Bessel function

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.^[1]

Bessel functions are solutions to a particular type of ordinary differential equation: $${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0,}$$ where ${\displaystyle \alpha }$ 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 ${\displaystyle \alpha }$ and ${\displaystyle -\alpha }$, mathematicians define separate Bessel functions for each to ensure the functions behave smoothly as the order changes.

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

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 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^[2]
• 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).

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.^[3][4]

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:^[5] $${\displaystyle J_{\alpha }(x)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{m!\,\Gamma (m+\alpha +1)}}{\left({\frac {x}{2}}\right)}^{2m+\alpha },}$$ 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 ${\displaystyle x/2}$;^[6] 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):^[7] $${\displaystyle J_{-n}(x)=(-1)^{n}J_{n}(x).}$$

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.

Another definition of the Bessel function, for integer values of n, is possible using an integral representation:^[8] $${\displaystyle J_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(n\tau -x\sin \tau )\,d\tau ={\frac {1}{\pi }}\operatorname {Re} \left(\int _{0}^{\pi }e^{i(n\tau -x\sin \tau )}\,d\tau \right),}$$ which is also called Hansen-Bessel formula.^[9]

This was the approach that Bessel used,^[10] 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:^{[8][11][12][13][14]} $${\displaystyle J_{\alpha }(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\alpha \tau -x\sin \tau )\,d\tau -{\frac {\sin(\alpha \pi )}{\pi }}\int _{0}^{\infty }e^{-x\sinh t-\alpha t}\,dt.}$$

The Bessel functions can be expressed in terms of the generalized hypergeometric series as^[15] $${\displaystyle J_{\alpha }(x)={\frac {\left({\frac {x}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}\;_{0}F_{1}\left(\alpha +1;-{\frac {x^{2}}{4}}\right).}$$

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

In terms of the Laguerre polynomials L_k and arbitrarily chosen parameter t, the Bessel function can be expressed as^[16] $${\displaystyle {\frac {J_{\alpha }(x)}{\left({\frac {x}{2}}\right)^{\alpha }}}={\frac {e^{-t}}{\Gamma (\alpha +1)}}\sum _{k=0}^{\infty }{\frac {L_{k}^{(\alpha )}\left({\frac {x^{2}}{4t}}\right)}{\binom {k+\alpha }{k}}}{\frac {t^{k}}{k!}}.}$$

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.^[17]

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: $${\displaystyle Y_{n}(x)=\lim _{\alpha \to n}Y_{\alpha }(x).}$$

If n is a nonnegative integer, we have the series^[18] $${\displaystyle Y_{n}(z)=-{\frac {\left({\frac {z}{2}}\right)^{-n}}{\pi }}\sum _{k=0}^{n-1}{\frac {(n-k-1)!}{k!}}\left({\frac {z^{2}}{4}}\right)^{k}+{\frac {2}{\pi }}J_{n}(z)\ln {\frac {z}{2}}-{\frac {\left({\frac {z}{2}}\right)^{n}}{\pi }}\sum _{k=0}^{\infty }(\psi (k+1)+\psi (n+k+1)){\frac {\left(-{\frac {z^{2}}{4}}\right)^{k}}{k!(n+k)!}}}$$ where ${\displaystyle \psi (z)}$ is the digamma function, the logarithmic derivative of the gamma function.^[4]

There is also a corresponding integral formula (for Re(x) > 0):^[19] $${\displaystyle Y_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\sin(x\sin \theta -n\theta )\,d\theta -{\frac {1}{\pi }}\int _{0}^{\infty }\left(e^{nt}+(-1)^{n}e^{-nt}\right)e^{-x\sinh t}\,dt.}$$

In the case where 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 is an example of the second kind of solution in Fuchs's theorem.

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^[20] $${\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(x)&=J_{\alpha }(x)+iY_{\alpha }(x),\\[5pt]H_{\alpha }^{(2)}(x)&=J_{\alpha }(x)-iY_{\alpha }(x),\end{aligned}}}$$ 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 $${\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(x)&={\frac {J_{-\alpha }(x)-e^{-\alpha \pi i}J_{\alpha }(x)}{i\sin \alpha \pi }},\\[5pt]H_{\alpha }^{(2)}(x)&={\frac {J_{-\alpha }(x)-e^{\alpha \pi i}J_{\alpha }(x)}{-i\sin \alpha \pi }}.\end{aligned}}}$$

If α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not:^[21] $${\displaystyle {\begin{aligned}H_{-\alpha }^{(1)}(x)&=e^{\alpha \pi i}H_{\alpha }^{(1)}(x),\\[6mu]H_{-\alpha }^{(2)}(x)&=e^{-\alpha \pi i}H_{\alpha }^{(2)}(x).\end{aligned}}}$$

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

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

The Hankel functions admit the following integral representations for Re(x) > 0:^[22] $${\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(x)&={\frac {1}{\pi i}}\int _{-\infty }^{+\infty +\pi i}e^{x\sinh t-\alpha t}\,dt,\\[5pt]H_{\alpha }^{(2)}(x)&=-{\frac {1}{\pi i}}\int _{-\infty }^{+\infty -\pi i}e^{x\sinh t-\alpha t}\,dt,\end{aligned}}}$$ 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.^[19]

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^[23] $${\displaystyle {\begin{aligned}I_{\alpha }(x)&=i^{-\alpha }J_{\alpha }(ix)=\sum _{m=0}^{\infty }{\frac {1}{m!\,\Gamma (m+\alpha +1)}}\left({\frac {x}{2}}\right)^{2m+\alpha },\\[5pt]K_{\alpha }(x)&={\frac {\pi }{2}}{\frac {I_{-\alpha }(x)-I_{\alpha }(x)}{\sin \alpha \pi }},\end{aligned}}}$$ 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: $${\displaystyle K_{\alpha }(x)={\begin{cases}{\frac {\pi }{2}}i^{\alpha +1}H_{\alpha }^{(1)}(ix)&-\pi <\arg x\leq {\frac {\pi }{2}}\\{\frac {\pi }{2}}(-i)^{\alpha +1}H_{\alpha }^{(2)}(-ix)&-{\frac {\pi }{2}}<\arg x\leq \pi \end{cases}}}$$

Using these two formulae the result to ${\displaystyle J_{\alpha }^{2}(z)}$+${\displaystyle Y_{\alpha }^{2}(z)}$, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following $${\displaystyle J_{\mathrm{\alpha <!--\; \alpha \; -->}}^2(x)+Y_{\mathrm{\alpha <!--\; \alpha \; -->}}^2(x)=\frac{8}{{\mathrm{\pi <!--\; \pi \; -->}}^2}}$$