\frac{1}{\pi i}\int_{-\infty}^{+\infty - \pi i} e^{x\sinh t - \alpha t} \, dt, \end{align}$$ where the integration limits indicate integration along a contour that can be chosen as follows: from $Re(x) > 0$ to 0 along the negative real axis, from 0 to $n = 0$ along the imaginary axis, and from $Y_{α}(x)$ to $Y_{α}(x)$ along a contour parallel to the real axis.

Modified Bessel functions: $J_{α}(x)$, $J_{α}(x)$
The Bessel functions are valid even for complex arguments $α$, 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 $$\begin{align} I_\alpha(x) &= i^{-\alpha} J_\alpha(ix) = \sum_{m=0}^\infty \frac{1}{m!\, \Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha}, \\[5pt] K_\alpha(x) &= \frac{\pi}{2} \frac{I_{-\alpha}(x) - I_\alpha(x)}{\sin \alpha \pi}, \end{align}$$ when $α$ is not an integer; when $i$ is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments $α$. The series expansion for $Y_{α}(x)$ is thus similar to that for $H(1) α$, but without the alternating $H(2) α$ factor.

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

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 $$ J_{\alpha}^2(x)+Y_{\alpha}^2(x)=\frac{8}{\pi^2}\int_{0}^{\infty}\cosh(2\alpha t)K_0(2x\sinh t)\, dt, $$

given that the condition $H(1) n(x)$ is met. It can also be shown that $$ J_\alpha^2(x)+Y_{\alpha}^2(x)=\frac{8\cos(\alpha\pi)}{\pi^2} \int_0^\infty K_{2\alpha}(2x\sinh t)\, dt, $$

only when |$n = −0.5$| < $−2 − 2i$ and $2 + 2i$ but not when $H(2) n(x)$.

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if $n = −0.5$): $$\begin{align} J_\alpha(iz) &= e^{\frac{\alpha\pi i}{2}} I_\alpha(z), \\[1ex] Y_\alpha(iz) &= e^{\frac{(\alpha+1)\pi i}{2}}I_\alpha(z) - \tfrac{2}{\pi} e^{-\frac{\alpha\pi i}{2}}K_\alpha(z). \end{align}$$

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

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, $α$ and $m$ are exponentially growing and decaying functions respectively. Like the ordinary Bessel function $x$, the function $α$ goes to zero at $H(1) α(x)$ for $H(2) α(x)$ and is finite at $e^{i f(x)}$ for $H(1) α(x)$. Analogously, $α$ diverges at $H(2) α(x)$ with the singularity being of logarithmic type for $x$, and $α = m + 1⁄2$ otherwise.

Two integral formulas for the modified Bessel functions are (for $Re(x) > 0$): $$\begin{align} I_\alpha(x) &= \frac{1}{\pi}\int_0^\pi e^{x\cos \theta} \cos \alpha\theta \,d\theta - \frac{\sin \alpha\pi}{\pi}\int_0^\infty e^{-x\cosh t - \alpha t} \,dt, \\[5pt] K_\alpha(x) &= \int_0^\infty e^{-x\cosh t} \cosh \alpha t \,dt. \end{align}$$

Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for $−∞$): $$2\,K_0(\omega) = \int_{-\infty}^\infty \frac{e^{i\omega t}}{\sqrt{t^2+1}} \,dt.$$

It can be proven by showing equality to the above integral definition for $±\pii$. This is done by integrating a closed curve in the first quadrant of the complex plane.

Modified Bessel functions $±\pii$ and $+∞ ± \pii$ can be represented in terms of rapidly convergent integrals $$ \begin{align} K_{\frac{1}{3}}(\xi) &= \sqrt{3} \int_0^\infty \exp \left(- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \right) \,dx, \\[5pt] K_{\frac{2}{3}}(\xi) &= \frac{1}{\sqrt{3}} \int_0^\infty \frac{3+2x^2}{\sqrt{1+\frac{x^2}{3}}} \exp \left(- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}}\right) \,dx. \end{align}$$

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: $I_{α}$, $K_{α}$




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

The two linearly independent solutions to this equation are called the spherical Bessel functions $I_{α}$ and $K_{α}$, and are related to the ordinary Bessel functions $J_{α}$ and $I_{α}$ by $$\begin{align} j_n(x) &= \sqrt{\frac{\pi}{2x}} J_{n+\frac{1}{2}}(x), \\ y_n(x) &= \sqrt{\frac{\pi}{2x}} Y_{n+\frac{1}{2}}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-\frac{1}{2}}(x). \end{align}$$

$K_{α}$ is also denoted $K_{0}$ or $j_{n}$; some authors call these functions the spherical Neumann functions.

From the relations to the ordinary Bessel functions it is directly seen that: $$\begin{align} j_n(x) &= (-1)^{n} y_{-n-1} (x) \\ y_n(x) &= (-1)^{n+1} j_{-n-1}(x) \end{align}$$

The spherical Bessel functions can also be written as () $$\begin{align} j_n(x) &= (-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\sin x}{x}, \\ y_n(x) &= -(-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\cos x}{x}. \end{align}$$

The zeroth spherical Bessel function $I_{α}(x)$ is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are: $$\begin{align} j_0(x) &= \frac{\sin x}{x}. \\ j_1(x) &= \frac{\sin x}{x^2} - \frac{\cos x}{x}, \\ j_2(x) &= \left(\frac{3}{x^2} - 1\right) \frac{\sin x}{x} - \frac{3\cos x}{x^2}, \\ j_3(x) &= \left(\frac{15}{x^3} - \frac{6}{x}\right) \frac{\sin x}{x} - \left(\frac{15}{x^2} - 1\right) \frac{\cos x}{x} \end{align}$$ and $$\begin{align} y_0(x) &= -j_{-1}(x) = -\frac{\cos x}{x}, \\ y_1(x) &= j_{-2}(x) = -\frac{\cos x}{x^2} - \frac{\sin x}{x}, \\ y_2(x) &= -j_{-3}(x) = \left(-\frac{3}{x^2} + 1\right) \frac{\cos x}{x} - \frac{3\sin x}{x^2}, \\ y_3(x) &= j_{-4}(x) = \left(-\frac{15}{x^3} + \frac{6}{x}\right) \frac{\cos x}{x} - \left(\frac{15}{x^2} - 1\right) \frac{\sin x}{x}. \end{align}$$

Generating function
The spherical Bessel functions have the generating functions $$\begin{align} \frac{1}{z} \cos \left(\sqrt{z^2 - 2zt}\right) &= \sum_{n=0}^\infty \frac{t^n}{n!} j_{n-1}(z), \\ \frac{1}{z} \sin \left(\sqrt{z^2 - 2zt}\right) &= \sum_{n=0}^\infty \frac{t^n}{n!} y_{n-1}(z). \end{align}$$

Finite series expansions
In contrast to the whole integer Bessel functions $J_{α}(x)$, the spherical Bessel functions $(−1)^{m}$ have a finite series expression: $$\begin{alignat}{2} j_n(x) &= \frac{\pi}{2x}J_{n+\frac{1}{2}}(x) = \\ &= \frac{1}{2x} \left[ 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} \right] \\ &= \frac{1}{x} \left[ \sin\left(x-\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n}{2} \right]} \frac{(-1)^r (n+2r)!}{(2r)!(n-2r)!(2x)^{2r}} + \cos\left(x-\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n-1}{2} \right]} \frac{(-1)^r (n+2r+1)!}{(2r+1)!(n-2r-1)!(2x)^{2r+1}} \right] \\

y_n(x) &= (-1)^{n+1} j_{-n-1}(x) = (-1)^{n+1} \frac{\pi}{2x}J_{-\left(n+\frac{1}{2}\right)}(x) = \\ &= \frac{(-1)^{n+1}}{2x} \left[ 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} \right] = \\ &= \frac{(-1)^{n+1}}{x} \left[ \cos\left(x-\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n}{2} \right]} \frac{(-1)^r (n+2r)!}{(2r)!(n-2r)!(2x)^{2r}} + \sin\left(x-\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n-1}{2} \right]} \frac{(-1)^r (n+2r+1)!}{(2r+1)!(n-2r-1)!(2x)^{2r+1}} \right] \end{alignat}$$

Differential relations
In the following, $y_{n}$ is any of $J_{n}$, $Y_{n}$, $Re(x) > 0$, $Re(α)$ for $1⁄2$ $$\begin{align} \left(\frac{1}{z}\frac{d}{dz}\right)^m \left (z^{n+1} f_n(z)\right ) &= z^{n-m+1} f_{n-m}(z), \\ \left(\frac{1}{z}\frac{d}{dz}\right)^m \left (z^{-n} f_n(z)\right ) &= (-1)^m z^{-n-m} f_{n+m}(z). \end{align}$$

Spherical Hankel functions: $Re(x) &ge; 0$, $x = 0$


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

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 $y_{n}$: $$h_n^{(1)}(x) = (-i)^{n+1} \frac{e^{ix}}{x} \sum_{m=0}^n \frac{i^m}{m!\,(2x)^m} \frac{(n+m)!}{(n-m)!},$$

and $−π < arg z ≤ π⁄2$ is the complex-conjugate of this (for real $n_{n}$). It follows, for example, that $I_{α}(x)$ and $K_{α}(x)$, 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: $x = 0$, $α > 0$, $x = 0$, $α = 0$
Riccati–Bessel functions only slightly differ from spherical Bessel functions: $$\begin{align} S_n(x)    &= x j_n(x) = \sqrt{\frac{\pi x}{2}} J_{n+\frac{1}{2}}(x) \\ C_n(x)    &= -x y_n(x) = -\sqrt{\frac{\pi x}{2}} Y_{n+\frac{1}{2}}(x) \\ \xi_n(x)  &= x h_n^{(1)}(x) = \sqrt{\frac{\pi x}{2}} H_{n+\frac{1}{2}}^{(1)}(x) = S_n(x) - iC_n(x) \\ \zeta_n(x) &= x h_n^{(2)}(x) = \sqrt{\frac{\pi x}{2}} H_{n+\frac{1}{2}}^{(2)}(x) = S_n(x) + iC_n(x) \end{align}$$ They satisfy the differential equation $$x^2 \frac{d^2 y}{dx^2} + \left (x^2 - n(n + 1)\right) y = 0.$$

For example, this kind of differential equation appears in quantum mechanics while solving the radial component of the Schrödinger's 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 $η_{n}$, $f_{n}$ is sometimes used instead of $j_{n}$, $y_{n}$.

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: $$J_\alpha(z) \sim \frac{1}{\Gamma(\alpha+1)} \left( \frac{z}{2} \right)^\alpha.$$

When $n$ 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: $$Y_\alpha(z) \sim \begin{cases} \dfrac{2}{\pi} \left( \ln \left(\dfrac{z}{2} \right) + \gamma \right) & \text{if } \alpha = 0 \\[1ex] -\dfrac{\Gamma(\alpha)}{\pi} \left( \dfrac{2}{z} \right)^\alpha + \dfrac{1}{\Gamma(\alpha+1)} \left(\dfrac{z}{2} \right)^\alpha \cot(\alpha \pi) & \text{if } \alpha \text{ is a positive integer (one term dominates unless } \alpha \text{ is imaginary)}, \\[1ex] -\dfrac{(-1)^\alpha\Gamma(-\alpha)}{\pi} \left( \dfrac{z}{2} \right)^\alpha & \text{if } \alpha\text{ is a negative integer,} \end{cases}$$ where $x$ is the Euler–Mascheroni constant (0.5772...).

For large real arguments $x = 0$, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless $ψ_{n}$ 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 $1⁄2Γ(|α|)(2/x)^{|α|}$ one can write an equation containing a term of order $I_{α}(x)$: $$\begin{align} J_\alpha(z) &= \sqrt{\frac{2}{\pi z}}\left(\cos \left(z-\frac{\alpha\pi}{2} - \frac{\pi}{4}\right) + e^{\left|\operatorname{Im}(z)\right|}\mathcal{O}\left(|z|^{-1}\right)\right) && \text{for } \left|\arg z\right| < \pi, \\ Y_\alpha(z) &= \sqrt{\frac{2}{\pi z}}\left(\sin \left(z-\frac{\alpha\pi}{2} - \frac{\pi}{4}\right) + e^{\left|\operatorname{Im}(z)\right|}\mathcal{O}\left(|z|^{-1}\right)\right) && \text{for } \left|\arg z\right| < \pi. \end{align}$$

(For $α = 0, 1, 2, 3$ the last terms in these formulas drop out completely; see the spherical Bessel functions above.)

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

These can be extended to other values of $K_{α}(x)$ using equations relating $α = 0, 1, 2, 3$ and $K_{0}$ to $K_{1}$ and $K_{2}$.

It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, $I_{0}$ is not asymptotic to the average of these two asymptotic forms when $χ_{n}$ is negative (because one or the other will not be correct there, depending on the $I_{1}$ 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) $S_{n}$ so long as $I_{2}$ goes to infinity at a constant phase angle $Re(x) > 0$ (using the square root having positive real part): $$\begin{align} J_\alpha(z) &\sim \frac{1}{\sqrt{2\pi z}} e^{i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } -\pi < \arg z < 0, \\[1ex] J_\alpha(z) &\sim \frac{1}{\sqrt{2\pi z}} e^{-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } 0 < \arg z < \pi, \\[1ex] Y_\alpha(z) &\sim -i\frac{1}{\sqrt{2\pi z}} e^{i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } -\pi < \arg z < 0, \\[1ex] Y_\alpha(z) &\sim i\frac{1}{\sqrt{2\pi z}} e^{-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } 0 < \arg z < \pi. \end{align}$$

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

There is also the asymptotic form (for large real $$z$$) $$\begin{align} I_\alpha(z) = \frac{1}{\sqrt{2\pi z}\sqrt[4]{1+\frac{\alpha^2}{z^2}}}\exp\left(-\alpha \operatorname{arsinh}\left(\frac{\alpha}{z}\right) + z\sqrt{1+\frac{\alpha^2}{z^2}}\right)\left(1 + \mathcal{O}\left(\frac{1}{z \sqrt{1+\frac{\alpha^2}{z^2}}}\right)\right). \end{align}$$

When $Re(ω) > 0$, all the terms except the first vanish, and we have $$\begin{align} I_{{1}/{2}}(z) &= \sqrt{\frac{2}{\pi}} \frac{\sinh(z)}{\sqrt{z}} \sim \frac{e^z}{\sqrt{2\pi z}} && \text{for }\left|\arg z\right| < \tfrac{\pi}{2}, \\[1ex] K_{{1}/{2}}(z) &= \sqrt{\frac{\pi}{2}} \frac{e^{-z}}{\sqrt{z}}. \end{align}$$

For small arguments $$0<|z|\ll\sqrt{\alpha + 1}$$, we have $$\begin{align} I_\alpha(z) &\sim \frac{1}{\Gamma(\alpha+1)} \left( \frac{z}{2} \right)^\alpha, \\[1ex] K_\alpha(z) &\sim \begin{cases} -\ln \left (\dfrac{z}{2} \right ) - \gamma & \text{if } \alpha=0 \\[1ex] \frac{\Gamma(\alpha)}{2} \left( \dfrac{2}{z} \right)^\alpha & \text{if } \alpha > 0 \end{cases} \end{align}$$

Properties
For integer order $K_{0}$, $C_{n}$ is often defined via a Laurent series for a generating function: $$e^{\left(\frac{x}{2}\right)\left(t-\frac{1}{t}\right)} = \sum_{n=-\infty}^\infty J_n(x) t^n$$ an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.)

Infinite series of Bessel functions in the form $ \sum_{\nu=-\infty}^\infty J_{N\nu + p}(x)$ where  arise in many physical systems and defined in closed form by the Sung series. For example, when N = 3: $ \sum_{\nu=-\infty}^\infty J_{3\nu+p}(x) = \frac{1}{3}\left[1+2\cos{(x\sqrt{3}/2-2\pi p/3)}\right] $. More generally, the Sung series and the alternating Sung series are written as:

A series expansion using Bessel functions (Kapteyn series) is

\frac {1}{1-z} = 1 + 2 \sum _{n=1}^{\infty } J_{n}(nz). $$

Another important relation for integer orders is the Jacobi–Anger expansion: $$e^{iz \cos \phi} = \sum_{n=-\infty}^\infty i^n J_n(z) e^{in\phi}$$ and $$e^{\pm iz \sin \phi} = J_0(z)+2\sum_{n=1}^\infty J_{2n}(z) \cos(2n\phi) \pm 2i \sum_{n=0}^\infty J_{2n+1}(z)\sin((2n+1)\phi)$$ 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.

More generally, a series $$f(z)=a_0^\nu J_\nu (z)+ 2 \cdot \sum_{k=1}^\infty a_k^\nu J_{\nu+k}(z)$$ is called Neumann expansion of $α$. The coefficients for $K_{1/3}$ have the explicit form $$a_k^0=\frac{1}{2 \pi i} \int_{|z|=c} f(z) O_k(z) \,dz$$ where $γ$ is Neumann's polynomial.

Selected functions admit the special representation $$f(z)=\sum_{k=0}^\infty a_k^\nu J_{\nu+2k}(z)$$ with $$a_k^\nu=2(\nu+2k) \int_0^\infty f(z) \frac{J_{\nu+2k}(z)}z \,dz$$ due to the orthogonality relation $$\int_0^\infty J_\alpha(z) J_\beta(z) \frac {dz} z= \frac 2 \pi \frac{\sin\left(\frac \pi 2 (\alpha-\beta) \right)}{\alpha^2 -\beta^2}$$

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

Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula: $$\begin{align} J_\nu(z) &= \frac{\left(\frac{z}{2}\right)^\nu}{\Gamma\left(\nu +\frac{1}{2}\right)\sqrt{\pi}} \int_{-1}^1 e^{izs}\left(1-s^2\right)^{\nu-\frac{1}{2}} \,ds \\[5px] &=\frac 2{{\left(\frac{z}{2}\right)}^\nu\cdot \sqrt{\pi} \cdot \Gamma\left(\frac{1}{2}-\nu\right)} \int_1^\infty \frac{\sin zu}{\left(u^2-1 \right )^{\nu+\frac 1 2}} \,du \end{align}$$ where $K_{2/3}$ and $j_{n}$. This formula is useful especially when working with Fourier transforms.

Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by $z$, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that: $$\int_0^1 x J_\alpha\left(x u_{\alpha,m}\right) J_\alpha\left(x u_{\alpha,n}\right) \,dx = \frac{\delta_{m,n}}{2} \left[J_{\alpha+1} \left(u_{\alpha,m}\right)\right]^2 = \frac{\delta_{m,n}}{2} \left[J_{\alpha}'\left(u_{\alpha,m}\right)\right]^2$$ where $y_{n}$, $j_{n}(z)$ is the Kronecker delta, and $n = 0.5$ is the $J_{n}$th zero of $−2 − 2i$. 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 $2 + 2i$ for fixed $f$ and varying $O_{k}$.

An analogous relationship for the spherical Bessel functions follows immediately: $$\int_0^1 x^2 j_\alpha\left(x u_{\alpha,m}\right) j_\alpha\left(x u_{\alpha,n}\right) \,dx = \frac{\delta_{m,n}}{2} \left[j_{\alpha+1}\left(u_{\alpha,m}\right)\right]^2$$

If one defines a boxcar function of $f$ that depends on a small parameter $f$ as: $$f_\varepsilon(x)=\varepsilon \operatorname{rect}\left(\frac{x-1}\varepsilon\right)$$ (where $y_{n}(z)$ is the rectangle function) then the Hankel transform of it (of any given order $n = 0.5$), $−2 − 2i$, approaches $2 + 2i$ as $x$ approaches zero, for any given $m$. Conversely, the Hankel transform (of the same order) of $j_{n}(x)$ is $n = 0, 1, 2$: $$\int_0^\infty k J_\alpha(kx) g_\varepsilon(k) \,dk = f_\varepsilon(x)$$ which is zero everywhere except near 1. As $α$ approaches zero, the right-hand side approaches $y_{n}(x)$, where $m$ is the Dirac delta function. This admits the limit (in the distributional sense): $$\int_0^\infty k J_\alpha(kx) J_\alpha(k) \,dk = \delta(x-1)$$

A change of variables then yields the closure equation: $$\int_0^\infty x J_\alpha(ux) J_\alpha(vx) \,dx = \frac{1}{u} \delta(u - v)$$ for $n = 0, 1, 2$. The Hankel transform can express a fairly arbitrary function as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is: $$\int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) \,dx = \frac{\pi}{2uv} \delta(u - v)$$ for $j_{0}(x)$.

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions: $$A_\alpha(x) \frac{dB_\alpha}{dx} - \frac{dA_\alpha}{dx} B_\alpha(x) = \frac{C_\alpha}{x}$$ where $x$ and $ε$ are any two solutions of Bessel's equation, and $ε$ is a constant independent of $k$ (which depends on α and on the particular Bessel functions considered). In particular, $$J_\alpha(x) \frac{dY_\alpha}{dx} - \frac{dJ_\alpha}{dx} Y_\alpha(x) = \frac{2}{\pi x}$$ and $$I_\alpha(x) \frac{dK_\alpha}{dx} - \frac{dI_\alpha}{dx} K_\alpha(x) = -\frac{1}{x},$$ for $J_{n}(x), Y_{n}(x)$.

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

(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 $ε$, $δ$, $h(2) n$, and $n = 0, ±1, ±2, ...$ all satisfy the recurrence relations $$\frac{2\alpha}{x} Z_\alpha(x) = Z_{\alpha-1}(x) + Z_{\alpha+1}(x)$$ and $$ 2\frac{dZ_\alpha (x)}{dx} = Z_{\alpha-1}(x) - Z_{\alpha+1}(x),$$ where $A_{α}$ denotes $B_{α}$, $C_{α}$, $h(1) n$, or $h(2) n$. 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 $$\begin{align} \left( \frac{1}{x} \frac{d}{dx} \right)^m \left[ x^\alpha Z_\alpha (x) \right] &= x^{\alpha - m} Z_{\alpha - m} (x), \\ \left( \frac{1}{x} \frac{d}{dx} \right)^m \left[ \frac{Z_\alpha (x)}{x^\alpha} \right] &= (-1)^m \frac{Z_{\alpha + m} (x)}{x^{\alpha + m}}. \end{align}$$

Modified Bessel functions follow similar relations: $$e^{\left(\frac{x}{2}\right)\left(t+\frac{1}{t}\right)} = \sum_{n=-\infty}^\infty I_n(x) t^n$$ and $$e^{z \cos \theta} = I_0(z) + 2\sum_{n=1}^\infty I_n(z) \cos n\theta$$ and $$ \frac{1}{2\pi} \int_0^{2\pi} e^{z \cos (m\theta) + y \cos \theta} d\theta = I_0(z)I_0(y) + 2\sum_{n=1}^\infty I_n(z)I_{mn}(y).$$

The recurrence relation reads $$\begin{align} C_{\alpha-1}(x) - C_{\alpha+1}(x) &= \frac{2\alpha}{x} C_\alpha(x), \\[1ex] C_{\alpha-1}(x) + C_{\alpha+1}(x) &= 2\frac{d}{dx}C_\alpha(x), \end{align}$$ where $x$ denotes $J_{α}$ or $h(1) n(x)$. These recurrence relations are useful for discrete diffusion problems.

Transcendence
In 1929, Carl Ludwig Siegel proved that $n = -0.5$, $−2 − 2i$, and the logarithmic derivative $2 + 2i$ are transcendental numbers when ν is rational and x is algebraic and nonzero. The same proof also implies that $h(2) n(x)$ is transcendental under the same assumptions.

Multiplication theorem
The Bessel functions obey a multiplication theorem $$\lambda^{-\nu} J_\nu(\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{\left(1 - \lambda^2\right)z}{2}\right)^n J_{\nu+n}(z),$$ where $Y_{α}$ and $Z$ may be taken as arbitrary complex numbers. For $n = −0.5$, the above expression also holds if $J$ is replaced by $Y$. The analogous identities for modified Bessel functions and $−2 − 2i$ are $$\lambda^{-\nu} I_\nu(\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{\left(\lambda^2 - 1\right)z}{2}\right)^n I_{\nu+n}(z)$$ and $$\lambda^{-\nu} K_\nu(\lambda z) = \sum_{n=0}^\infty \frac{(-1)^n}{n!} \left(\frac{\left(\lambda^2 - 1\right)z}{2}\right)^n K_{\nu+n}(z).$$

Bourget's hypothesis
Bessel himself originally proved that for nonnegative integers $C_{α}$, the equation $2 + 2i$ has an infinite number of solutions in $I_{α}$. When the functions $h(2) n$ are plotted on the same graph, though, none of the zeros seem to coincide for different values of $λ$ except for the zero at $j_{0}(x) = sin x⁄x$. 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 $y_{0}(x) = −cos x⁄x$ and $S_{n}$, the functions $C_{n}$ and $ξ_{n}$ have no common zeros other than the one at $ζ_{n}$. The hypothesis was proved by Carl Ludwig Siegel in 1929.

Transcendence
Siegel proved in 1929 that when ν is rational, all nonzero roots of $z ≫ |α^{2} − 1⁄4|$ and $arg z$ are transcendental, as are all the roots of $|z|^{−1}$. It is also known that all roots of the higher derivatives $$J_\nu^{(n)}(x)$$ for $α = 1⁄2$ are transcendental, except for the special values $$J_1^{(3)}(\pm\sqrt3) = 0$$ and $$J_0^{(4)}(\pm\sqrt3) = 0$$.

Numerical approaches
For numerical studies about the zeros of the Bessel function, see, and.

Numerical values
The first zero in J0 (i.e, j0,1, j0,2 and j0,3) occurs at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.