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. Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by and , respectively, rather than and . J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!\, \Gamma(m+\alpha+1)} {\left(\frac{x}{2}\right)}^{2m + \alpha}, where 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 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 . (The series indicates that is the derivative of , much like is the derivative of ; more generally, the derivative of can be expressed in terms of by the identities
below.) For non-integer , the functions and are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order , 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 , 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}{\pi} \operatorname{Re}\left(\int_{0}^\pi 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 : 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 and arbitrarily chosen parameter , 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: Yα The Bessel functions of the second kind, denoted by , occasionally denoted instead by , are solutions of the Bessel differential equation that have a singularity at the origin () and are
multivalued. These are sometimes called
Weber functions, as they were introduced by , and also
Neumann functions after
Carl Neumann. For non-integer , is related to by Y_\alpha(x) = \frac{J_\alpha(x) \cos (\alpha \pi) - J_{-\alpha}(x)}{\sin (\alpha \pi)}. In the case of integer order , the function is defined by taking the limit as a non-integer tends to : Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x). If 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 ): 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 : (with \gamma being
Euler's constant)Y_{0}\left(x\right)=\frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos\left(x\cos\theta\right)\left(\gamma+\ln\left(2x\sin^2\theta\right)\right)\, d\theta. is necessary as the second linearly independent solution of the Bessel's equation when is an integer. But has more meaning than that. It can be considered as a "natural" partner of . See also the subsection on Hankel functions below. When 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 and are
holomorphic functions of on the
complex plane cut along the negative real axis. When is an integer, the Bessel functions are
entire functions of . If is held fixed at a non-zero value, then the Bessel functions are entire functions of . The Bessel functions of the second kind, when is an integer, are an example of the second kind of solution in
Fuchs's theorem.
Hankel functions: H, H Another important formulation of the two linearly independent solutions to Bessel's equation are the
Hankel functions of the first and second kind, and , 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 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 . 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 , 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 is an integer, the limit has to be calculated. The following relationships are valid, whether 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 with 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 : \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 to 0 along the negative real axis, from 0 to along the imaginary axis, and from to along a contour parallel to the real axis. \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 is not an integer. When is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments . The series expansion for is thus similar to that for , but without the alternating 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 Using these two formulae the result to {{nowrap|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 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 and but not when . We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if ): \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} and 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, and are
exponentially growing and
decaying functions respectively. Like the ordinary Bessel function , the function goes to zero at for and is finite at for . Analogously, diverges at with the singularity being of logarithmic type for , and otherwise. Plot of six modified Bessel functions. In solid line , , and . In dashed line: , , and . --> Two integral formulas for the modified Bessel functions are (for ): \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 . 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 \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: jn, yn 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 and , and are related to the ordinary Bessel functions and 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} is also denoted or ; 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 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} The first few non-zero roots of the first few spherical Bessel functions are:
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 , the spherical Bessel functions have a finite series expression: \begin{alignat}{2} j_n(x) &= \sqrt{\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 \lfloor \frac{n}{2} \right \rfloor} \frac{(-1)^r (n+2r)!}{(2r)!(n-2r)!(2x)^{2r}} + \cos\left(x-\frac{n\pi}{2}\right) \sum_{r=0}^{\left \lfloor \frac{n-1}{2} \right \rfloor} \frac{(-1)^r (n+2r+1)!}{(2r+1)!(n-2r-1)!(2x)^{2r+1}} \right] \\ \end{alignat} \begin{alignat}{2} 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 \lfloor \frac{n}{2} \right \rfloor} \frac{(-1)^r (n+2r)!}{(2r)!(n-2r)!(2x)^{2r}} - \sin\left(x+\frac{n\pi}{2}\right) \sum_{r=0}^{\left \lfloor \frac{n-1}{2} \right \rfloor} \frac{(-1)^r (n+2r+1)!}{(2r+1)!(n-2r-1)!(2x)^{2r+1}} \right] \end{alignat}
Differential relations In the following, is any of , , , for \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: h, h 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} 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 : 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 is the complex-conjugate of this (for real ). It follows, for example, that and , 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: Sn, Cn, ξn, ζn 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 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 , is sometimes used instead of , . == Asymptotic forms ==