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Bessel function

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Families of solutions to related differential equations

Bessel functions describe the radial part ofvibrations of a circular membrane.

Bessel functions aremathematical special functions that commonly appear in problems involvingwave motion,heat conduction, and other physical phenomena withcircular symmetry orcylindrical symmetry. They are named after the German astronomer and mathematicianFriedrich Bessel, who studied them systematically in 1824.^[1]

Bessel functions are solutions to a particular type ofordinary differential equation: $x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0,$ where $\alpha$ is a number that determines the shape of the solution. This number is called theorder of the Bessel function and can be any complex number. Although the same equation arises for both $\alpha$ and $-\alpha$ , mathematicians define separate Bessel functions for each to ensure the functions behave smoothly as the order changes.

The most important cases are when $\alpha$ is an integer or a half-integer. When $\alpha$ is an integer, the resulting Bessel functions are often calledcylinder functions orcylindrical harmonics because they naturally arise when solving problems (like Laplace's equation) incylindrical coordinates. When $\alpha$ is a half-integer, the solutions are calledspherical Bessel functions and are used in spherical systems, such as in solving theHelmholtz equation inspherical coordinates.

Applications

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Bessel's equation arises when finding separable solutions toLaplace's equation and theHelmholtz equation in cylindrical orspherical coordinates. Bessel functions are therefore especially important for many problems ofwave 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 cylindricalwaveguide
Pressure amplitudes ofinviscid rotational flows
Heat conduction in a cylindrical object
Modes of vibration of a thin circular or annularacoustic membrane (such as adrumhead or othermembranophone) or thicker plates such as sheet metal (seeKirchhoff–Love plate theory,Mindlin–Reissner plate theory)
Diffusion problems on a lattice
Solutions to theSchrödinger equation in spherical and cylindrical coordinates for a free particle
Position space representation of the Feynmanpropagator inquantum field theory
Solving for patterns of acoustical radiation
Frequency-dependent friction in circular pipelines
Dynamics of floating bodies
Angular resolution
Diffraction from helical objects, includingDNA
Probability density function of product of two normally distributed random variables^[2]
Analyzing of the surface waves generated by microtremors, ingeophysics andseismology.

Bessel functions also appear in other problems, such as signal processing (e.g., seeFM audio synthesis,Kaiser window, orBessel filter).

Definitions

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Because this is alinear 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 assolutions to definite integrals rather than solutions to differential equations. Because the differential equation is second-order, there must be twolinearly 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 subscriptn is typically used in place of $\alpha$ when $\alpha$ is known to be an integer.

Type	First kind	Second kind
Bessel functions	J_α	Y_α
Modified Bessel functions	I_α	K_α
Hankel functions	H⁽¹⁾ _α =J_α +iY_α	H⁽²⁾ _α =J_α −iY_α
Spherical Bessel functions	j_n	y_n
Modified spherical Bessel functions	i_n	k_n
Spherical Hankel functions	h⁽¹⁾ _n =j_n +iy_n	h⁽²⁾ _n =j_n −iy_n

Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted byN_n andn_n, respectively, rather thanY_n andy_n.^[3]^[4]

Bessel functions of the first kind:J_α

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Plot of Bessel function of the first kind, $J_{\alpha }(x)$ , for integer orders $\alpha =0,1,2$ .

{\displaystyle J_{\alpha }(x)} — Plot of Bessel function of the first kind, $J_{\alpha }(x)$ , for integer orders $\alpha =0,1,2$ .

Plot of Bessel function of the first kind $J_{\alpha }(z)$ with $\alpha =0.5$ in the plane from $-4-4i$ to $4+4i$ .

{\displaystyle J_{\alpha }(z)} — Plot of Bessel function of the first kind $J_{\alpha }(z)$ with $\alpha =0.5$ in the plane from $-4-4i$ to $4+4i$ .

Bessel functions of the first kind, denoted asJ_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 asx approaches zero. It is possible to define the function by $x^{\alpha }$ times aMaclaurin series (note thatα need not be an integer, and non-integer powers are not permitted in a Taylor series), which can be found by applying theFrobenius method to Bessel's equation:^[5] $J_{\alpha }(x)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{m!\,\Gamma (m+\alpha +1)}}{\left({\frac {x}{2}}\right)}^{2m+\alpha },$ whereΓ(z) is thegamma function, a shifted generalization of thefactorial function to non-integer values. Some earlier authors define the Bessel function of the first kind differently, essentially without the division by $2 {\displaystyle 2}$ in $x/2$ ;^[6] this definition is not used in this article. The Bessel function of the first kind is anentire function ifα is an integer, otherwise it is amultivalued function with singularity at zero. The graphs of Bessel functions look roughly like oscillatingsine orcosine functions that decay proportionally to $x^{-{1}/{2}}$ (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for largex. (The series indicates that−J₁(x) is the derivative ofJ₀(x), much like−sinx is the derivative ofcosx; more generally, the derivative ofJ_n(x) can be expressed in terms ofJ_{n ± 1}(x) by the identitiesbelow.)

For non-integerα, the functionsJ_α(x) andJ_−α(x) are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer ordern, the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):^[7] $J_{-n}(x)=(-1)^{n}J_{n}(x).$

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

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Another definition of the Bessel function, for integer values ofn, is possible using an integral representation:^[8] $J_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(n\tau -x\sin \tau )\,d\tau ={\frac {1}{\pi }}\operatorname {Re} \left(\int _{0}^{\pi }e^{i(n\tau -x\sin \tau )}\,d\tau \right),$ 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, forRe(x) > 0:^[8]^[11]^[12]^[13]^[14] $J_{\alpha }(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\alpha \tau -x\sin \tau )\,d\tau -{\frac {\sin(\alpha \pi )}{\pi }}\int _{0}^{\infty }e^{-x\sinh t-\alpha t}\,dt.$

Relation to hypergeometric series

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

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

Relation to Laguerre polynomials

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

Bessel functions of the second kind:Y_α

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Plot of Bessel function of the second kind, $Y_{\alpha }(x)$ , for integer orders $\alpha =0,1,2$

{\displaystyle Y_{\alpha }(x)} — Plot of Bessel function of the second kind, $Y_{\alpha }(x)$ , for integer orders $\alpha =0,1,2$

The Bessel functions of the second kind, denoted byY_α(x), occasionally denoted instead byN_α(x), are solutions of the Bessel differential equation that have a singularity at the origin (x = 0) and aremultivalued. These are sometimes calledWeber functions, as they were introduced byH. M. Weber (1873), and alsoNeumann functions afterCarl Neumann.^[17]

For non-integerα,Y_α(x) is related toJ_α(x) by $Y_{\alpha }(x)={\frac {J_{\alpha }(x)\cos(\alpha \pi )-J_{-\alpha }(x)}{\sin(\alpha \pi )}}.$

In the case of integer ordern, the function is defined by taking the limit as a non-integerα tends ton: $Y_{n}(x)=\lim _{\alpha \to n}Y_{\alpha }(x).$

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

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

In the case wheren = 0: (with $\gamma$ beingEuler's constant) $Y_{0}\left(x\right)={\frac {4}{\pi ^{2}}}\int _{0}^{{\frac {1}{2}}\pi }\cos \left(x\cos \theta \right)\left(\gamma +\ln \left(2x\sin ^{2}\theta \right)\right)\,d\theta .$

Plot of the Bessel function of the second kind $Y_{\alpha }(z)$ with $\alpha =0.5$ in the complex plane from $-2-2i$ to $2+2i$ .

{\displaystyle Y_{\alpha }(z)} — Plot of the Bessel function of the second kind $Y_{\alpha }(z)$ with $\alpha =0.5$ in the complex plane from $-2-2i$ to $2+2i$ .

Y_α(x) is necessary as the second linearly independent solution of the Bessel's equation whenα is an integer. ButY_α(x) has more meaning than that. It can be considered as a "natural" partner ofJ_α(x). See also the subsection on Hankel functions below.

Whenα is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid: $Y_{-n}(x)=(-1)^{n}Y_{n}(x).$

BothJ_α(x) andY_α(x) areholomorphic functions ofx on thecomplex plane cut along the negative real axis. Whenα is an integer, the Bessel functionsJ areentire functions ofx. Ifx is held fixed at a non-zero value, then the Bessel functions are entire functions ofα.

The Bessel functions of the second kind whenα is an integer is an example of the second kind of solution inFuchs's theorem.

Hankel functions:H⁽¹⁾
_α,H⁽²⁾
_α

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Plot of the Hankel function of the first kindH⁽¹⁾
_n(x) withn = −0.5 in the complex plane from−2 − 2i to2 + 2i

Plot of the Hankel function of the second kindH⁽²⁾
_n(x) withn = −0.5 in the complex plane from−2 − 2i to2 + 2i

Another important formulation of the two linearly independent solutions to Bessel's equation are theHankel functions of the first and second kind,H⁽¹⁾
_α(x) andH⁽²⁾
_α(x), defined as^[20] ${\begin{aligned}H_{\alpha }^{(1)}(x)&=J_{\alpha }(x)+iY_{\alpha }(x),\\[5pt]H_{\alpha }^{(2)}(x)&=J_{\alpha }(x)-iY_{\alpha }(x),\end{aligned}}$ wherei is theimaginary unit. These linear combinations are also known asBessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. They are named afterHermann Hankel.

These forms oflinear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of a factor of the forme^if(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 ofEuler's formula, substitutingH⁽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 theasymptotic 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 thesign convention for thefrequency).

Using the previous relationships, they can be expressed as ${\begin{aligned}H_{\alpha }^{(1)}(x)&={\frac {J_{-\alpha }(x)-e^{-\alpha \pi i}J_{\alpha }(x)}{i\sin \alpha \pi }},\\[5pt]H_{\alpha }^{(2)}(x)&={\frac {J_{-\alpha }(x)-e^{\alpha \pi i}J_{\alpha }(x)}{-i\sin \alpha \pi }}.\end{aligned}}$

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

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

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

The Hankel functions admit the following integral representations forRe(x) > 0:^[22] ${\begin{aligned}H_{\alpha }^{(1)}(x)&={\frac {1}{\pi i}}\int _{-\infty }^{+\infty +\pi i}e^{x\sinh t-\alpha t}\,dt,\\[5pt]H_{\alpha }^{(2)}(x)&=-{\frac {1}{\pi i}}\int _{-\infty }^{+\infty -\pi i}e^{x\sinh t-\alpha t}\,dt,\end{aligned}}$ where the integration limits indicate integration along acontour 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]

Modified Bessel functions:I_α,K_α

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The Bessel functions are valid even forcomplex argumentsx, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called themodified Bessel functions (or occasionally thehyperbolic Bessel functions)of the first and second kind and are defined as^[23] ${\begin{aligned}I_{\alpha }(x)&=i^{-\alpha }J_{\alpha }(ix)=\sum _{m=0}^{\infty }{\frac {1}{m!\,\Gamma (m+\alpha +1)}}\left({\frac {x}{2}}\right)^{2m+\alpha },\\[5pt]K_{\alpha }(x)&={\frac {\pi }{2}}{\frac {I_{-\alpha }(x)-I_{\alpha }(x)}{\sin \alpha \pi }},\end{aligned}}$ whenα is not an integer. Whenα is an integer, then the limit is used. These are chosen to be real-valued for real and positive argumentsx. The series expansion forI_α(x) is thus similar to that forJ_α(x), but without the alternating(−1)^m 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 conditionRe(x) > 0 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|Re(α)| <⁠1/2⁠ andRe(x) ≥ 0 but not whenx = 0.^[24]

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if−π < argz ≤⁠π/2⁠):^[25] ${\begin{aligned}J_{\alpha }(iz)&=e^{\frac {\alpha \pi i}{2}}I_{\alpha }(z),\\[1ex]Y_{\alpha }(iz)&=e^{\frac {(\alpha +1)\pi i}{2}}I_{\alpha }(z)-{\tfrac {2}{\pi }}e^{-{\frac {\alpha \pi i}{2}}}K_{\alpha }(z).\end{aligned}}$

I_α(x) andK_α(x) are the two linearly independent solutions to themodified Bessel's equation:^[26] $x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-\left(x^{2}+\alpha ^{2}\right)y=0.$

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument,I_α andK_α areexponentially growing anddecaying functions respectively. Like the ordinary Bessel functionJ_α, the functionI_α goes to zero atx = 0 forα > 0 and is finite atx = 0 forα = 0. Analogously,K_α diverges atx = 0 with the singularity being of logarithmic type forK₀, and⁠1/2⁠Γ(|α|)(2/x)^|α| otherwise.^[27]

Modified Bessel functions of the first kind, $I_{\alpha }(x)$ , for $\alpha =0,1,2,3$ .

{\displaystyle I_{\alpha }(x)} — Modified Bessel functions of the first kind, $I_{\alpha }(x)$ , for $\alpha =0,1,2,3$ .

Modified Bessel functions of the second kind, $K_{\alpha }(x)$ , for $\alpha =0,1,2,3$ .

{\displaystyle K_{\alpha }(x)} — Modified Bessel functions of the second kind, $K_{\alpha }(x)$ , for $\alpha =0,1,2,3$ .

Two integral formulas for the modified Bessel functions are (forRe(x) > 0):^[28] ${\begin{aligned}I_{\alpha }(x)&={\frac {1}{\pi }}\int _{0}^{\pi }e^{x\cos \theta }\cos \alpha \theta \,d\theta -{\frac {\sin \alpha \pi }{\pi }}\int _{0}^{\infty }e^{-x\cosh t-\alpha t}\,dt,\\[5pt]K_{\alpha }(x)&=\int _{0}^{\infty }e^{-x\cosh t}\cosh \alpha t\,dt.\end{aligned}}$

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

It can be proven by showing equality to the above integral definition forK₀. 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^[29] $K_{n}(xz)={\frac {\Gamma {\left(n+{\frac {1}{2}}\right)}(2z)^{n}}{{\sqrt {\pi }}x^{n}}}\int _{0}^{\infty }{\frac {\cos(xt)\,dt}{(t^{2}+z^{2})^{n+{\frac {1}{2}}}}}.$

Modified Bessel functionsK_1/3 andK_2/3 can be represented in terms of rapidly convergent integrals^[30] ${\begin{aligned}K_{\frac {1}{3}}(\xi )&={\sqrt {3}}\int _{0}^{\infty }\exp \left(-\xi \left(1+{\frac {4x^{2}}{3}}\right){\sqrt {1+{\frac {x^{2}}{3}}}}\right)\,dx,\\[5pt]K_{\frac {2}{3}}(\xi )&={\frac {1}{\sqrt {3}}}\int _{0}^{\infty }{\frac {3+2x^{2}}{\sqrt {1+{\frac {x^{2}}{3}}}}}\exp \left(-\xi \left(1+{\frac {4x^{2}}{3}}\right){\sqrt {1+{\frac {x^{2}}{3}}}}\right)\,dx.\end{aligned}}$

The modified Bessel function $K_{\frac {1}{2}}(\xi )=(2\xi /\pi )^{-1/2}\exp(-\xi )$ is useful to represent theLaplace distribution as an Exponential-scale mixture of normal distributions.

Themodified Bessel function of the second kind has also been called by the following names (now rare):

Basset function afterAlfred Barnard Basset
Modified Bessel function of the third kind
Modified Hankel function^[31]
Macdonald function afterHector Munro Macdonald

Spherical Bessel functions:j_n,y_n

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Plot of the spherical Bessel function of the first kindj_n(z) withn = 0.5 in the complex plane from−2 − 2i to2 + 2i

Plot of the spherical Bessel function of the second kindy_n(z) withn = 0.5 in the complex plane from−2 − 2i to2 + 2i

Spherical Bessel functions of the first kind $j_{\alpha }(x)$ , for $\alpha =0,1,2$ .

Spherical Bessel functions of the second kind $y_{\alpha }(x)$ , for $\alpha =0,1,2$ .

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

The two linearly independent solutions to this equation are called thespherical Bessel functionsj_n andy_n, and are related to the ordinary Bessel functionsJ_n andY_n by^[32] ${\begin{aligned}j_{n}(x)&={\sqrt {\frac {\pi }{2x}}}J_{n+{\frac {1}{2}}}(x),\\y_{n}(x)&={\sqrt {\frac {\pi }{2x}}}Y_{n+{\frac {1}{2}}}(x)=(-1)^{n+1}{\sqrt {\frac {\pi }{2x}}}J_{-n-{\frac {1}{2}}}(x).\end{aligned}}$

y_n is also denotedn_n orη_n; some authors call these functions thespherical Neumann functions.

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

The spherical Bessel functions can also be written as (Rayleigh's formulas)^[33] ${\begin{aligned}j_{n}(x)&=(-x)^{n}\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{n}{\frac {\sin x}{x}},\\y_{n}(x)&=-(-x)^{n}\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{n}{\frac {\cos x}{x}}.\end{aligned}}$

The zeroth spherical Bessel functionj₀(x) is also known as the (unnormalized)sinc function. The first few spherical Bessel functions are:^[34] ${\begin{aligned}j_{0}(x)&={\frac {\sin x}{x}}.\\j_{1}(x)&={\frac {\sin x}{x^{2}}}-{\frac {\cos x}{x}},\\j_{2}(x)&=\left({\frac {3}{x^{2}}}-1\right){\frac {\sin x}{x}}-{\frac {3\cos x}{x^{2}}},\\j_{3}(x)&=\left({\frac {15}{x^{3}}}-{\frac {6}{x}}\right){\frac {\sin x}{x}}-\left({\frac {15}{x^{2}}}-1\right){\frac {\cos x}{x}}\end{aligned}}$ and^[35] ${\begin{aligned}y_{0}(x)&=-j_{-1}(x)=-{\frac {\cos x}{x}},\\y_{1}(x)&=j_{-2}(x)=-{\frac {\cos x}{x^{2}}}-{\frac {\sin x}{x}},\\y_{2}(x)&=-j_{-3}(x)=\left(-{\frac {3}{x^{2}}}+1\right){\frac {\cos x}{x}}-{\frac {3\sin x}{x^{2}}},\\y_{3}(x)&=j_{-4}(x)=\left(-{\frac {15}{x^{3}}}+{\frac {6}{x}}\right){\frac {\cos x}{x}}-\left({\frac {15}{x^{2}}}-1\right){\frac {\sin x}{x}}.\end{aligned}}$

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

Non-zero Roots of the Spherical Bessel Function (first kind)
Order	Root 1	Root 2	Root 3	Root 4	Root 5
$j_{0}$	3.141593	6.283185	9.424778	12.566371	15.707963
$j_{1}$	4.493409	7.725252	10.904122	14.066194	17.220755
$j_{2}$	5.763459	9.095011	12.322941	15.514603	18.689036
$j_{3}$	6.987932	10.417119	13.698023	16.923621	20.121806
$j_{4}$	8.182561	11.704907	15.039665	18.301256	21.525418

Non-zero Roots of the Spherical Bessel Function (second kind)
Order	Root 1	Root 2	Root 3	Root 4	Root 5
$y_{0}$	1.570796	4.712389	7.853982	10.995574	14.137167
$y_{1}$	2.798386	6.121250	9.317866	12.486454	15.644128
$y_{2}$	3.959528	7.451610	10.715647	13.921686	17.103359
$y_{3}$	5.088498	8.733710	12.067544	15.315390	18.525210
$y_{4}$	6.197831	9.982466	13.385287	16.676625	19.916796

There are simple closed-form expressions for the Bessel functions ofhalf-integer order in terms of the standardtrigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integersn: $h_{n}^{(1)}(x)=(-i)^{n+1}{\frac {e^{ix}}{x}}\sum _{m=0}^{n}{\frac {i^{m}}{m!\,(2x)^{m}}}{\frac {(n+m)!}{(n-m)!}},$ andh⁽²⁾
_n is the complex-conjugate of this (for realx). It follows, for example, thatj₀(x) =⁠sinx/x⁠ andy₀(x) = −⁠cosx/x⁠, and so on.

The spherical Hankel functions appear in problems involvingspherical wave propagation, for example in themultipole expansion of the electromagnetic field.

Riccati–Bessel functions:S_n,C_n,ξ_n,ζ_n

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Riccati–Bessel functions only slightly differ from spherical Bessel functions: ${\begin{aligned}S_{n}(x)&=xj_{n}(x)={\sqrt {\frac {\pi x}{2}}}J_{n+{\frac {1}{2}}}(x)\\C_{n}(x)&=-xy_{n}(x)=-{\sqrt {\frac {\pi x}{2}}}Y_{n+{\frac {1}{2}}}(x)\\\xi _{n}(x)&=xh_{n}^{(1)}(x)={\sqrt {\frac {\pi x}{2}}}H_{n+{\frac {1}{2}}}^{(1)}(x)=S_{n}(x)-iC_{n}(x)\\\zeta _{n}(x)&=xh_{n}^{(2)}(x)={\sqrt {\frac {\pi x}{2}}}H_{n+{\frac {1}{2}}}^{(2)}(x)=S_{n}(x)+iC_{n}(x)\end{aligned}}$

Riccati–Bessel functions Sn complex plot from -2-2i to 2+2i — Riccati–Bessel functions Sn complex plot from −2 − 2i to 2 + 2i

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 inquantum mechanics while solving the radial component of theSchrödinger equation with hypothetical cylindrical infinite potential barrier.^[39] This differential equation, and the Riccati–Bessel solutions, also arises in the problem of scattering of electromagnetic waves by a sphere, known asMie scattering after the first published solution by Mie (1908). See e.g., Du (2004)^[40] for recent developments and references.

FollowingDebye (1909), the notationψ_n,χ_n is sometimes used instead ofS_n,C_n.

Asymptotic forms

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The Bessel functions have the followingasymptotic forms. For small arguments $0<z\ll {\sqrt {\alpha +1}}$ , one obtains, when $\alpha$ is not a negative integer:^[5] $J_{\alpha }(z)\sim {\frac {1}{\Gamma (\alpha +1)}}\left({\frac {z}{2}}\right)^{\alpha }.$

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

For large real argumentsz ≫ |α² −⁠1/4⁠|, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unlessα ishalf-integer) because they havezeros all the way out to infinity, which would have to be matched exactly by any asymptotic expansion. However, for a given value ofargz one can write an equation containing a term of order|z|⁻¹:^[41] ${\begin{aligned}J_{\alpha }(z)&={\sqrt {\frac {2}{\pi z}}}\left(\cos \left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)+e^{\left|\operatorname {Im} (z)\right|}{\mathcal {O}}\left(|z|^{-1}\right)\right)&&{\text{for }}\left|\arg z\right|<\pi ,\\Y_{\alpha }(z)&={\sqrt {\frac {2}{\pi z}}}\left(\sin \left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)+e^{\left|\operatorname {Im} (z)\right|}{\mathcal {O}}\left(|z|^{-1}\right)\right)&&{\text{for }}\left|\arg z\right|<\pi .\end{aligned}}$

(Forα =⁠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: ${\begin{aligned}H_{\alpha }^{(1)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-\pi <\arg z<2\pi ,\\H_{\alpha }^{(2)}(z)&\sim {\sqrt {\frac {2}{\pi z}}}e^{-i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-2\pi <\arg z<\pi .\end{aligned}}$

These can be extended to other values ofargz using equations relatingH⁽¹⁾
_α(ze^imπ) andH⁽²⁾
_α(ze^imπ) toH⁽¹⁾
_α(z) andH⁽²⁾
_α(z).^[42]

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 whenz is negative (because one or the other will not be correct there, depending on theargz used). But the asymptotic forms for the Hankel functions permit us to write asymptotic forms for the Bessel functions of first and second kinds forcomplex (non-real)z so long as|z| goes to infinity at a constant phase angleargz (using the square root having positive real part): ${\begin{aligned}J_{\alpha }(z)&\sim {\frac {1}{\sqrt {2\pi z}}}e^{i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-\pi <\arg z<0,\\[1ex]J_{\alpha }(z)&\sim {\frac {1}{\sqrt {2\pi z}}}e^{-i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}0<\arg z<\pi ,\\[1ex]Y_{\alpha }(z)&\sim -i{\frac {1}{\sqrt {2\pi z}}}e^{i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-\pi <\arg z<0,\\[1ex]Y_{\alpha }(z)&\sim i{\frac {1}{\sqrt {2\pi z}}}e^{-i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}0<\arg z<\pi .\end{aligned}}$

For the modified Bessel functions,Hankel developedasymptotic expansions as well:^[43]^[44] ${\begin{aligned}I_{\alpha }(z)&\sim {\frac {e^{z}}{\sqrt {2\pi z}}}\left(1-{\frac {4\alpha ^{2}-1}{8z}}+{\frac {\left(4\alpha ^{2}-1\right)\left(4\alpha ^{2}-9\right)}{2!(8z)^{2}}}-{\frac {\left(4\alpha ^{2}-1\right)\left(4\alpha ^{2}-9\right)\left(4\alpha ^{2}-25\right)}{3!(8z)^{3}}}+\cdots \right)&&{\text{for }}\left|\arg z\right|<{\frac {\pi }{2}},\\K_{\alpha }(z)&\sim {\sqrt {\frac {\pi }{2z}}}e^{-z}\left(1+{\frac {4\alpha ^{2}-1}{8z}}+{\frac {\left(4\alpha ^{2}-1\right)\left(4\alpha ^{2}-9\right)}{2!(8z)^{2}}}+{\frac {\left(4\alpha ^{2}-1\right)\left(4\alpha ^{2}-9\right)\left(4\alpha ^{2}-25\right)}{3!(8z)^{3}}}+\cdots \right)&&{\text{for }}\left|\arg z\right|<{\frac {3\pi }{2}}.\end{aligned}}$

There is also the asymptotic form (for large real $z {\displaystyle z}$ )^[45] ${\begin{aligned}I_{\alpha }(z)={\frac {1}{{\sqrt {2\pi z}}{\sqrt[{4}]{1+{\frac {\alpha ^{2}}{z^{2}}}}}}}\exp \left(-\alpha \operatorname {arcsinh} \left({\frac {\alpha }{z}}\right)+z{\sqrt {1+{\frac {\alpha ^{2}}{z^{2}}}}}\right)\left(1+{\mathcal {O}}\left({\frac {1}{z{\sqrt {1+{\frac {\alpha ^{2}}{z^{2}}}}}}}\right)\right).\end{aligned}}$

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

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

Properties

[edit]

For integer orderα =n,J_n is often defined via aLaurent series for agenerating function: $e^{{\frac {x}{2}}\left(t-{\frac {1}{t}}\right)}=\sum _{n=-\infty }^{\infty }J_{n}(x)t^{n}$ an approach used byP. A. Hansen in 1843. (This can be generalized to non-integer order bycontour integration or other methods.)

Infinite series of Bessel functions in the form ${\textstyle \sum _{\nu =-\infty }^{\infty }J_{N\nu +p}(x)}$ where $\nu ,p\in \mathbb {Z} ,\ N\in \mathbb {Z} ^{+}$ arise in many physical systems and are defined in closed form by theSung series.^[46] For example, when N = 3: ${\textstyle \sum _{\nu =-\infty }^{\infty }J_{3\nu +p}(x)={\frac {1}{3}}\left[1+2\cos {(x{\sqrt {3}}/2-2\pi p/3)}\right]}$ . More generally, the Sung series and the alternating Sung series are written as: $\sum _{\nu =-\infty }^{\infty }J_{N\nu +p}(x)={\frac {1}{N}}\sum _{q=0}^{N-1}e^{ix\sin {2\pi q/N}}e^{-i2\pi pq/N}$ $\sum _{\nu =-\infty }^{\infty }(-1)^{\nu }J_{N\nu +p}(x)={\frac {1}{N}}\sum _{q=0}^{N-1}e^{ix\sin {(2q+1)\pi /N}}e^{-i(2q+1)\pi p/N}$

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

Another important relation for integer orders is theJacobi–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 aplane wave as asum of cylindrical waves, or to find theFourier series of a tone-modulatedFM 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 off. The coefficients forν = 0 have the explicit form $a_{k}^{0}={\frac {1}{2\pi i}}\int _{|z|=c}f(z)O_{k}(z)\,dz$ whereO_k isNeumann's polynomial.^[47]

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, iff 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 theLaplace transform off.^[48]

Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula: ${\begin{aligned}J_{\nu }(z)&={\frac {\left({\frac {z}{2}}\right)^{\nu }}{\Gamma \left(\nu +{\frac {1}{2}}\right){\sqrt {\pi }}}}\int _{-1}^{1}e^{izs}\left(1-s^{2}\right)^{\nu -{\frac {1}{2}}}\,ds\\[5px]&={\frac {2}{{\left({\frac {z}{2}}\right)}^{\nu }\cdot {\sqrt {\pi }}\cdot \Gamma \left({\frac {1}{2}}-\nu \right)}}\int _{1}^{\infty }{\frac {\sin zu}{\left(u^{2}-1\right)^{\nu +{\frac {1}{2}}}}}\,du\end{aligned}}$ whereν > −⁠1/2⁠ andz ∈C.^[49]This formula is useful especially when working withFourier transforms.

Because Bessel's equation becomesHermitian (self-adjoint) if it is divided byx, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that: $\int _{0}^{1}xJ_{\alpha }\left(xu_{\alpha ,m}\right)J_{\alpha }\left(xu_{\alpha ,n}\right)\,dx={\frac {\delta _{m,n}}{2}}\left[J_{\alpha +1}\left(u_{\alpha ,m}\right)\right]^{2}={\frac {\delta _{m,n}}{2}}\left[J_{\alpha }'\left(u_{\alpha ,m}\right)\right]^{2}$ whereα > −1,δ_m,n is theKronecker delta, andu_α,m is themthzero ofJ_α(x). This orthogonality relation can then be used to extract the coefficients in theFourier–Bessel series, where a function is expanded in the basis of the functionsJ_α(xu_α,m) for fixedα and varyingm.

An analogous relationship for the spherical Bessel functions follows immediately: $\int _{0}^{1}x^{2}j_{\alpha }\left(xu_{\alpha ,m}\right)j_{\alpha }\left(xu_{\alpha ,n}\right)\,dx={\frac {\delta _{m,n}}{2}}\left[j_{\alpha +1}\left(u_{\alpha ,m}\right)\right]^{2}$

If one defines aboxcar function ofx that depends on a small parameterε as: $f_{\varepsilon }(x)={\frac {1}{\varepsilon }}\operatorname {rect} \left({\frac {x-1}{\varepsilon }}\right)$ (whererect is therectangle function) then theHankel transform of it (of any given orderα > −⁠1/2⁠),g_ε(k), approachesJ_α(k) asε approaches zero, for any givenk. Conversely, the Hankel transform (of the same order) ofg_ε(k) isf_ε(x): $\int _{0}^{\infty }kJ_{\alpha }(kx)g_{\varepsilon }(k)\,dk=f_{\varepsilon }(x)$ which is zero everywhere except near 1. Asε approaches zero, the right-hand side approachesδ(x − 1), whereδ is theDirac delta function. This admits the limit (in thedistributional sense): $\int _{0}^{\infty }kJ_{\alpha }(kx)J_{\alpha }(k)\,dk=\delta (x-1)$

A change of variables then yields theclosure equation:^[50] $\int _{0}^{\infty }xJ_{\alpha }(ux)J_{\alpha }(vx)\,dx={\frac {1}{u}}\delta (u-v)$ forα > −⁠1/2⁠. For the spherical Bessel functions the orthogonality relation is: $\int _{0}^{\infty }x^{2}j_{\alpha }(ux)j_{\alpha }(vx)\,dx={\frac {\pi }{2uv}}\delta (u-v)$ forα > −1.

Another important property of Bessel's equations, which follows fromAbel's identity, involves theWronskian of the solutions: $A_{\alpha }(x){\frac {dB_{\alpha }}{dx}}-{\frac {dA_{\alpha }}{dx}}B_{\alpha }(x)={\frac {C_{\alpha }}{x}}$ whereA_α andB_α are any two solutions of Bessel's equation, andC_α is a constant independent ofx (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 $I_{\alpha }(x){\frac {dK_{\alpha }}{dx}}-{\frac {dI_{\alpha }}{dx}}K_{\alpha }(x)=-{\frac {1}{x}},$ forα > −1.

Forα > −1, the even entire function of genus 1,x^−αJ_α(x), 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

[edit]

The functionsJ_α,Y_α,H⁽¹⁾
_α, andH⁽²⁾
_α all satisfy therecurrence relations^[51] ${\frac {2\alpha }{x}}Z_{\alpha }(x)=Z_{\alpha -1}(x)+Z_{\alpha +1}(x)$ and $2{\frac {dZ_{\alpha }(x)}{dx}}=Z_{\alpha -1}(x)-Z_{\alpha +1}(x),$ whereZ denotesJ,Y,H⁽¹⁾, orH⁽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^[52] ${\begin{aligned}\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{m}\left[x^{\alpha }Z_{\alpha }(x)\right]&=x^{\alpha -m}Z_{\alpha -m}(x),\\\left({\frac {1}{x}}{\frac {d}{dx}}\right)^{m}\left[{\frac {Z_{\alpha }(x)}{x^{\alpha }}}\right]&=(-1)^{m}{\frac {Z_{\alpha +m}(x)}{x^{\alpha +m}}}.\end{aligned}}$

Using the previous relations one can arrive to similar relations for theSpherical Bessel functions:

${\frac {2\alpha +1}{x}}j_{\alpha }(x)=j_{\alpha -1}+j_{\alpha +1}$

and

${\frac {dj_{\alpha }(x)}{dx}}=j_{\alpha -1}-{\frac {\alpha +1}{x}}j_{\alpha }$

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 ${\begin{aligned}C_{\alpha -1}(x)-C_{\alpha +1}(x)&={\frac {2\alpha }{x}}C_{\alpha }(x),\\[1ex]C_{\alpha -1}(x)+C_{\alpha +1}(x)&=2{\frac {d}{dx}}C_{\alpha }(x),\end{aligned}}$ whereC_α denotesI_α ore^αiπK_α. These recurrence relations are useful for discrete diffusion problems.

Transcendence

[edit]

In 1929,Carl Ludwig Siegel proved thatJ_ν(x),J'_ν(x), and thelogarithmic derivative⁠J'_ν(x)/J_ν(x)⁠ aretranscendental numbers whenν is rational andx is algebraic and nonzero.^[53] The same proof also implies that $\Gamma (v+1)(2/x)^{v}J_{v}(x)$ is transcendental under the same assumptions.^[54]

Sums with Bessel functions

[edit]

The product of two Bessel functions admits the following sum: $\sum _{\nu =-\infty }^{\infty }J_{\nu }(x)J_{n-\nu }(y)=J_{n}(x+y),$ $\sum _{\nu =-\infty }^{\infty }J_{\nu }(x)J_{\nu +n}(y)=J_{n}(y-x).$ From these equalities it follows that $\sum _{\nu =-\infty }^{\infty }J_{\nu }(x)J_{\nu +n}(x)=\delta _{n,0}$ and as a consequence $\sum _{\nu =-\infty }^{\infty }J_{\nu }^{2}(x)=1.$

These sums can be extended to include a term multiplier that is a polynomial function of the index. For example, $\sum _{\nu =-\infty }^{\infty }\nu J_{\nu }(x)J_{\nu +n}(x)={\frac {x}{2}}\left(\delta _{n,1}+\delta _{n,-1}\right),$ $\sum _{\nu =-\infty }^{\infty }\nu J_{\nu }^{2}(x)=0,$ $\sum _{\nu =-\infty }^{\infty }\nu ^{2}J_{\nu }(x)J_{\nu +n}(x)={\frac {x}{2}}\left(\delta _{n,-1}-\delta _{n,1}\right)+{\frac {x^{2}}{4}}\left(\delta _{n,-2}+2\delta _{n,0}+\delta _{n,2}\right),$ $\sum _{\nu =-\infty }^{\infty }\nu ^{2}J_{\nu }^{2}(x)={\frac {x^{2}}{2}}.$

Multiplication theorem

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The Bessel functions obey amultiplication theorem $\lambda ^{-\nu }J_{\nu }(\lambda z)=\sum _{n=0}^{\infty }{\frac {1}{n!}}\left({\frac {\left(1-\lambda ^{2}\right)z}{2}}\right)^{n}J_{\nu +n}(z),$ whereλ andν may be taken as arbitrary complex numbers.^[55]^[56] For|λ² − 1| < 1,^[55] the above expression also holds ifJ is replaced byY. The analogous identities for modified Bessel functions and|λ² − 1| < 1 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 $\lambda ^{-\nu }K_{\nu }(\lambda z)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\left({\frac {\left(\lambda ^{2}-1\right)z}{2}}\right)^{n}K_{\nu +n}(z).$

Zeros of the Bessel function

[edit]

Bourget's hypothesis

[edit]

Bessel himself originally proved that for nonnegative integersn, the equationJ_n(x) = 0 has an infinite number of solutions inx.^[57] When the functionsJ_n(x) are plotted on the same graph, though, none of the zeros seem to coincide for different values ofn except for the zero atx = 0. This phenomenon is known asBourget's hypothesis after the 19th-century French mathematician who studied Bessel functions. Specifically it states that for any integersn ≥ 0 andm ≥ 1, the functionsJ_n(x) andJ_{n +m}(x) have no common zeros other than the one atx = 0. The hypothesis was proved byCarl Ludwig Siegel in 1929.^[58]

Transcendence

[edit]

Siegel proved in 1929 that whenν is rational, all nonzero roots ofJ_ν(x) andJ'_ν(x) aretranscendental,^[59] as are all the roots ofK_ν(x).^[54] It is also known that all roots of the higher derivatives $J_{\nu }^{(n)}(x)$ forn ≤ 18 are transcendental, except for the special values $J_{1}^{(3)}(\pm {\sqrt {3}})=0$ and $J_{0}^{(4)}(\pm {\sqrt {3}})=0$ .^[59]

Numerical approaches

[edit]

For numerical studies about the zeros of the Bessel function, seeGil, Segura & Temme (2007),Kravanja et al. (1998) andMoler (2004).

Numerical values

[edit]

The first zeros in J₀ (i.e., j_0,1, j_0,2 and j_0,3) occur at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.^[60]

History

[edit]

Waves and elasticity problems

[edit]

The first appearance of a Bessel function appears in the work ofDaniel Bernoulli in 1732, while working on the analysis of avibrating string, a problem that was tackled before by his fatherJohann Bernoulli.^[1] Daniel considered a flexible chain suspended from a fixed point above and free at its lower end.^[1] The solution of the differential equation led to the introduction of a function that is now considered $J_{0}(x)$ . Bernoulli also developed a method to find the zeros of the function.^[1]

Leonhard Euler in 1736, found a link between other functions (now known asLaguerre polynomials) and Bernoulli's solution. Euler also introduced a non-uniform chain that lead to the introduction of functions now related to modified Bessel functions $I_{n}(x)$ .^[1]

In the middle of the eighteen century,Jean le Rond d'Alembert had found aformula to solve thewave equation. By 1771 there was dispute between Bernoulli, Euler, d'Alembert andJoseph-Louis Lagrange on the nature of the solutions of vibrating strings.^[1]

Euler worked in 1778 onbuckling, introducing the concept ofEuler's critical load. To solve the problem he introduced the series for $J_{\pm 1/3}(x)$ .^[1] Euler also worked out the solutions of vibrating 2D membranes in cylindrical coordinates in 1780. In order to solve his differential equation he introduced a power series associated to $J_{n}(x)$ , for integern.^[1]

During the end of the 19th century Lagrange,Pierre-Simon Laplace andMarc-Antoine Parseval also found equivalents to the Bessel functions.^[1] Parseval for example found an integral representation of $J_{0}(x)$ using cosine.^[1]

At the beginning of the 1800s,Joseph Fourier used $J_{0}(x)$ to solve theheat equation in a problem with cylindrical symmetry.^[1] Fourier won a prize of theFrench Academy of Sciences for this work in 1811.^[1] But most of the details of his work, including the use of aFourier series, remained unpublished until 1822.^[1] Poisson in rivalry with Fourier, extended Fourier's work in 1823, introducing new properties of Bessel functions including Bessel functions of half-integer order (now known as spherical Bessel functions).^[1]

Astronomical problems

[edit]

In 1770, Lagrange introduced the series expansion of Bessel functions to solveKepler's equation, a transcendental equation in astronomy.Friedrich Wilhelm Bessel had seen Lagrange's solution but found it difficult to handle. In 1813 in a letter toCarl Friedrich Gauss, Bessel simplified the calculation using trigonometric functions.^[1] Bessel published his work in 1819, independently introducing the method of Fourier series unaware of the work of Fourier which was published later.^[1]In 1824, Bessel carried out a systematic investigation of the functions, which earned the functions his name.^[1] In older literature the functions were called cylindrical functions or even Bessel–Fourier functions.^[1]

Notes

[edit]

^^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q ^rDutka, Jacques (1995). "On the early history of Bessel functions".Archive for History of Exact Sciences.49 (2):105–134.doi:10.1007/BF00376544.
^Wilensky, Michael; Brown, Jordan; Hazelton, Bryna (June 2023)."Why and when to expect Gaussian error distributions in epoch of reionization 21-cm power spectrum measurements".Monthly Notices of the Royal Astronomical Society.521 (4):5191–5206.arXiv:2211.13576.doi:10.1093/mnras/stad863.
^Weisstein, Eric W."Spherical Bessel Function of the Second Kind".MathWorld.
^^a ^bWeisstein, Eric W."Bessel Function of the Second Kind".MathWorld.
^^a ^bAbramowitz and Stegun,p. 360, 9.1.10.
^Whittaker, Edmund Taylor;Watson, George Neville (1927).A Course of Modern Analysis (4th ed.). Cambridge University Press. p. 356. For example, Hansen (1843) and Schlömilch (1857).
^Abramowitz and Stegun,p. 358, 9.1.5.
^^a ^bTemme, Nico M. (1996).Special Functions: An introduction to the classical functions of mathematical physics (2nd print ed.). New York: Wiley. pp. 228–231.ISBN 0471113131.
^Weisstein, Eric W."Hansen-Bessel Formula".MathWorld.
^Bessel, F. (1824). The relevant integral is an unnumbered equation between equations 28 and 29. Note that Bessel's $I_{k}^{h}$ would today be written $J_{h}(k)$ .
^Watson,p. 176
^"Properties of Hankel and Bessel Functions". Archived fromthe original on 2010-09-23. Retrieved2010-10-18.
^"Integral representations of the Bessel function".www.nbi.dk. Archived fromthe original on 3 October 2022. Retrieved25 March 2018.
^Arfken & Weber, exercise 11.1.17.
^Abramowitz and Stegun,p. 362, 9.1.69.
^Szegő, Gábor (1975).Orthogonal Polynomials (4th ed.). Providence, RI: AMS.
^"Bessel Functions of the First and Second Kind"(PDF).mhtlab.uwaterloo.ca. p. 3.Archived(PDF) from the original on 2022-10-09. Retrieved24 May 2022.
^NIST Digital Library of Mathematical Functions, (10.8.1). Accessed on line Oct. 25, 2016.
^^a ^bWatson,p. 178.
^Abramowitz and Stegun,p. 358, 9.1.3, 9.1.4.
^Abramowitz and Stegun,p. 358, 9.1.6.
^Abramowitz and Stegun,p. 360, 9.1.25.
^Abramowitz and Stegun,p. 375, 9.6.2, 9.6.10, 9.6.11.
^Dixon; Ferrar, W.L. (1930). "A direct proof of Nicholson's integral".The Quarterly Journal of Mathematics. Oxford:236–238.doi:10.1093/qmath/os-1.1.236.
^Abramowitz and Stegun,p. 375, 9.6.3, 9.6.5.
^Abramowitz and Stegun,p. 374, 9.6.1.
^Greiner, Walter; Reinhardt, Joachim (2009).Quantum Electrodynamics. Springer. p. 72.ISBN 978-3-540-87561-1.
^Watson,p. 181.
^"Modified Bessel Functions §10.32 Integral Representations".NIST Digital Library of Mathematical Functions. NIST. Retrieved2024-11-20.
^Khokonov, M. Kh. (2004). "Cascade Processes of Energy Loss by Emission of Hard Photons".Journal of Experimental and Theoretical Physics.99 (4):690–707.Bibcode:2004JETP...99..690K.doi:10.1134/1.1826160.S2CID 122599440.. Derived from formulas sourced toI. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, and Products (Fizmatgiz, Moscow, 1963; Academic Press, New York, 1980).
^Referred to as such in:Teichroew, D. (1957)."The Mixture of Normal Distributions with Different Variances"(PDF).The Annals of Mathematical Statistics.28 (2):510–512.doi:10.1214/aoms/1177706981.
^Abramowitz and Stegun,p. 437, 10.1.1.
^Abramowitz and Stegun,p. 439, 10.1.25, 10.1.26.
^Abramowitz and Stegun,p. 438, 10.1.11.
^Abramowitz and Stegun,p. 438, 10.1.12.
^Abramowitz and Stegun,p. 439, 10.1.39.
^L.V. Babushkina, M.K. Kerimov, A.I. Nikitin, Algorithms for computing Bessel functions of half-integer order with complex arguments,p. 110, p. 111.
^Abramowitz and Stegun,p. 439, 10.1.23, 10.1.24.
^Griffiths. Introduction to Quantum Mechanics, 2nd edition, p. 154.
^Du, Hong (2004). "Mie-scattering calculation".Applied Optics.43 (9):1951–1956.Bibcode:2004ApOpt..43.1951D.doi:10.1364/ao.43.001951.PMID 15065726.
^Abramowitz and Stegun,p. 364, 9.2.1.
^NIST Digital Library of Mathematical Functions, Section10.11.
^Abramowitz and Stegun,p. 377, 9.7.1.
^Abramowitz and Stegun,p. 378, 9.7.2.
^Fröhlich and Spencer 1981 Appendix B
^Sung, S.; Hovden, R. (2022). "On Infinite Series of Bessel functions of the First Kind".arXiv:2211.01148 [math-ph].
^Abramowitz and Stegun,p. 363, 9.1.82 ff.
^Watson, G. N. (25 August 1995).A Treatise on the Theory of Bessel Functions. Cambridge University Press.ISBN 9780521483919. Retrieved25 March 2018 – via Google Books.
^Gradshteyn, Izrail Solomonovich;Ryzhik, Iosif Moiseevich;Geronimus, Yuri Veniaminovich;Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "8.411.10.". In Zwillinger, Daniel;Moll, Victor Hugo (eds.).Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.).Academic Press, Inc.ISBN 978-0-12-384933-5.LCCN 2014010276.
^Arfken & Weber, section 11.2
^Abramowitz and Stegun,p. 361, 9.1.27.
^Abramowitz and Stegun,p. 361, 9.1.30.
^Siegel, Carl L. (2014)."Über einige Anwendungen diophantischer Approximationen".On Some Applications of Diophantine Approximations: a translation of Carl Ludwig Siegel's Über einige Anwendungen diophantischer Approximationen by Clemens Fuchs, with a commentary and the article Integral points on curves: Siegel's theorem after Siegel's proof by Clemens Fuchs and Umberto Zannier (in German). Scuola Normale Superiore. pp. 81–138.doi:10.1007/978-88-7642-520-2_2.ISBN 978-88-7642-520-2.
^^a ^bJames, R. D. (November 1950)."Review: Carl Ludwig Siegel, Transcendental numbers".Bulletin of the American Mathematical Society.56 (6):523–526.doi:10.1090/S0002-9904-1950-09435-X.
^^a ^bAbramowitz and Stegun,p. 363, 9.1.74.
^Truesdell, C. (1950)."On the Addition and Multiplication Theorems for the Special Functions".Proceedings of the National Academy of Sciences.1950 (12):752–757.Bibcode:1950PNAS...36..752T.doi:10.1073/pnas.36.12.752.PMC 1063284.PMID 16578355.
^Bessel, F. (1824), article 14.
^Watson, pp. 484–485.
^^a ^bLorch, Lee; Muldoon, Martin E. (1995)."Transcendentality of zeros of higher dereivatives of functions involving Bessel functions".International Journal of Mathematics and Mathematical Sciences.18 (3):551–560.doi:10.1155/S0161171295000706.
^Abramowitz & Stegun, p409

References

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Abramowitz, Milton;Stegun, Irene Ann, eds. (1983) [June 1964]."Chapter 9".Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 355, 435.ISBN 978-0-486-61272-0.LCCN 64-60036.MR 0167642.LCCN 65-12253. See alsochapter 10.
Arfken, George B. and Hans J. Weber,Mathematical Methods for Physicists, 6th edition (Harcourt: San Diego, 2005).ISBN 0-12-059876-0.
Bessel, Friedrich (1824). "Untersuchung des Theils der planetarischen Störungen, welcher aus der Bewegung der Sonne entsteht" [Investigation of the part of the planetary disturbances which arise from the movement of the sun].Berlin Abhandlungen. Reproduced as pages 84 to 109 inAbhandlungen von Friedrich Wilhelm Bessel. Leipzig: Engelmann. 1875.English translation of the text.
Bowman, FrankIntroduction to Bessel Functions (Dover: New York, 1958).ISBN 0-486-60462-4.
Gil, A.; Segura, J.; Temme, N. M. (2007).Numerical methods for special functions. Society for Industrial and Applied Mathematics.
Kravanja, P.; Ragos, O.; Vrahatis, M.N.; Zafiropoulos, F.A. (1998), "ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument",Computer Physics Communications,113 (2–3):220–238,Bibcode:1998CoPhC.113..220K,doi:10.1016/S0010-4655(98)00064-2
Mie, G. (1908)."Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen".Annalen der Physik.25 (3): 377.Bibcode:1908AnP...330..377M.doi:10.1002/andp.19083300302.
Olver, F. W. J.; Maximon, L. C. (2010),"Bessel function", inOlver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),NIST Handbook of Mathematical Functions, Cambridge University Press,ISBN 978-0-521-19225-5,MR 2723248..
Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007),"Section 6.5. Bessel Functions of Integer Order",Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press,ISBN 978-0-521-88068-8, archived fromthe original on 2021-02-03, retrieved2022-09-28.
B Spain, M. G. Smith,Functions of mathematical physics, Van Nostrand Reinhold Company, London, 1970. Chapter 9 deals with Bessel functions.
N. M. Temme,Special Functions. An Introduction to the Classical Functions of Mathematical Physics, John Wiley and Sons, Inc., New York, 1996.ISBN 0-471-11313-1. Chapter 9 deals with Bessel functions.
Watson, G. N.,A Treatise on the Theory of Bessel Functions, Second Edition, (1995) Cambridge University Press.ISBN 0-521-48391-3.
Weber, Heinrich (1873), "Ueber eine Darstellung willkürlicher Functionen durch Bessel'sche Functionen",Mathematische Annalen,6 (2):146–161,doi:10.1007/BF01443190,S2CID 122409461.

External links

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Lizorkin, P. I. (2001) [1994],"Bessel functions",Encyclopedia of Mathematics,EMS Press.
Karmazina, L. N.; Prudnikov, A.P. (2001) [1994],"Cylinder function",Encyclopedia of Mathematics,EMS Press.
Rozov, N. Kh. (2001) [1994],"Bessel equation",Encyclopedia of Mathematics,EMS Press.
Wolfram function pages on BesselJ andY functions, and modified BesselI andK functions. Pages include formulas, function evaluators, and plotting calculators.
Weisstein, Eric W."Bessel functions of the first kind".MathWorld.
Bessel functionsJ_ν,Y_ν,I_ν andK_ν in LibrowFunction handbook.
F. W. J. Olver, L. C. Maximon,Bessel Functions (chapter 10 of the Digital Library of Mathematical Functions).
Moler, C. B. (2004).Numerical Computing with MATLAB(PDF). Society for Industrial and Applied Mathematics. Archived fromthe original(PDF) on 2017-08-08.