Bessel function

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 ${\displaystyle \alpha }$ when ${\displaystyle \alpha }$ is known to be an integer.

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

Bessel functions of the first kind: Jα

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

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

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 n, is possible using an integral representation: $${\displaystyle J_n(x)=\frac{1}{\pi }{\int }_0^{\pi }\cos (n\tau -x\sin \tau )d\tau =\frac{1}{\pi }\mathrm{Re}\left({\int }_0^{\pi }{e}^{i(n\tau -x\sin \tau )}d\tau \right),}$$ 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 Re(x) > 0: $${\displaystyle J_{\alpha }(x)=\frac{1}{\pi }{\int }_0^{\pi }\cos (\alpha \tau -x\sin \tau )d\tau -\frac{\sin (\alpha \pi )}{\pi }{\int }_0^{\mathrm{\infty }}{e}^{-x\sinh t-\alpha t}dt.}$$

Relation to hypergeometric series

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

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

Bessel functions of the second kind: Yα

The Bessel functions of the second kind, denoted by Y_α(x), occasionally denoted instead by N_α(x), are solutions of the Bessel differential equation that have a singularity at the origin (x = 0) and are multivalued. These are sometimes called Weber functions, as they were introduced by H. M. Weber (1873), and also Neumann functions after Carl Neumann.

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

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

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

There is also a corresponding integral formula (for Re(x) > 0): $${\displaystyle Y_n(x)=\frac{1}{\pi }{\int }_0^{\pi }\sin (x\sin \theta -n\theta )d\theta -\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\left(e^{nt}+(-1{)}^n{e}^{-nt}\right)e^{-x\sinh t}dt.}$$

In the case where n = 0: (with ${\displaystyle \gamma }$ being Euler's constant)$${\displaystyle Y_0\left(x\right)=\frac{4}{{\pi }^2}{\int }_0^{\frac{1}{2}\pi }\cos \left(x\cos \theta \right)\left(\gamma +\ln \left(2x{\sin }^2\theta \right)\right)d\theta .}$$

Y_α(x) is necessary as the second linearly independent solution of the Bessel's equation when α is an integer. But Y_α(x) has more meaning than that. It can be considered as a "natural" partner of J_α(x). 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: $${\displaystyle Y_{-n}(x)=(-1{)}^n{Y}_n(x).}$$

Both J_α(x) and Y_α(x) are holomorphic functions of x on the complex plane cut along the negative real axis. When α is an integer, the Bessel functions J are entire functions of x. If x 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 α is an integer, are an example of the second kind of solution in Fuchs's theorem.

Hankel functions: H(1)α, H(2)α

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, H⁽¹⁾
_α(x) and H⁽²⁾
_α(x), defined as $${\displaystyle \begin{array}{cc}{H}_{\alpha }^{(1)}(x)& =J_{\alpha }(x)+i{Y}_{\alpha }(x),\\ H_{\alpha }^{(2)}(x)& =J_{\alpha }(x)-i{Y}_{\alpha }(x),\end{array}}$$ where i is the imaginary unit. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. 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 e^{i f(x)}. For real ${\displaystyle x>0}$ where ${\displaystyle J_{\alpha }(x)}$, ${\displaystyle Y_{\alpha }(x)}$ 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 H⁽¹⁾
_α(x), H⁽²⁾
_α(x) for ${\displaystyle e^{\pm ix}}$ and ${\displaystyle J_{\alpha }(x)}$, ${\displaystyle Y_{\alpha }(x)}$ for ${\displaystyle \cos (x)}$, ${\displaystyle \sin (x)}$, 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 $${\displaystyle \begin{array}{cc}{H}_{\alpha }^{(1)}(x)& =\frac{J_{-\alpha }(x)-e^{-\alpha \pi i}{J}_{\alpha }(x)}{i\sin \alpha \pi },\\ H_{\alpha }^{(2)}(x)& =\frac{J_{-\alpha }(x)-e^{\alpha \pi i}{J}_{\alpha }(x)}{-i\sin \alpha \pi }.\end{array}}$$

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

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

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

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

Modified Bessel functions: Iα, Kα

The Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as $${\displaystyle \begin{array}{cc}{I}_{\alpha }(x)& =i^{-\alpha }{J}_{\alpha }(ix)=\sum _{m=0}^{\mathrm{\infty }}\frac{1}{m!\mathrm{\Gamma }(m+\alpha +1)}{\left(\frac{x}{2}\right)}^{2m+\alpha },\\ K_{\alpha }(x)& =\frac{\pi }{2}\frac{I_{-\alpha }(x)-I_{\alpha }(x)}{\sin \alpha \pi },\end{array}}$$ when α is not an integer. When α is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments x. The series expansion for I_α(x) is thus similar to that for J_α(x), but without the alternating (−1)^m factor.

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

Using these two formulae the result to ${\displaystyle J_{\alpha }^2(z)+Y_{\alpha }^2(z)}$, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following $${\displaystyle J_{\alpha }^2(x)+Y_{\alpha }^2(x)=\frac{8}{{\pi }^2}{\int }_0^{\mathrm{\infty }}\cosh (2\alpha t)K_0(2x\sinh t)dt,}$$

given that the condition Re(x) > 0 is met. It can also be shown that $${\displaystyle J_{\alpha }^2(x)+Y_{\alpha }^2(x)=\frac{8\cos (\alpha \pi )}{{\pi }^2}{\int }_0^{\mathrm{\infty }}{K}_{2\alpha }(2x\sinh t)dt,}$$ only when |Re(α)| < ⁠1/2⁠ and Re(x) ≥ 0 but not when x = 0.

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

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

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, I_α and K_α are exponentially growing and decaying functions respectively. Like the ordinary Bessel function J_α, the function I_α goes to zero at x = 0 for α > 0 and is finite at x = 0 for α = 0. Analogously, K_α diverges at x = 0 with the singularity being of logarithmic type for K₀, and ⁠1/2⁠Γ(|α|)(2/x)^|α| otherwise.

Two integral formulas for the modified Bessel functions are (for Re(x) > 0): $${\displaystyle \begin{array}{cc}{I}_{\alpha }(x)& =\frac{1}{\pi }{\int }_0^{\pi }{e}^{x\cos \theta }\cos \alpha \theta d\theta -\frac{\sin \alpha \pi }{\pi }{\int }_0^{\mathrm{\infty }}{e}^{-x\cosh t-\alpha t}dt,\\ K_{\alpha }(x)& ={\int }_0^{\mathrm{\infty }}{e}^{-x\cosh t}\cosh \alpha tdt.\end{array}}$$

Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for Re(ω) > 0): $${\displaystyle 2K_0(\omega )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{e^{i\omega t}}{\sqrt{t^2+1}}dt.}$$

It can be proven by showing equality to the above integral definition for K₀. 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 $${\displaystyle K_n(xz)=\frac{\mathrm{\Gamma }\left(n+\frac{1}{2}\right)(2z{)}^n}{\sqrt{\pi }{x}^n}{\int }_0^{\mathrm{\infty }}\frac{\cos (xt)dt}{(t^2+z^2{)}^{n+\frac{1}{2}}}.}$$

Modified Bessel functions K_1/3 and K_2/3 can be represented in terms of rapidly convergent integrals $${\displaystyle \begin{array}{cc}{K}_{\frac{1}{3}}(\xi )& =\sqrt{3}{\int }_0^{\mathrm{\infty }}\exp \left(-\xi \left(1+\frac{4x^2}{3}\right)\sqrt{1+\frac{x^2}{3}}\right)dx,\\ K_{\frac{2}{3}}(\xi )& =\frac{1}{\sqrt{3}}{\int }_0^{\mathrm{\infty }}\frac{3+2x^2}{\sqrt{1+\frac{x^2}{3}}}\exp \left(-\xi \left(1+\frac{4x^2}{3}\right)\sqrt{1+\frac{x^2}{3}}\right)dx.\end{array}}$$

The modified Bessel function ${\displaystyle K_{\frac{1}{2}}(\xi )=(2\xi /\pi {)}^{-1/2}\exp (-\xi )}$ 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 $${\displaystyle x^2\frac{d^{2}y}{d{x}^2}+2x\frac{dy}{dx}+\left(x^2-n(n+1)\right)y=0.}$$

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

y_n is also denoted n_n or η_n; some authors call these functions the spherical Neumann functions.

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

The spherical Bessel functions can also be written as (Rayleigh's formulas) $${\displaystyle \begin{array}{cc}{j}_n(x)& =(-x{)}^n{\left(\frac{1}{x}\frac{d}{dx}\right)}^n\frac{\sin x}{x},\\ y_n(x)& =-(-x{)}^n{\left(\frac{1}{x}\frac{d}{dx}\right)}^n\frac{\cos x}{x}.\end{array}}$$

The zeroth spherical Bessel function j₀(x) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are: $${\displaystyle \begin{array}{cc}{j}_0(x)& =\frac{\sin x}{x}.\\ j_1(x)& =\frac{\sin x}{x^2}-\frac{\cos x}{x},\\ j_2(x)& =\left(\frac{3}{x^2}-1\right)\frac{\sin x}{x}-\frac{3\cos x}{x^2},\\ j_3(x)& =\left(\frac{15}{x^3}-\frac{6}{x}\right)\frac{\sin x}{x}-\left(\frac{15}{x^2}-1\right)\frac{\cos x}{x}\end{array}}$$ and $${\displaystyle \begin{array}{cc}{y}_0(x)& =-j_{-1}(x)=-\frac{\cos x}{x},\\ y_1(x)& =j_{-2}(x)=-\frac{\cos x}{x^2}-\frac{\sin x}{x},\\ y_2(x)& =-j_{-3}(x)=\left(-\frac{3}{x^2}+1\right)\frac{\cos x}{x}-\frac{3\sin x}{x^2},\\ y_3(x)& =j_{-4}(x)=\left(-\frac{15}{x^3}+\frac{6}{x}\right)\frac{\cos x}{x}-\left(\frac{15}{x^2}-1\right)\frac{\sin x}{x}.\end{array}}$$

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

Generating function

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

Finite series expansions

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

Differential relations

In the following, f_n is any of j_n, y_n, h⁽¹⁾
_n, h⁽²⁾
_n for n = 0, ±1, ±2, ... $${\displaystyle \begin{array}{cc}{\left(\frac{1}{z}\frac{d}{dz}\right)}^m\left(z^{n+1}{f}_n(z)\right)& =z^{n-m+1}{f}_{n-m}(z),\\ {\left(\frac{1}{z}\frac{d}{dz}\right)}^m\left(z^{-n}{f}_n(z)\right)& =(-1{)}^m{z}^{-n-m}{f}_{n+m}(z).\end{array}}$$

Spherical Hankel functions: h(1)n, h(2)n

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

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 n: $${\displaystyle h_n^{(1)}(x)=(-i{)}^{n+1}\frac{e^{ix}}{x}\sum _{m=0}^n\frac{i^m}{m!(2x{)}^m}\frac{(n+m)!}{(n-m)!},}$$ and h⁽²⁾
_n is the complex-conjugate of this (for real x). It follows, for example, that j₀(x) = ⁠sin x/x⁠ and y₀(x) = −⁠cos x/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: Sn, Cn, ξn, ζn

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

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

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 ψ_n, χ_n is sometimes used instead of S_n, C_n.

Asymptotic forms

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

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

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

For large real arguments z ≫ |α² − ⁠1/4⁠|, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless α 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 arg z one can write an equation containing a term of order |z|⁻¹: $${\displaystyle \begin{array}{cccc}{J}_{\alpha }(z)& =\sqrt{\frac{2}{\pi z}}\left(\cos \left(z-\frac{\alpha \pi }{2}-\frac{\pi }{4}\right)+e^{\left|\mathrm{Im}(z)\right|}\mathcal{O}\left(|z{|}^{-1}\right)\right)& & \text{for }\left|\arg z\right|<\pi ,\\ Y_{\alpha }(z)& =\sqrt{\frac{2}{\pi z}}\left(\sin \left(z-\frac{\alpha \pi }{2}-\frac{\pi }{4}\right)+e^{\left|\mathrm{Im}(z)\right|}\mathcal{O}\left(|z{|}^{-1}\right)\right)& & \text{for }\left|\arg z\right|<\pi .\end{array}}$$

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

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

These can be extended to other values of arg z using equations relating H⁽¹⁾
_α(ze^imπ) and H⁽²⁾
_α(ze^imπ) to H⁽¹⁾
_α(z) and H⁽²⁾
_α(z).

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

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

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

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

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

Properties

For any Bessel function whose order ${\displaystyle \alpha }$ is not a negative integer, the derivatives of the function can be computed as:

${\displaystyle \frac{d}{dx}{J}_{\alpha }(x)=J_{\alpha -1}(x)-\frac{\alpha }{x}{J}_{\alpha }(x)}$

or, equivalently,

${\displaystyle \frac{d}{dx}{J}_{\alpha }(x)=\frac{\alpha }{x}{J}_{\alpha }(x)-J_{\alpha +1}(x)}$

These formulas can be used to determine a recurrence relation for ${\displaystyle J_{\alpha }(x)}$, a more general form of which is given below.

For integer order α = n, J_n is often defined via a Laurent series for a generating function: $${\displaystyle e^{\frac{x}{2}\left(t-\frac{1}{t}\right)}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_n(x)t^n}$$ 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 $\nu ,p\in \mathbb{Z},N\in {\mathbb{Z}}^{+}$ arise in many physical systems and are defined in closed form by the Sung series. For example, when N = 3: $\sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{3\nu +p}(x)=\frac{1}{3}\left[1+2\cos (x\sqrt{3}/2-2\pi p/3)\right]$. More generally, the Sung series and the alternating Sung series are written as: $${\displaystyle \sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}{J}_{N\nu +p}(x)=\frac{1}{N}\sum _{q=0}^{N-1}{e}^{ix\sin 2\pi q/N}{e}^{-i2\pi pq/N}}$$ $${\displaystyle \sum _{\nu =-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^{\nu }{J}_{N\nu +p}(x)=\frac{1}{N}\sum _{q=0}^{N-1}{e}^{ix\sin (2q+1)\pi /N}{e}^{-i(2q+1)\pi p/N}}$$

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

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

${\displaystyle e^{iz\sin \theta }\equiv \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_n(z)e^{in\theta }.}$

The latter is equivalent to $${\displaystyle e^{\pm iz\sin \varphi }=J_0(z)+2\sum _{n=1}^{\mathrm{\infty }}{J}_{2n}(z)\cos (2n\varphi )\pm 2i\sum _{n=0}^{\mathrm{\infty }}{J}_{2n+1}(z)\sin ((2n+1)\varphi )}$$ 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 $${\displaystyle f(z)=a_0^{\nu }{J}_{\nu }(z)+2\cdot \sum _{k=1}^{\mathrm{\infty }}{a}_k^{\nu }{J}_{\nu +k}(z)}$$ is called Neumann expansion of f. The coefficients for ν = 0 have the explicit form $${\displaystyle a_k^0=\frac{1}{2\pi i}{\int }_{|z|=c}f(z)O_k(z)dz}$$ where O_k is Neumann's polynomial.

Selected functions admit the special representation $${\displaystyle f(z)=\sum _{k=0}^{\mathrm{\infty }}{a}_k^{\nu }{J}_{\nu +2k}(z)}$$ with $${\displaystyle a_k^{\nu }=2(\nu +2k){\int }_0^{\mathrm{\infty }}f(z)\frac{J_{\nu +2k}(z)}{z}dz}$$ due to the orthogonality relation $${\displaystyle {\int }_0^{\mathrm{\infty }}{J}_{\alpha }(z)J_{\beta }(z)\frac{dz}{z}=\frac{2}{\pi }\frac{\sin \left(\frac{\pi }{2}(\alpha -\beta )\right)}{{\alpha }^2-{\beta }^2}}$$

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

Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula: $${\displaystyle \begin{array}{cc}{J}_{\nu }(z)& =\frac{{\left(\frac{z}{2}\right)}^{\nu }}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)\sqrt{\pi }}{\int }_{-1}^1{e}^{izs}{\left(1-s^2\right)}^{\nu -\frac{1}{2}}ds\\ & =\frac{2}{{\left(\frac{z}{2}\right)}^{\nu }\cdot \sqrt{\pi }\cdot \mathrm{\Gamma }\left(\frac{1}{2}-\nu \right)}{\int }_1^{\mathrm{\infty }}\frac{\sin zu}{{\left(u^2-1\right)}^{\nu +\frac{1}{2}}}du\end{array}}$$ where ν > −⁠1/2⁠ and z ∈ C. This formula is useful especially when working with Fourier transforms.

Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by x, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that: $${\displaystyle {\int }_0^{1}x{J}_{\alpha }\left(x{u}_{\alpha ,m}\right)J_{\alpha }\left(x{u}_{\alpha ,n}\right)dx=\frac{{\delta }_{m,n}}{2}{\left[J_{\alpha +1}\left(u_{\alpha ,m}\right)\right]}^2=\frac{{\delta }_{m,n}}{2}{\left[J_{\alpha }^{\prime }\left(u_{\alpha ,m}\right)\right]}^2}$$ where α > −1, δ_m,n is the Kronecker delta, and u_α,m is the mth zero of J_α(x). This orthogonality relation can then be used to extract the coefficients in the Fourier–Bessel series, where a function is expanded in the basis of the functions J_α(x u_α,m) for fixed α and varying m.

An analogous relationship for the spherical Bessel functions follows immediately: $${\displaystyle {\int }_0^1{x}^2{j}_{\alpha }\left(x{u}_{\alpha ,m}\right)j_{\alpha }\left(x{u}_{\alpha ,n}\right)dx=\frac{{\delta }_{m,n}}{2}{\left[j_{\alpha +1}\left(u_{\alpha ,m}\right)\right]}^2}$$

If one defines a boxcar function of x that depends on a small parameter ε as: $${\displaystyle f_{\epsilon }(x)=\frac{1}{\epsilon }\mathrm{rect}\left(\frac{x-1}{\epsilon }\right)}$$ (where rect is the rectangle function) then the Hankel transform of it (of any given order α > −⁠1/2⁠), g_ε(k), approaches J_α(k) as ε approaches zero, for any given k. Conversely, the Hankel transform (of the same order) of g_ε(k) is f_ε(x): $${\displaystyle {\int }_0^{\mathrm{\infty }}k{J}_{\alpha }(kx)g_{\epsilon }(k)dk=f_{\epsilon }(x)}$$ which is zero everywhere except near 1. As ε approaches zero, the right-hand side approaches δ(x − 1), where δ is the Dirac delta function. This admits the limit (in the distributional sense): $${\displaystyle {\int }_0^{\mathrm{\infty }}k{J}_{\alpha }(kx)J_{\alpha }(k)dk=\delta (x-1)}$$

A change of variables then yields the closure equation: $${\displaystyle {\int }_0^{\mathrm{\infty }}x{J}_{\alpha }(ux)J_{\alpha }(vx)dx=\frac{1}{u}\delta (u-v)}$$ for α > −⁠1/2⁠. For the spherical Bessel functions the orthogonality relation is: $${\displaystyle {\int }_0^{\mathrm{\infty }}{x}^2{j}_{\alpha }(ux)j_{\alpha }(vx)dx=\frac{\pi }{2uv}\delta (u-v)}$$ for α > −1.

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

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