for an arbitrary complex number$\alpha$, 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 Bessel function is a generalization of the sine function. It can be interpreted as the vibration of a string with variable thickness, variable tension (or both conditions simultaneously); vibrations in a medium with variable properties; vibrations of the disc membrane, etc.
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:
Because this is a second-order linear differential equation, 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 N_{n} and n_{n}, respectively, rather than Y_{n} and y_{n}.^{[1]}^{[2]}
Bessel functions of the first kind: J_{α}Edit
Plot of the Bessel function of the first kind J n(z) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of Bessel function of the first kind, J_{α}(x), for integer orders α = 0, 1, 2
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 its series expansion around x = 0, which can be found by applying the Frobenius method to Bessel's equation:^{[3]}
where Γ(z) 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 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 $x^{-{\frac {1}{2}}}$ (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large x. (The series indicates that −J_{1}(x) is the derivative of J_{0}(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):^{[4]}
$J_{-n}(x)=(-1)^{n}J_{n}(x).$
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 integralsEdit
Another definition of the Bessel function, for integer values of n, is possible using an integral representation:^{[5]}
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:^{[5]}^{[7]}^{[8]}^{[9]}^{[10]}
Plot of the Bessel function of the second kind Y n(z) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of Bessel function of the second kind, Y_{α}(x), for integer orders α = 0, 1, 2
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.^{[13]}
For non-integer α, Y_{α}(x) is related to J_{α}(x) by
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:
$Y_{-n}(x)=(-1)^{n}Y_{n}(x).$
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.
Hankel functions: H^{(1)} _{α}, H^{(2)} _{α}Edit
Plot of the Hankel function of the first kind H n^(1)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of the Hankel function of the second kind H n^(2)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
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^{[17]}
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 $x>0$ 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 H^{(1)} _{α}(x), H^{(2)} _{α}(x) for $e^{\pm ix}$ and $J_{\alpha }(x)$, $Y_{\alpha }(x)$ for $\cos(x)$, $\sin(x)$, as explicitly shown in the asymptotic expansion.
The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the sign convention for the frequency).
Using the previous relationships, they can be expressed as
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.^{[16]}
Modified Bessel functions: I_{α}, K_{α}Edit
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^{[20]}
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.
$K_{\alpha }$ can be expressed in terms of Hankel functions:
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
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 K_{0}, and 1/2Γ(|α|)(2/x)^{|α|} otherwise.^{[24]}
Modified Bessel functions of the first kind, I_{α}(x), for α = 0, 1, 2, 3
Modified Bessel functions of the second kind, K_{α}(x), for α = 0, 1, 2, 3
Two integral formulas for the modified Bessel functions are (for Re(x) > 0):^{[25]}
It can be proven by showing equality to the above integral definition for K_{0}. This is done by integrating a closed curve in the first quadrant of the complex plane.
Modified Bessel functions K_{1/3} and K_{2/3} can be represented in terms of rapidly convergent integrals^{[26]}
The modified Bessel function $K_{\frac {1}{2}}(\xi )=\xi ^{-1/2}\exp(-\xi )$ 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):
Plot of the spherical Bessel function of the first kind j n(z) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of the spherical Bessel function of the second kind y n(z) with n=0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Spherical Bessel functions of the first kind, j_{n}(x), for n = 0, 1, 2
Spherical Bessel functions of the second kind, y_{n}(x), for n = 0, 1, 2
When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form
The two linearly independent solutions to this equation are called the spherical Bessel functionsj_{n} and y_{n}, and are related to the ordinary Bessel functions J_{n} and Y_{n} by^{[28]}
Plot of the spherical Hankel function of the first kind h n^(1)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of the spherical Hankel function of the second kind h n^(2)(z) with n=-0.5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
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 n:
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.^{[34]} 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)^{[35]} for recent developments and references.
Following Debye (1909), the notation ψ_{n}, χ_{n} is sometimes used instead of S_{n}, C_{n}.
Asymptotic formsEdit
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:^{[3]}
For large real arguments z ≫ |α^{2} − 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}:^{[36]}
(For α = 1/2 the last terms in these formulas drop out completely; see the spherical Bessel functions above.) Even though these equations are true, better approximations may be available for complex z. For example, J_{0}(z) when z is near the negative real line is approximated better by
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).^{[37]}
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):
where ν > −1/2 and z ∈ C.^{[43]}
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:
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}(xu_{α,m}) for fixed α and varying m.
An analogous relationship for the spherical Bessel functions follows immediately:
(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):
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):
for α > −1/2. The Hankel transform can express a fairly arbitrary function^{[clarification needed]}as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is:
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,
where Z denotes J, Y, H^{(1)}, or H^{(2)}. 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^{[46]}
where C_{α} denotes I_{α} or e^{αiπ}K_{α}. These recurrence relations are useful for discrete diffusion problems.
TranscendenceEdit
In 1929, Carl Ludwig Siegel proved that J_{ν}(x), J'_{ν}(x), and the quotient J'_{ν}(x)/J_{ν}(x) are transcendental numbers when ν is rational and x is algebraic and nonzero.^{[47]} The same proof also implies that K_{ν}(x) is transcendental under the same assumptions.^{[48]}
where λ and ν may be taken as arbitrary complex numbers.^{[49]}^{[50]} For |λ^{2} − 1| < 1,^{[49]} the above expression also holds if J is replaced by Y. The analogous identities for modified Bessel functions and |λ^{2} − 1| < 1 are
Bessel himself originally proved that for nonnegative integers n, the equation J_{n}(x) = 0 has an infinite number of solutions in x.^{[51]} When the functions J_{n}(x) are plotted on the same graph, though, none of the zeros seem to coincide for different values of n except for the zero at x = 0. 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 n ≥ 0 and m ≥ 1, the functions J_{n}(x) and J_{n + m}(x) have no common zeros other than the one at x = 0. The hypothesis was proved by Carl Ludwig Siegel in 1929.^{[52]}
TranscendenceEdit
Siegel proved in 1929 that when ν is rational, all nonzero roots of J_{ν}(x) and J'_{ν}(x) are transcendental,^{[53]} as are all the roots of K_{ν}(x).^{[48]} It is also known that all roots of the higher derivatives $J_{\nu }^{(n)}(x)$ for n ≤ 18 are transcendental, except for the special values $J_{1}^{(3)}(\pm {\sqrt {3}})=0$ and $J_{0}^{(4)}(\pm {\sqrt {3}})=0$.^{[53]}
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