Bessel function#Bessel functions of the first kind : J.CE.B1

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 ({{math|1=α = n}}); in spherical problems, one obtains half-integer orders ({{math|1=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 radial 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{{cite journal |last1=Wilensky |first1=Michael |last2=Brown |first2=Jordan |last3=Hazelton |first3=Bryna |title=Why and when to expect Gaussian error distributions in epoch of reionization 21-cm power spectrum measurements |journal=Monthly Notices of the Royal Astronomical Society |date=June 2023 |volume=521 |issue=4 |pages=5191–5206 |doi=10.1093/mnras/stad863|doi-access=free |arxiv=2211.13576 }}
• 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. 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 {{mvar|N_n}} and {{mvar|n_n}}, respectively, rather than {{mvar|Y_n}} and {{mvar|y_n}}.{{MathWorld|id=SphericalBesselFunctionoftheSecondKind|title=Spherical Bessel Function of the Second Kind}}

File:BesselJ.png

File:Bessel half.png

Bessel functions of the first kind, denoted as {{math|J_α(x)}}, are solutions of Bessel's differential equation. For integer or positive {{mvar|α}}, Bessel functions of the first kind are finite at the origin ({{math|1=x = 0}}); while for negative non-integer {{mvar|α}}, Bessel functions of the first kind diverge as {{mvar|x}} approaches zero. It is possible to define the function by $x^\backslash alpha$ times a Maclaurin series (note that {{mvar|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:Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_360.htm p. 360, 9.1.10].

$$J\_\backslash alpha(x)\; =\; \backslash sum\_\{m=0\}^\backslash infty\; \backslash frac\{(-1)^m\}\{m!\backslash ,\; \backslash Gamma(m+\backslash alpha+1)\}\; \{\backslash left(\backslash frac\{x\}\{2\}\backslash right)\}^\{2m\; +\; \backslash alpha\},$$

where {{math|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 $2$ in $x/2$;{{cite book |last1=Whittaker |first1=Edmund Taylor |authorlink1=Edmund T. Whittaker |last2=Watson |first2=George Neville |authorlink2=George N. Watson |date= 1927 |page=356 |edition=4th |title=A Course of Modern Analysis |title-link=A Course of Modern Analysis |publisher= Cambridge University Press}} For example, Hansen (1843) and Schlömilch (1857). this definition is not used in this article. The Bessel function of the first kind is an entire function if {{mvar|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 $x^\{-\{1\}/\{2\}\}$ (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large {{mvar|x}}. (The series indicates that {{math|−J₁(x)}} is the derivative of {{math|J₀(x)}}, much like {{math|−sin x}} is the derivative of {{math|cos x}}; more generally, the derivative of {{math|J_n(x)}} can be expressed in terms of {{math|J_{n ± 1}(x)}} by the identities below.)

For non-integer {{mvar|α}}, the functions {{math|J_α(x)}} and {{math|J_−α(x)}} are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order {{mvar|n}}, the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_358.htm p. 358, 9.1.5].

$$J\_\{-n\}(x)\; =\; (-1)^n\; J\_n(x).$$

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 {{mvar|n}}, is possible using an integral representation:{{cite book |last=Temme |first=Nico M. |title=Special Functions: An introduction to the classical functions of mathematical physics |year=1996 |publisher=Wiley |location=New York |isbn=0471113131 |pages=228–231 |edition=2nd print}}

$$J\_n(x)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\backslash pi\; \backslash cos\; (n\; \backslash tau\; -\; x\; \backslash sin\; \backslash tau)\; \backslash ,d\backslash tau\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash operatorname\{Re\}\backslash left(\backslash int\_\{0\}^\backslash pi\; e^\{i(n\; \backslash tau-x\; \backslash sin\; \backslash tau\; )\}\; \backslash ,d\backslash tau\backslash right),$$

which is also called Hansen-Bessel formula.{{MathWorld|id=Hansen-BesselFormula|title=Hansen-Bessel Formula}}

This was the approach that Bessel used,Bessel, F. (1824). The relevant integral is an unnumbered equation between equations 28 and 29. Note that Bessel's $I^h\_k$ would today be written $J\_h(k)$. 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 {{math|Re(x) > 0}}:Watson, [https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA176 p. 176]{{cite web |url=http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html |title=Properties of Hankel and Bessel Functions |access-date=2010-10-18 |url-status=dead |archive-url=https://web.archive.org/web/20100923194031/http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html |archive-date=2010-09-23}}{{cite web |url=https://www.nbi.dk/~polesen/borel/node15.html |title=Integral representations of the Bessel function |website=www.nbi.dk |access-date=25 March 2018 |archive-date=3 October 2022 |archive-url=https://web.archive.org/web/20221003054117/https://www.nbi.dk/~polesen/borel/node15.html |url-status=dead }}Arfken & Weber, exercise 11.1.17.

$$J\_\backslash alpha(x)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\backslash pi\; \backslash cos(\backslash alpha\backslash tau\; -\; x\; \backslash sin\backslash tau)\backslash ,d\backslash tau\; -\; \backslash frac\{\backslash sin(\backslash alpha\backslash pi)\}\{\backslash pi\}\; \backslash int\_0^\backslash infty\; e^\{-x\; \backslash sinh\; t\; -\; \backslash alpha\; t\}\; \backslash ,\; dt.$$

The Bessel functions can be expressed in terms of the generalized hypergeometric series asAbramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_362.htm p. 362, 9.1.69].

$$J\_\backslash alpha(x)\; =\; \backslash frac\{\backslash left(\backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash ;\_0F\_1\; \backslash left(\backslash alpha+1;\; -\backslash frac\{x^2\}\{4\}\backslash right).$$

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

In terms of the Laguerre polynomials {{mvar|L_k}} and arbitrarily chosen parameter {{mvar|t}}, the Bessel function can be expressed as{{cite book |author-link=Gábor Szegő |last=Szegő |first=Gábor |title=Orthogonal Polynomials |edition=4th |location=Providence, RI |publisher=AMS |date=1975}}

$$\backslash frac\{J\_\backslash alpha(x)\}\{\backslash left(\; \backslash frac\{x\}\{2\}\backslash right)^\backslash alpha\}\; =\; \backslash frac\{e^\{-t\}\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{L\_k^\{(\backslash alpha)\}\backslash left(\; \backslash frac\{x^2\}\{4\; t\}\backslash right)\}\{\backslash binom\{k+\backslash alpha\}\{k\}\}\; \backslash frac\{t^k\}\{k!\}.$$

File:Besselyn.png

The Bessel functions of the second kind, denoted by {{math|Y_α(x)}}, occasionally denoted instead by {{math|N_α(x)}}, are solutions of the Bessel differential equation that have a singularity at the origin ({{math|1=x = 0}}) and are multivalued. These are sometimes called Weber functions, as they were introduced by {{harvs|txt|authorlink=Heinrich Martin Weber|first=H. M.|last=Weber|year=1873}}, and also Neumann functions after Carl Neumann.{{cite web |url=http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf |archive-date=2022-10-09 |url-status=live |title=Bessel Functions of the First and Second Kind |website=mhtlab.uwaterloo.ca |access-date=24 May 2022 |page=3}}

For non-integer {{mvar|α}}, {{math|Y_α(x)}} is related to {{math|J_α(x)}} by

$$Y\_\backslash alpha(x)\; =\; \backslash frac\{J\_\backslash alpha(x)\; \backslash cos\; (\backslash alpha\; \backslash pi)\; -\; J\_\{-\backslash alpha\}(x)\}\{\backslash sin\; (\backslash alpha\; \backslash pi)\}.$$

In the case of integer order {{mvar|n}}, the function is defined by taking the limit as a non-integer {{mvar|α}} tends to {{mvar|n}}:

$$Y\_n(x)\; =\; \backslash lim\_\{\backslash alpha\; \backslash to\; n\}\; Y\_\backslash alpha(x).$$

If {{mvar|n}} is a nonnegative integer, we have the series[https://dlmf.nist.gov/10.8#E1 NIST Digital Library of Mathematical Functions], (10.8.1). Accessed on line Oct. 25, 2016.

$$Y\_n(z)\; =-\backslash frac\{\backslash left(\backslash frac\{z\}\{2\}\backslash right)^\{-n\}\}\{\backslash pi\}\backslash sum\_\{k=0\}^\{n-1\}\; \backslash frac\{(n-k-1)!\}\{k!\}\backslash left(\backslash frac\{z^2\}\{4\}\backslash right)^k\; +\backslash frac\{2\}\{\backslash pi\}\; J\_n(z)\; \backslash ln\; \backslash frac\{z\}\{2\}\; -\backslash frac\{\backslash left(\backslash frac\{z\}\{2\}\backslash right)^n\}\{\backslash pi\}\backslash sum\_\{k=0\}^\backslash infty\; (\backslash psi(k+1)+\backslash psi(n+k+1))\; \backslash frac\{\backslash left(-\backslash frac\{z^2\}\{4\}\backslash right)^k\}\{k!(n+k)!\}$$

where $\backslash psi(z)$ is the digamma function, the logarithmic derivative of the gamma function.{{MathWorld|id=BesselFunctionoftheSecondKind|title=Bessel Function of the Second Kind}}

There is also a corresponding integral formula (for {{math|Re(x) > 0}}):Watson, [https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA178 p. 178].

$$Y\_n(x)\; =\; \backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\backslash pi\; \backslash sin(x\; \backslash sin\backslash theta\; -\; n\backslash theta)\; \backslash ,\; d\backslash theta\; -\backslash frac\{1\}\{\backslash pi\}\; \backslash int\_0^\backslash infty\; \backslash left(e^\{nt\}\; +\; (-1)^n\; e^\{-nt\}\; \backslash right)\; e^\{-x\; \backslash sinh\; t\}\; \backslash ,\; dt.$$

In the case where {{math|n {{=}} 0}}: (with $\backslash gamma$ being Euler's constant)$$Y\_\{0\}\backslash left(x\backslash right)=\backslash frac\{4\}\{\backslash pi^\{2\}\}\backslash int\_\{0\}^\{\backslash frac\{1\}\{2\}\backslash pi\}\backslash cos\backslash left(x\backslash cos\backslash theta\backslash right)\backslash left(\backslash gamma+\backslash ln\backslash left(2x\backslash sin^2\backslash theta\backslash right)\backslash right)\backslash ,\; d\backslash theta.$$

File:Besselyhalf.png

{{math|Y_α(x)}} is necessary as the second linearly independent solution of the Bessel's equation when {{mvar|α}} is an integer. But {{math|Y_α(x)}} has more meaning than that. It can be considered as a "natural" partner of {{math|J_α(x)}}. See also the subsection on Hankel functions below.

When {{mvar|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 {{math|J_α(x)}} and {{math|Y_α(x)}} are holomorphic functions of {{mvar|x}} on the complex plane cut along the negative real axis. When {{mvar|α}} is an integer, the Bessel functions {{mvar|J}} are entire functions of {{mvar|x}}. If {{mvar|x}} is held fixed at a non-zero value, then the Bessel functions are entire functions of {{mvar|α}}.

The Bessel functions of the second kind when {{mvar|α}} is an integer is an example of the second kind of solution in Fuchs's theorem.

File: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.svg

File: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.svg

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, {{math|H{{su|b=α|p=(1)}}(x)}} and {{math|H{{su|b=α|p=(2)}}(x)}}, defined asAbramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_358.htm p. 358, 9.1.3, 9.1.4].

$$\backslash begin\{align\}$$

H_\alpha^{(1)}(x) &= J_\alpha(x) + iY_\alpha(x), \\[5pt]

H_\alpha^{(2)}(x) &= J_\alpha(x) - iY_\alpha(x),

\end{align}

where {{mvar|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 {{math|e^{i f(x)}}}. For real $x>0$ where $J\_\backslash alpha(x)$, $Y\_\backslash alpha(x)$ 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 {{math|H{{su|b=α|p=(1)}}(x)}}, {{math|H{{su|b=α|p=(2)}}(x)}} for $e^\{\backslash pm\; i\; x\}$ and $J\_\backslash alpha(x)$, $Y\_\backslash alpha(x)$ for $\backslash cos(x)$, $\backslash sin(x)$, 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

$$\backslash begin\{align\}$$

H_\alpha^{(1)}(x) &= \frac{J_{-\alpha}(x) - e^{-\alpha \pi i} J_\alpha(x)}{i \sin \alpha\pi}, \\[5pt]

H_\alpha^{(2)}(x) &= \frac{J_{-\alpha}(x) - e^{\alpha \pi i} J_\alpha(x)}{- i \sin \alpha\pi}.

\end{align}

If {{mvar|α}} is an integer, the limit has to be calculated. The following relationships are valid, whether {{mvar|α}} is an integer or not:Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_358.htm p. 358, 9.1.6].

$$\backslash begin\{align\}$$

H_{-\alpha}^{(1)}(x) &= e^{\alpha \pi i} H_\alpha^{(1)} (x), \\[6mu]

H_{-\alpha}^{(2)}(x) &= e^{-\alpha \pi i} H_\alpha^{(2)} (x).

\end{align}

In particular, if {{math|1=α = m + {{sfrac|1|2}}}} with {{mvar|m}} a nonnegative integer, the above relations imply directly that

$$\backslash begin\{align\}$$

J_{-(m+\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\frac{1}{2}}(x), \\[5pt]

Y_{-(m+\frac{1}{2})}(x) &= (-1)^m J_{m+\frac{1}{2}}(x).

\end{align}

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

The Hankel functions admit the following integral representations for {{math|Re(x) > 0}}:Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_360.htm p. 360, 9.1.25].

$$\backslash begin\{align\}$$

H_\alpha^{(1)}(x) &= \frac{1}{\pi i}\int_{-\infty}^{+\infty + \pi i} e^{x\sinh t - \alpha t} \, dt, \\[5pt]

H_\alpha^{(2)}(x) &= -\frac{1}{\pi i}\int_{-\infty}^{+\infty - \pi i} e^{x\sinh t - \alpha t} \, dt,

\end{align}

where the integration limits indicate integration along a contour that can be chosen as follows: from {{math|−∞}} to 0 along the negative real axis, from 0 to {{math|±{{pi}}i}} along the imaginary axis, and from {{math|±{{pi}}i}} to {{math|+∞ ± {{pi}}i}} along a contour parallel to the real axis.

The Bessel functions are valid even for complex arguments {{mvar|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 asAbramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_375.htm p. 375, 9.6.2, 9.6.10, 9.6.11].

$$\backslash begin\{align\}$$

I_\alpha(x) &= i^{-\alpha} J_\alpha(ix) = \sum_{m=0}^\infty \frac{1}{m!\, \Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha}, \\[5pt]

K_\alpha(x) &= \frac{\pi}{2} \frac{I_{-\alpha}(x) - I_\alpha(x)}{\sin \alpha \pi},

\end{align}

when {{mvar|α}} is not an integer; when {{mvar|α}} is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments {{mvar|x}}. The series expansion for {{math|I_α(x)}} is thus similar to that for {{math|J_α(x)}}, but without the alternating {{math|(−1)^m}} factor.

$K\_\{\backslash alpha\}$ can be expressed in terms of Hankel functions:

$$K\_\{\backslash alpha\}(x)\; =\; \backslash begin\{cases\}$$

\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 $J\_\{\backslash alpha\}^2(z)$+$Y\_\{\backslash alpha\}^2(z)$, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following

J_{\alpha}^2(x)+Y_{\alpha}^2(x)=\frac{8}{\pi^2}\int_{0}^{\infty}\cosh(2\alpha t)K_0(2x\sinh t)\, dt,

given that the condition {{math|Re(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 |{{math|Re(α)}}| < {{math|{{sfrac|1|2}}}} and {{math|Re(x) ≥ 0}} but not when {{math|1=x = 0}}.{{cite journal |last1=Dixon |last2=Ferrar |first2=W.L. |date=1930 |title=A direct proof of Nicholson's integral |journal=The Quarterly Journal of Mathematics |location=Oxford |pages=236–238 |doi=10.1093/qmath/os-1.1.236}}

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if {{math|−π < arg z ≤ {{sfrac|π|2}}}}):Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_375.htm p. 375, 9.6.3, 9.6.5].

$$\backslash begin\{align\}$$

J_\alpha(iz) &= e^{\frac{\alpha\pi i}{2}} I_\alpha(z), \\[1ex]

Y_\alpha(iz) &= e^{\frac{(\alpha+1)\pi i}{2}}I_\alpha(z) - \tfrac{2}{\pi} e^{-\frac{\alpha\pi i}{2}}K_\alpha(z).

\end{align}

{{math|I_α(x)}} and {{math|K_α(x)}} are the two linearly independent solutions to the modified Bessel's equation:Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_374.htm p. 374, 9.6.1].

$$x^2\; \backslash frac\{d^2\; y\}\{dx^2\}\; +\; x\; \backslash frac\{dy\}\{dx\}\; -\; \backslash left(x^2\; +\; \backslash alpha^2\; \backslash right)y\; =\; 0.$$

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, {{mvar|I_α}} and {{mvar|K_α}} are exponentially growing and decaying functions respectively. Like the ordinary Bessel function {{mvar|J_α}}, the function {{mvar|I_α}} goes to zero at {{math|1=x = 0}} for {{math|α > 0}} and is finite at {{math|1=x = 0}} for {{math|1=α = 0}}. Analogously, {{mvar|K_α}} diverges at {{math|1=x = 0}} with the singularity being of logarithmic type for {{mvar|K₀}}, and {{math|1={{sfrac|1|2}}Γ({{abs|α}})(2/x)^{{abs|α}}}} otherwise.{{cite book |title=Quantum Electrodynamics |last1=Greiner |first1=Walter |last2=Reinhardt |first2=Joachim |date=2009 |publisher=Springer |page=72 |isbn=978-3-540-87561-1}}

Two integral formulas for the modified Bessel functions are (for {{math|Re(x) > 0}}):Watson, [https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA181 p. 181].

$$\backslash begin\{align\}$$

I_\alpha(x) &= \frac{1}{\pi}\int_0^\pi e^{x\cos \theta} \cos \alpha\theta \,d\theta - \frac{\sin \alpha\pi}{\pi}\int_0^\infty e^{-x\cosh t - \alpha t} \,dt, \\[5pt]

K_\alpha(x) &= \int_0^\infty e^{-x\cosh t} \cosh \alpha t \,dt.

\end{align}

Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for {{math|Re(ω) > 0}}):

$$2\backslash ,K\_0(\backslash omega)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash frac\{e^\{i\backslash omega\; t\}\}\{\backslash sqrt\{t^2+1\}\}\; \backslash ,dt.$$

It can be proven by showing equality to the above integral definition for {{math|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 {{cite web |url=http://dlmf.nist.gov/10.32.E11

|title=Modified Bessel Functions §10.32 Integral Representations |author= |date= |website=NIST Digital Library of Mathematical Functions |publisher=NIST |access-date=2024-11-20}}

$$K\_n(xz)\; =\; \backslash frac\{\backslash Gamma\backslash left(n+\backslash frac\{1\}\{2\}\backslash right)(2z)^\{n\}\}\{\backslash sqrt\{\backslash pi\}\; x^\{n\}\}\; \backslash int\_0^\backslash infty\; \backslash frac\{\backslash cos\; (xt)\backslash ,dt\}\{(t^2+z^2)^\{n+\backslash frac\{1\}\{2\}\}\}.$$

Modified Bessel functions {{math|K_1/3}} and {{math|K_2/3}} can be represented in terms of rapidly convergent integrals{{cite journal |first=M. Kh. |last=Khokonov |title=Cascade Processes of Energy Loss by Emission of Hard Photons |journal=Journal of Experimental and Theoretical Physics |volume=99 |issue=4 |pages=690–707 |date=2004 |doi=10.1134/1.1826160 |bibcode=2004JETP...99..690K |s2cid=122599440}}. Derived from formulas sourced to I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Fizmatgiz, Moscow, 1963; Academic Press, New York, 1980).

$$\backslash begin\{align\}$$

K_{\frac{1}{3}}(\xi) &= \sqrt{3} \int_0^\infty \exp \left(- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \right) \,dx, \\[5pt]

K_{\frac{2}{3}}(\xi) &= \frac{1}{\sqrt{3}} \int_0^\infty \frac{3+2x^2}{\sqrt{1+\frac{x^2}{3}}} \exp \left(- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}}\right) \,dx.

\end{align}

The modified Bessel function $K\_\{\backslash frac\{1\}\{2\}\}(\backslash xi)=(2\; \backslash xi\; /\; \backslash pi)^\{-1/2\}\backslash exp(-\backslash xi)$ 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 functionReferred to as such in: {{cite journal |last=Teichroew |first=D. |title=The Mixture of Normal Distributions with Different Variances |journal=The Annals of Mathematical Statistics |volume=28 |issue=2 |date=1957 |pages=510–512 |doi=10.1214/aoms/1177706981 |url=https://dml.cz/bitstream/handle/10338.dmlcz/103973/AplMat_27-1982-4_7.pdf |doi-access=free}}
• Macdonald function after Hector Munro Macdonald

File: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.svg

File: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.svg

File:Sphericalbesselj.png

File:Sphericalbessely.png

When solving the Helmholtz equation in spherical coordinates by separation of variables, the radial equation has the form

$$x^2\; \backslash frac\{d^2\; y\}\{dx^2\}\; +\; 2x\; \backslash frac\{d\; y\}\{dx\}\; +\backslash left(x^2\; -\; n(n\; +\; 1)\backslash right)\; y\; =\; 0.$$

The two linearly independent solutions to this equation are called the spherical Bessel functions {{mvar|j_n}} and {{mvar|y_n}}, and are related to the ordinary Bessel functions {{mvar|J_n}} and {{mvar|Y_n}} byAbramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_437.htm p. 437, 10.1.1].

$$\backslash begin\{align\}$$

j_n(x) &= \sqrt{\frac{\pi}{2x}} J_{n+\frac{1}{2}}(x), \\

y_n(x) &= \sqrt{\frac{\pi}{2x}} Y_{n+\frac{1}{2}}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-\frac{1}{2}}(x).

\end{align}

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

From the relations to the ordinary Bessel functions it is directly seen that:

$$\backslash begin\{align\}$$

j_n(x) &= (-1)^{n} y_{-n-1} (x)

\\

y_n(x) &= (-1)^{n+1} j_{-n-1}(x)

\end{align}

The spherical Bessel functions can also be written as ({{va|Rayleigh's formulas}})Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_439.htm p. 439, 10.1.25, 10.1.26].

$$\backslash begin\{align\}$$

j_n(x) &= (-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\sin x}{x}, \\

y_n(x) &= -(-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\cos x}{x}.

\end{align}

The zeroth spherical Bessel function {{math|j₀(x)}} is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_438.htm p. 438, 10.1.11].

$$\backslash begin\{align\}$$

j_0(x) &= \frac{\sin x}{x}. \\

j_1(x) &= \frac{\sin x}{x^2} - \frac{\cos x}{x}, \\

j_2(x) &= \left(\frac{3}{x^2} - 1\right) \frac{\sin x}{x} - \frac{3\cos x}{x^2}, \\

j_3(x) &= \left(\frac{15}{x^3} - \frac{6}{x}\right) \frac{\sin x}{x} - \left(\frac{15}{x^2} - 1\right) \frac{\cos x}{x}

\end{align}

andAbramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_438.htm p. 438, 10.1.12].

$$\backslash begin\{align\}$$

y_0(x) &= -j_{-1}(x) = -\frac{\cos x}{x}, \\

y_1(x) &= j_{-2}(x) = -\frac{\cos x}{x^2} - \frac{\sin x}{x}, \\

y_2(x) &= -j_{-3}(x) = \left(-\frac{3}{x^2} + 1\right) \frac{\cos x}{x} - \frac{3\sin x}{x^2}, \\

y_3(x) &= j_{-4}(x) = \left(-\frac{15}{x^3} + \frac{6}{x}\right) \frac{\cos x}{x} - \left(\frac{15}{x^2} - 1\right) \frac{\sin x}{x}.

\end{align}

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

The spherical Bessel functions have the generating functionsAbramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_439.htm p. 439, 10.1.39].

$$\backslash begin\{align\}$$

\frac{1}{z} \cos \left(\sqrt{z^2 - 2zt}\right) &= \sum_{n=0}^\infty \frac{t^n}{n!} j_{n-1}(z), \\

\frac{1}{z} \sin \left(\sqrt{z^2 - 2zt}\right) &= \sum_{n=0}^\infty \frac{t^n}{n!} y_{n-1}(z).

\end{align}

In contrast to the whole integer Bessel functions {{math|J_n(x), Y_n(x)}}, the spherical Bessel functions {{math|j_n(x), y_n(x)}} have a finite series expression:L.V. Babushkina, M.K. Kerimov, A.I. Nikitin, Algorithms for computing Bessel functions of half-integer order with complex arguments, [https://www.sciencedirect.com/science/article/pii/0041555388900183 p. 110, p. 111].

$$\backslash begin\{alignat\}\{2\}$$

j_n(x) &= \sqrt{\frac{\pi}{2x}}J_{n+\frac{1}{2}}(x) = \\

&= \frac{1}{2x} \left[ e^{ix} \sum_{r=0}^n \frac{i^{r-n-1}(n+r)!}{r!(n-r)!(2x)^r} + e^{-ix} \sum_{r=0}^n \frac{(-i)^{r-n-1}(n+r)!}{r!(n-r)!(2x)^r} \right] \\

&= \frac{1}{x} \left[ \sin\left(x-\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n}{2} \right]} \frac{(-1)^r (n+2r)!}{(2r)!(n-2r)!(2x)^{2r}} + \cos\left(x-\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n-1}{2} \right]} \frac{(-1)^r (n+2r+1)!}{(2r+1)!(n-2r-1)!(2x)^{2r+1}} \right] \\

y_n(x) &= (-1)^{n+1} j_{-n-1}(x) = (-1)^{n+1} \frac{\pi}{2x}J_{-\left(n+\frac{1}{2}\right)}(x) = \\

&= \frac{(-1)^{n+1}}{2x} \left[ e^{ix} \sum_{r=0}^n \frac{i^{r+n}(n+r)!}{r!(n-r)!(2x)^r} + e^{-ix} \sum_{r=0}^n \frac{(-i)^{r+n}(n+r)!}{r!(n-r)!(2x)^r} \right] = \\

&= \frac{(-1)^{n+1}}{x} \left[ \cos\left(x+\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n}{2} \right]} \frac{(-1)^r (n+2r)!}{(2r)!(n-2r)!(2x)^{2r}} - \sin\left(x+\frac{n\pi}{2}\right) \sum_{r=0}^{\left[ \frac{n-1}{2} \right]} \frac{(-1)^r (n+2r+1)!}{(2r+1)!(n-2r-1)!(2x)^{2r+1}} \right]

\end{alignat}

In the following, {{mvar|f_n}} is any of {{mvar|j_n}}, {{mvar|y_n}}, {{math|h{{su|b=n|p=(1)}}}}, {{math|h{{su|b=n|p=(2)}}}} for {{math|1=n = 0, ±1, ±2, ...}}Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_439.htm p. 439, 10.1.23, 10.1.24].

$$\backslash begin\{align\}$$

\left(\frac{1}{z}\frac{d}{dz}\right)^m \left (z^{n+1} f_n(z)\right ) &= z^{n-m+1} f_{n-m}(z), \\

\left(\frac{1}{z}\frac{d}{dz}\right)^m \left (z^{-n} f_n(z)\right ) &= (-1)^m z^{-n-m} f_{n+m}(z).

\end{align}

File: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.svg

File: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.svg

There are also spherical analogues of the Hankel functions:

$$\backslash begin\{align\}$$

h_n^{(1)}(x) &= j_n(x) + i y_n(x), \\

h_n^{(2)}(x) &= j_n(x) - i y_n(x).

\end{align}

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 {{mvar|n}}:

$$h\_n^\{(1)\}(x)\; =\; (-i)^\{n+1\}\; \backslash frac\{e^\{ix\}\}\{x\}\; \backslash sum\_\{m=0\}^n\; \backslash frac\{i^m\}\{m!\backslash ,(2x)^m\}\; \backslash frac\{(n+m)!\}\{(n-m)!\},$$

and {{math|h{{su|b=n|p=(2)}}}} is the complex-conjugate of this (for real {{mvar|x}}). It follows, for example, that {{math|1=j₀(x) = {{sfrac|sin x|x}}}} and {{math|1=y₀(x) = −{{sfrac|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 only slightly differ from spherical Bessel functions:

$$\backslash begin\{align\}$$

S_n(x) &= x j_n(x) = \sqrt{\frac{\pi x}{2}} J_{n+\frac{1}{2}}(x) \\

C_n(x) &= -x y_n(x) = -\sqrt{\frac{\pi x}{2}} Y_{n+\frac{1}{2}}(x) \\

\xi_n(x) &= x h_n^{(1)}(x) = \sqrt{\frac{\pi x}{2}} H_{n+\frac{1}{2}}^{(1)}(x) = S_n(x) - iC_n(x) \\

\zeta_n(x) &= x h_n^{(2)}(x) = \sqrt{\frac{\pi x}{2}} H_{n+\frac{1}{2}}^{(2)}(x) = S_n(x) + iC_n(x)

\end{align}

File:Riccati Bessel Function S 3D Complex Color Plot with Mathematica 13.2.svg

They satisfy the differential equation

$$x^2\; \backslash frac\{d^2\; y\}\{dx^2\}\; +\; \backslash left\; (x^2\; -\; n(n\; +\; 1)\backslash right)\; y\; =\; 0.$$

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.Griffiths. Introduction to Quantum Mechanics, 2nd edition, p. 154. 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){{cite journal |first=Hong |last=Du |title=Mie-scattering calculation |journal=Applied Optics |volume=43 |issue=9 |pages=1951–1956 |date=2004 |doi=10.1364/ao.43.001951 |pmid=15065726 |bibcode=2004ApOpt..43.1951D}} for recent developments and references.

Following Debye (1909), the notation {{mvar|ψ_n}}, {{mvar|χ_n}} is sometimes used instead of {{mvar|S_n}}, {{mvar|C_n}}.

The Bessel functions have the following asymptotic forms. For small arguments

$$J\_\backslash alpha(z)\; \backslash sim\; \backslash frac\{1\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash left(\; \backslash frac\{z\}\{2\}\; \backslash right)^\backslash alpha.$$

When {{mvar|α}} is a negative integer, we have

$$J\_\backslash alpha(z)\; \backslash sim\; \backslash frac\{(-1)^\{\backslash alpha\}\}\{(-\backslash alpha)!\}\; \backslash left(\; \backslash frac\{2\}\{z\}\; \backslash right)^\backslash alpha.$$

For the Bessel function of the second kind we have three cases:

$$Y\_\backslash alpha(z)\; \backslash sim\; \backslash begin\{cases\}$$

\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}

where {{mvar|γ}} is the Euler–Mascheroni constant (0.5772...).

For large real arguments {{math|z ≫ {{abs|α² − {{sfrac|1|4}}}}}}, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless {{mvar|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 {{math|arg z}} one can write an equation containing a term of order {{math|{{abs|z}}⁻¹}}:Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_364.htm p. 364, 9.2.1].

$$\backslash begin\{align\}$$

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{align}

(For {{math|1=α = {{sfrac|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:

$$\backslash begin\{align\}$$

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{align}

These can be extended to other values of {{math|arg z}} using equations relating {{math|H{{su|b=α|p=(1)}}(ze^imπ)}} and {{math|H{{su|b=α|p=(2)}}(ze^imπ)}} to {{math|H{{su|b=α|p=(1)}}(z)}} and {{math|H{{su|b=α|p=(2)}}(z)}}.NIST Digital Library of Mathematical Functions, Section [https://dlmf.nist.gov/10.11#E1 10.11].

It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, {{math|J_α(z)}} is not asymptotic to the average of these two asymptotic forms when {{mvar|z}} is negative (because one or the other will not be correct there, depending on the {{math|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) {{mvar|z}} so long as {{math|{{abs|z}}}} goes to infinity at a constant phase angle {{math|arg z}} (using the square root having positive real part):

$$\backslash begin\{align\}$$

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{align}

For the modified Bessel functions, Hankel developed asymptotic expansions as well:Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_377.htm p. 377, 9.7.1].Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_378.htm p. 378, 9.7.2].

$$\backslash begin\{align\}$$

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{align}

There is also the asymptotic form (for large real $z$)[https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-81/issue-4/The-Kosterlitz-Thouless-transition-in-two-dimensional-abelian-spin-systems/cmp/1103920388.full Fröhlich and Spencer 1981 Appendix B]

$$\backslash begin\{align\}$$

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{align}

When {{math|1=α = {{sfrac|1|2}}}}, all the terms except the first vanish, and we have

$$\backslash begin\{align\}$$

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{align}

For small arguments , we have

$$\backslash begin\{align\}$$

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{align}

For integer order {{math|1=α = n}}, {{mvar|J_n}} is often defined via a Laurent series for a generating function:

$$e^\{\backslash frac\{x\}\{2\}\backslash left(t-\backslash frac\{1\}\{t\}\backslash right)\}\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; J\_n(x)\; t^n$$

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 $\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; J\_\{N\backslash nu\; +\; p\}(x)$ where \backslash nu,\; p\; \backslash in\; \backslash mathbb\{Z\},\; \backslash \; N\; \backslash in\; \backslash mathbb\{Z\}^+ arise in many physical systems and are defined in closed form by the Sung series.{{cite arXiv |last1=Sung |first1=S. |last2=Hovden |first2=R. |title=On Infinite Series of Bessel functions of the First Kind |year=2022 |class=math-ph |eprint=2211.01148}} For example, when N = 3: $\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; J\_\{3\backslash nu+p\}(x)\; =\; \backslash frac\{1\}\{3\}\backslash left[1+2\backslash cos\{(x\backslash sqrt\{3\}/2-2\backslash pi\; p/3)\}\backslash right]$. More generally, the Sung series and the alternating Sung series are written as:

$$\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; J\_\{N\backslash nu+p\}(x)\; =\; \backslash frac\{1\}\{N\}\backslash sum\_\{q=0\}^\{N-1\}\; e^\{ix\backslash sin\{2\backslash pi\; q/N\}\}e^\{-i2\backslash pi\; pq/N\}$$

$$\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; (-1)^\backslash nu\; J\_\{N\backslash nu+p\}(x)\; =\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{q=0\}^\{N-1\}e^\{ix\backslash sin\{(2q+1)\backslash pi/N\}\}e^\{-i(2q+1)\backslash pi\; p/N\}$$

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 the Jacobi–Anger expansion:

$$e^\{iz\; \backslash cos\; \backslash phi\}\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; i^n\; J\_n(z)\; e^\{in\backslash phi\}$$

and

$$e^\{\backslash pm\; iz\; \backslash sin\; \backslash phi\}\; =\; J\_0(z)+2\backslash sum\_\{n=1\}^\backslash infty\; J\_\{2n\}(z)\; \backslash cos(2n\backslash phi)\; \backslash pm\; 2i\; \backslash sum\_\{n=0\}^\backslash infty\; J\_\{2n+1\}(z)\backslash sin((2n+1)\backslash phi)$$

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

$$f(z)=a\_0^\backslash nu\; J\_\backslash nu\; (z)+\; 2\; \backslash cdot\; \backslash sum\_\{k=1\}^\backslash infty\; a\_k^\backslash nu\; J\_\{\backslash nu+k\}(z)$$

is called Neumann expansion of {{mvar|f}}. The coefficients for {{math|1=ν = 0}} have the explicit form

$$a\_k^0=\backslash frac\{1\}\{2\; \backslash pi\; i\}\; \backslash int\_\{|z|=c\}\; f(z)\; O\_k(z)\; \backslash ,dz$$

where {{mvar|O_k}} is Neumann's polynomial.Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_363.htm p. 363, 9.1.82] ff.

Selected functions admit the special representation

$$f(z)=\backslash sum\_\{k=0\}^\backslash infty\; a\_k^\backslash nu\; J\_\{\backslash nu+2k\}(z)$$

with

$$a\_k^\backslash nu=2(\backslash nu+2k)\; \backslash int\_0^\backslash infty\; f(z)\; \backslash frac\{J\_\{\backslash nu+2k\}(z)\}z\; \backslash ,dz$$

due to the orthogonality relation

$$\backslash int\_0^\backslash infty\; J\_\backslash alpha(z)\; J\_\backslash beta(z)\; \backslash frac\; \{dz\}\; z=\; \backslash frac\; 2\; \backslash pi\; \backslash frac\{\backslash sin\backslash left(\backslash frac\; \backslash pi\; 2\; (\backslash alpha-\backslash beta)\; \backslash right)\}\{\backslash alpha^2\; -\backslash beta^2\}$$

More generally, if {{mvar|f}} has a branch-point near the origin of such a nature that

$$f(z)=\; \backslash sum\_\{k=0\}\; a\_k\; J\_\{\backslash nu+k\}(z)$$

then

$$\backslash mathcal\{L\}\backslash left\backslash \{\backslash sum\_\{k=0\}\; a\_k\; J\_\{\backslash nu+k\}\backslash right\backslash \}(s)=\backslash frac\{1\}\{\backslash sqrt\{1+s^2\}\}\backslash sum\_\{k=0\}\backslash frac\{a\_k\}\{\backslash left(s+\backslash sqrt\{1+s^2\}\; \backslash right)\; ^\{\backslash nu+k\}\}$$

or

$$\backslash sum\_\{k=0\}\; a\_k\; \backslash xi^\{\backslash nu+k\}=\; \backslash frac\{1+\backslash xi^2\}\{2\backslash xi\}\; \backslash mathcal\{L\}\backslash \{f\; \backslash \}\; \backslash left(\; \backslash frac\{1-\backslash xi^2\}\{2\backslash xi\}\; \backslash right)$$

where $\backslash mathcal\{L\}\backslash \{f\; \backslash \}$ is the Laplace transform of {{mvar|f}}.{{cite book |url=https://books.google.com/books?id=Mlk3FrNoEVoC&q=bessel+neumann+series&pg=PA536 |title=A Treatise on the Theory of Bessel Functions |first=G. N. |last=Watson |date=25 August 1995 |publisher=Cambridge University Press |access-date=25 March 2018 |via=Google Books |isbn=9780521483919}}

Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula:

$$\backslash begin\{align\}$$

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{align}

where {{math|ν > −{{sfrac|1|2}}}} and {{math|z ∈ C}}.{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=Academic Press, Inc. |date=2015 |orig-year=October 2014 |edition=8 |language=en |isbn=978-0-12-384933-5 |lccn=2014010276 |title-link=Gradshteyn and Ryzhik |chapter=8.411.10.}}

This formula is useful especially when working with Fourier transforms.

Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by {{mvar|x}}, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that:

$$\backslash int\_0^1\; x\; J\_\backslash alpha\backslash left(x\; u\_\{\backslash alpha,m\}\backslash right)\; J\_\backslash alpha\backslash left(x\; u\_\{\backslash alpha,n\}\backslash right)\; \backslash ,dx\; =\; \backslash frac\{\backslash delta\_\{m,n\}\}\{2\}\; \backslash left[J\_\{\backslash alpha+1\}\; \backslash left(u\_\{\backslash alpha,m\}\backslash right)\backslash right]^2\; =\; \backslash frac\{\backslash delta\_\{m,n\}\}\{2\}\; \backslash left[J\_\{\backslash alpha\}\text{'}\backslash left(u\_\{\backslash alpha,m\}\backslash right)\backslash right]^2$$

where {{math|α > −1}}, {{math|δ_m,n}} is the Kronecker delta, and {{math|u_α,m}} is the {{mvar|m}}th zero of {{math|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 {{math|J_α(x u_α,m)}} for fixed {{mvar|α}} and varying {{mvar|m}}.

An analogous relationship for the spherical Bessel functions follows immediately:

$$\backslash int\_0^1\; x^2\; j\_\backslash alpha\backslash left(x\; u\_\{\backslash alpha,m\}\backslash right)\; j\_\backslash alpha\backslash left(x\; u\_\{\backslash alpha,n\}\backslash right)\; \backslash ,dx\; =\; \backslash frac\{\backslash delta\_\{m,n\}\}\{2\}\; \backslash left[j\_\{\backslash alpha+1\}\backslash left(u\_\{\backslash alpha,m\}\backslash right)\backslash right]^2$$

If one defines a boxcar function of {{mvar|x}} that depends on a small parameter {{mvar|ε}} as:

$$f\_\backslash varepsilon(x)=\backslash frac\; 1\backslash varepsilon\; \backslash operatorname\{rect\}\backslash left(\backslash frac\{x-1\}\backslash varepsilon\backslash right)$$

(where {{math|rect}} is the rectangle function) then the Hankel transform of it (of any given order {{math|α > −{{sfrac|1|2}}}}), {{math|g_ε(k)}}, approaches {{math|J_α(k)}} as {{mvar|ε}} approaches zero, for any given {{mvar|k}}. Conversely, the Hankel transform (of the same order) of {{math|g_ε(k)}} is {{math|f_ε(x)}}:

$$\backslash int\_0^\backslash infty\; k\; J\_\backslash alpha(kx)\; g\_\backslash varepsilon(k)\; \backslash ,dk\; =\; f\_\backslash varepsilon(x)$$

which is zero everywhere except near 1. As {{mvar|ε}} approaches zero, the right-hand side approaches {{math|δ(x − 1)}}, where {{mvar|δ}} is the Dirac delta function. This admits the limit (in the distributional sense):

$$\backslash int\_0^\backslash infty\; k\; J\_\backslash alpha(kx)\; J\_\backslash alpha(k)\; \backslash ,dk\; =\; \backslash delta(x-1)$$

A change of variables then yields the closure equation:Arfken & Weber, section 11.2

$$\backslash int\_0^\backslash infty\; x\; J\_\backslash alpha(ux)\; J\_\backslash alpha(vx)\; \backslash ,dx\; =\; \backslash frac\{1\}\{u\}\; \backslash delta(u\; -\; v)$$

for {{math|α > −{{sfrac|1|2}}}}. The Hankel transform can express a fairly arbitrary function{{Clarify|reason=This can probably be precisely qualified e.g. square integrable etc.|date=June 2018}} as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is:

$$\backslash int\_0^\backslash infty\; x^2\; j\_\backslash alpha(ux)\; j\_\backslash alpha(vx)\; \backslash ,dx\; =\; \backslash frac\{\backslash pi\}\{2uv\}\; \backslash delta(u\; -\; v)$$

for {{math|α > −1}}.

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions:

$$A\_\backslash alpha(x)\; \backslash frac\{dB\_\backslash alpha\}\{dx\}\; -\; \backslash frac\{dA\_\backslash alpha\}\{dx\}\; B\_\backslash alpha(x)\; =\; \backslash frac\{C\_\backslash alpha\}\{x\}$$

where {{mvar|A_α}} and {{mvar|B_α}} are any two solutions of Bessel's equation, and {{mvar|C_α}} is a constant independent of {{mvar|x}} (which depends on α and on the particular Bessel functions considered). In particular,

$$J\_\backslash alpha(x)\; \backslash frac\{dY\_\backslash alpha\}\{dx\}\; -\; \backslash frac\{dJ\_\backslash alpha\}\{dx\}\; Y\_\backslash alpha(x)\; =\; \backslash frac\{2\}\{\backslash pi\; x\}$$

and

$$I\_\backslash alpha(x)\; \backslash frac\{dK\_\backslash alpha\}\{dx\}\; -\; \backslash frac\{dI\_\backslash alpha\}\{dx\}\; K\_\backslash alpha(x)\; =\; -\backslash frac\{1\}\{x\},$$

for {{math|α > −1}}.

For {{math|α > −1}}, the even entire function of genus 1, {{math|x^−αJ_α(x)}}, has only real zeros. Let

be all its positive zeros, then

$$J\_\{\backslash alpha\}(z)=\backslash frac\{\backslash left(\backslash frac\{z\}\{2\}\backslash right)^\backslash alpha\}\{\backslash Gamma(\backslash alpha+1)\}\backslash prod\_\{n=1\}^\{\backslash infty\}\backslash left(1-\backslash frac\{z^2\}\{j\_\{\backslash alpha,n\}^2\}\backslash right)$$

(There are a large number of other known integrals and identities that are not reproduced here, but which can be found in the references.)

The functions {{mvar|J_α}}, {{mvar|Y_α}}, {{math|H{{su|b=α|p=(1)}}}}, and {{math|H{{su|b=α|p=(2)}}}} all satisfy the recurrence relationsAbramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_361.htm p. 361, 9.1.27].

$$\backslash frac\{2\backslash alpha\}\{x\}\; Z\_\backslash alpha(x)\; =\; Z\_\{\backslash alpha-1\}(x)\; +\; Z\_\{\backslash alpha+1\}(x)$$

and

$$2\backslash frac\{dZ\_\backslash alpha\; (x)\}\{dx\}\; =\; Z\_\{\backslash alpha-1\}(x)\; -\; Z\_\{\backslash alpha+1\}(x),$$

where {{mvar|Z}} denotes {{mvar|J}}, {{mvar|Y}}, {{math|H⁽¹⁾}}, or {{math|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 thatAbramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_361.htm p. 361, 9.1.30].

$$\backslash begin\{align\}$$

\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{align}

Modified Bessel functions follow similar relations:

$$e^\{\backslash left(\backslash frac\{x\}\{2\}\backslash right)\backslash left(t+\backslash frac\{1\}\{t\}\backslash right)\}\; =\; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; I\_n(x)\; t^n$$

and

$$e^\{z\; \backslash cos\; \backslash theta\}\; =\; I\_0(z)\; +\; 2\backslash sum\_\{n=1\}^\backslash infty\; I\_n(z)\; \backslash cos\; n\backslash theta$$

and

$$\backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_0^\{2\backslash pi\}\; e^\{z\; \backslash cos\; (m\backslash theta)\; +\; y\; \backslash cos\; \backslash theta\}\; d\backslash theta\; =\; I\_0(z)I\_0(y)\; +\; 2\backslash sum\_\{n=1\}^\backslash infty\; I\_n(z)I\_\{mn\}(y).$$

The recurrence relation reads

$$\backslash begin\{align\}$$

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{align}

where {{mvar|C_α}} denotes {{mvar|I_α}} or {{math|e^αiπK_α}}. These recurrence relations are useful for discrete diffusion problems.

In 1929, Carl Ludwig Siegel proved that {{math|J_ν(x)}}, {{math|J{{'}}_ν(x)}}, and the logarithmic derivative {{math|{{sfrac|J'_ν(x)|J_ν(x)}}}} are transcendental numbers when ν is rational and x is algebraic and nonzero.{{cite book |last1=Siegel |first1=Carl L. |title=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 |date=2014 |publisher=Scuola Normale Superiore |isbn=978-88-7642-520-2 |pages=81–138 |chapter-url=https://link.springer.com/chapter/10.1007/978-88-7642-520-2_2 |language=de |chapter=Über einige Anwendungen diophantischer Approximationen |doi=10.1007/978-88-7642-520-2_2}} The same proof also implies that $\backslash Gamma(v+1)(2/x)^v\; J\_\{v\}(x)$ is transcendental under the same assumptions.{{cite journal |last1=James |first1=R. D. |title=Review: Carl Ludwig Siegel, Transcendental numbers |journal=Bulletin of the American Mathematical Society |date=November 1950 |volume=56 |issue=6 |pages=523–526 |doi=10.1090/S0002-9904-1950-09435-X |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-56/issue-6/Review-Carl-Ludwig-Siegel-Transcendental-numbers/bams/1183515049.full|doi-access=free }}

The product of two Bessel functions admits the following sum:

$$\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; J\_\backslash nu(x)\; J\_\{n\; -\; \backslash nu\}(y)\; =\; J\_\{n\}(x\; +\; y),$$

$$\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; J\_\backslash nu(x)\; J\_\{\backslash nu\; +\; n\}(y)\; =\; J\_\{n\}(y\; -\; x).$$

From these equalities it follows that

$$\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; J\_\backslash nu(x)\; J\_\{\backslash nu\; +\; n\}(x)\; =\; \backslash delta\_\{n,\; 0\}$$

and as a consequence

$$\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; J\_\{\backslash nu\}^2(x)\; =\; 1.$$

These sums can be extended for a polynomial prefactor. For example,

$$\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; \backslash nu\; J\_\backslash nu(x)\; J\_\{\backslash nu\; +\; n\}(x)\; =\; \backslash frac\{x\}\{2\}\; \backslash left(\; \backslash delta\_\{n,\; 1\}\; +\; \backslash delta\_\{n,\; -1\}\; \backslash right),$$

$$\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; \backslash nu\; J\_\{\backslash nu\}^2(x)\; =\; 0,$$

$$\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; \backslash nu^2\; J\_\backslash nu(x)\; J\_\{\backslash nu\; +\; n\}(x)\; =\; \backslash frac\{x\}\{2\}\; \backslash left(\; \backslash delta\_\{n,\; -1\}\; -\; \backslash delta\_\{n,\; 1\}\; \backslash right)\; +\; \backslash frac\{x^2\}\{4\}\; \backslash left(\; \backslash delta\_\{n,\; -2\}\; +\; 2\; \backslash delta\_\{n,\; 0\}\; +\; \backslash delta\_\{n,\; 2\}\; \backslash right),$$

$$\backslash sum\_\{\backslash nu=-\backslash infty\}^\backslash infty\; \backslash nu^2\; J\_\{\backslash nu\}^2(x)\; =\; \backslash frac\{x^2\}\{2\}.$$

The Bessel functions obey a multiplication theorem

$$\backslash lambda^\{-\backslash nu\}\; J\_\backslash nu(\backslash lambda\; z)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{1\}\{n!\}\; \backslash left(\backslash frac\{\backslash left(1\; -\; \backslash lambda^2\backslash right)z\}\{2\}\backslash right)^n\; J\_\{\backslash nu+n\}(z),$$

where {{mvar|λ}} and {{mvar|ν}} may be taken as arbitrary complex numbers.Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_363.htm p. 363, 9.1.74].{{cite journal |last=Truesdell |first=C. |title=On the Addition and Multiplication Theorems for the Special Functions |journal=Proceedings of the National Academy of Sciences |volume=1950 |issue=12 |pages=752–757 |date=1950 |doi=10.1073/pnas.36.12.752 |pmid=16578355 |pmc=1063284 |bibcode=1950PNAS...36..752T |doi-access=free}} For {{math|{{abs|λ² − 1}} < 1}}, the above expression also holds if {{mvar|J}} is replaced by {{mvar|Y}}. The analogous identities for modified Bessel functions and {{math|{{abs|λ² − 1}} < 1}} are

$$\backslash lambda^\{-\backslash nu\}\; I\_\backslash nu(\backslash lambda\; z)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{1\}\{n!\}\; \backslash left(\backslash frac\{\backslash left(\backslash lambda^2\; -\; 1\backslash right)z\}\{2\}\backslash right)^n\; I\_\{\backslash nu+n\}(z)$$

and

$$\backslash lambda^\{-\backslash nu\}\; K\_\backslash nu(\backslash lambda\; z)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-1)^n\}\{n!\}\; \backslash left(\backslash frac\{\backslash left(\backslash lambda^2\; -\; 1\backslash right)z\}\{2\}\backslash right)^n\; K\_\{\backslash nu+n\}(z).$$

Bessel himself originally proved that for nonnegative integers {{mvar|n}}, the equation {{math|1=J_n(x) = 0}} has an infinite number of solutions in {{mvar|x}}.Bessel, F. (1824), article 14. When the functions {{math|J_n(x)}} are plotted on the same graph, though, none of the zeros seem to coincide for different values of {{mvar|n}} except for the zero at {{math|1=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 {{math|n ≥ 0}} and {{math|m ≥ 1}}, the functions {{math|J_n(x)}} and {{math|J_{n + m}(x)}} have no common zeros other than the one at {{math|1=x = 0}}. The hypothesis was proved by Carl Ludwig Siegel in 1929.Watson, pp. 484–485.

Siegel proved in 1929 that when ν is rational, all nonzero roots of {{math|J_ν(x)}} and {{math|J{{'}}_ν(x)}} are transcendental, as are all the roots of {{math|K_ν(x)}}. It is also known that all roots of the higher derivatives $J\_\backslash nu^\{(n)\}(x)$ for {{math|n ≤ 18}} are transcendental, except for the special values $J\_1^\{(3)\}(\backslash pm\backslash sqrt3)\; =\; 0$ and $J\_0^\{(4)\}(\backslash pm\backslash sqrt3)\; =\; 0$.{{cite journal |last1=Lorch |first1=Lee |last2=Muldoon |first2=Martin E. |title=Transcendentality of zeros of higher dereivatives of functions involving Bessel functions |journal=International Journal of Mathematics and Mathematical Sciences |date=1995 |volume=18 |issue=3 |pages=551–560 |doi=10.1155/S0161171295000706 |doi-access=free}}

For numerical studies about the zeros of the Bessel function, see {{harvtxt|Gil|Segura|Temme|2007}}, {{harvtxt|Kravanja|Ragos|Vrahatis|Zafiropoulos|1998}} and {{harvtxt|Moler|2004}}.

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.Abramowitz & Stegun, p409

The first appearance of a Bessel function appears in the work of Daniel Bernoulli in 1732, while working on the analysis of a vibrating string, a problem that was tackled before by his father Johann Bernoulli. Daniel considered a flexible chain suspended from a fixed point above and free at its lower end. 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.

Leonhard Euler in 1736, found a link between other functions (now known as Laguerre 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 $I\_n(x)$.

In the middle of the eighteen century, Jean le Rond d'Alembert had found a formula to solve the wave equation. By 1771 there was dispute between Bernoulli, Euler, d'Alembert and Joseph-Louis Lagrange on the nature of the solutions vibrating strings.

Euler worked in 1778 on buckling, introducing the concept of Euler's critical load. To solve the problem he introduced the series for $J\_\{\backslash pm\; 1/3\}(x)$. 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 integer n.

During the end of the 19th century Lagrange, Pierre-Simon Laplace and Marc-Antoine Parseval also found equivalents to the Bessel functions. Parseval for example found an integral represantion of $J\_0(x)$ using cosine.

At the beginning of the 1800s, Joseph Fourier used $J\_0(x)$ to solve the heat equation in a problem with cylindrical symmetry. Fourier won a prize of the French Academy of Sciences for this work in 1811. But most of the details of his work, including the use of a Fourier series, remained unpublished until 1822. 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).

In 1770, Lagrangre introduced the series expansion of Bessel functions to solve Kepler's equation, a trascendental equation in astronomy. Friedrich Wilhelm Bessel had seen Lagrange solution but found it difficult to handle. In 1813 in a letter to Carl Friedrich Gauss, Bessel simplified the calculation using trigonometric functions. Bessel published his work in 1819, independently introducing the method of Fourier series unaware of the work of Fourier who was published later.

In 1824, Bessel carried out a systematic investigation of the Bessel functions which earned the function his name. In older literature the functions were called cylindrical functions or even Bessel–Fourier functions.

{{Div col|colwidth=20em}}

• Anger function
• Bessel polynomials
• Bessel–Clifford function
• Bessel–Maitland function
• Fourier–Bessel series
• Hahn–Exton q-Bessel function
• Hankel transform
• Incomplete Bessel functions
• Jackson q-Bessel function
• Kelvin functions
• Kontorovich–Lebedev transform
• Lentz's algorithm
• Lerche–Newberger sum rule
• Lommel function
• Lommel polynomial
• Neumann polynomial
• [https://mathworld.wolfram.com/Riccati-BesselFunctions.html Riccati-Bessel Functions]
• Schlömilch's series
• Sonine formula
• Struve function
• Vibrations of a circular membrane
• Weber function (defined at Anger function)
• Gauss' circle problem

{{Div col end}}

{{Reflist|30em}}

{{Refbegin|30em}}

• {{Abramowitz_Stegun_ref2|9|355|10|435}}
• Arfken, George B. and Hans J. Weber, Mathematical Methods for Physicists, 6th edition (Harcourt: San Diego, 2005). {{ISBN|0-12-059876-0}}.
• {{cite journal |last1=Bessel |first1=Friedrich |title=Untersuchung des Theils der planetarischen Störungen, welcher aus der Bewegung der Sonne entsteht |journal=Berlin Abhandlungen |date=1824 |trans-title=Investigation of the part of the planetary disturbances which arise from the movement of the sun}} Reproduced as pages 84 to 109 in {{cite book |title=Abhandlungen von Friedrich Wilhelm Bessel |date=1875 |publisher=Engelmann |location=Leipzig |url=https://play.google.com/store/books/details?id=Un4EAAAAYAAJ}} [https://drive.google.com/file/d/1W-z4BNN4s7nfGC9IbCkhQksHzGQG-wox/view?usp=sharing English translation of the text].
• Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). {{ISBN|0-486-60462-4}}.
• {{cite book |last1=Gil |first1=A. |last2=Segura |first2=J. |last3=Temme |first3=N. M. |year=2007 |title=Numerical methods for special functions |publisher=Society for Industrial and Applied Mathematics}}
• {{citation |title=ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument |journal=Computer Physics Communications |volume=113 |issue=2–3 |pages=220–238 |year=1998 |doi=10.1016/S0010-4655(98)00064-2 |first1=P. |last1=Kravanja |first2=O. |last2=Ragos |first3=M.N. |last3=Vrahatis |first4=F.A. |last4=Zafiropoulos |bibcode=1998CoPhC.113..220K |author-link=P. Kravanja}}
• {{cite journal |last1=Mie |first1=G. |author-link=Gustav Mie |year=1908 |title=Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen |journal=Annalen der Physik |volume=25 |issue=3 |page=377 |doi=10.1002/andp.19083300302 |bibcode=1908AnP...330..377M |doi-access=free}}
• {{dlmf|first=F. W. J. |last=Olver|authorlink=Frank W. J. Olver|first2=L. C. |last2=Maximon|id=10}}.
• {{Citation |last1=Press |first1=W. H. |author-link=William H. Press |last2=Teukolsky |first2=S. A. |last3=Vetterling |first3=W. T. |last4=Flannery |first4=B. P. |year=2007 |title=Numerical Recipes: The Art of Scientific Computing |edition=3rd |publisher=Cambridge University Press |location=New York |isbn=978-0-521-88068-8 |chapter=Section 6.5. Bessel Functions of Integer Order |chapter-url=http://numerical.recipes/book/book.html |access-date=2022-09-28 |archive-date=2021-02-03 |archive-url=https://web.archive.org/web/20210203001225/http://numerical.recipes/book/book.html |url-status=dead }}.
• B Spain, M. G. Smith, [https://books.google.com/books?id=kYgZAQAAIAAJ&q=Bessel 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}}.
• {{citation |last=Weber |first=Heinrich |author-link=Heinrich Friedrich Weber |title=Ueber eine Darstellung willkürlicher Functionen durch Bessel'sche Functionen |journal=Mathematische Annalen |volume=6 |issue=2 |pages=146–161 |year=1873 |doi=10.1007/BF01443190 |s2cid=122409461}}.

{{Refend}}

{{Refbegin|30em}}

• {{SpringerEOM|first=P. I. |last=Lizorkin|id=b/b015840|title=Bessel functions}}.
• {{SpringerEOM|first=L. N. |last=Karmazina|first2=A.P. |last2=Prudnikov|id=c/c027610|title=Cylinder function}}.
• {{SpringerEOM|first=N. Kh.|last=Rozov|id=B/b015830|title=Bessel equation}}.
• Wolfram function pages on Bessel [https://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/ J] and [https://functions.wolfram.com/Bessel-TypeFunctions/BesselY/ Y] functions, and modified Bessel [https://functions.wolfram.com/Bessel-TypeFunctions/BesselI/ I] and [https://functions.wolfram.com/Bessel-TypeFunctions/BesselK/ K] functions. Pages include formulas, function evaluators, and plotting calculators.
• {{MathWorld|id=BesselFunctionoftheFirstKind|title=Bessel functions of the first kind}}
• Bessel functions [http://www.librow.com/articles/article-11/appendix-a-34 J_ν], [http://www.librow.com/articles/article-11/appendix-a-35 Y_ν], [http://www.librow.com/articles/article-11/appendix-a-36 I_ν] and [http://www.librow.com/articles/article-11/appendix-a-37 K_ν] in Librow [http://www.librow.com/articles/article-11 Function handbook].
• F. W. J. Olver, L. C. Maximon, [https://dlmf.nist.gov/10 Bessel Functions] (chapter 10 of the Digital Library of Mathematical Functions).
• {{cite book |last=Moler |first=C. B. |year=2004 |title=Numerical Computing with MATLAB |publisher=Society for Industrial and Applied Mathematics |url=http://tocs.ulb.tu-darmstadt.de/124154883.pdf |url-status=dead |archive-url=https://web.archive.org/web/20170808214249/http://tocs.ulb.tu-darmstadt.de/124154883.pdf |archive-date=2017-08-08}}

{{Refend}}

{{Authority control}}

Category:Special hypergeometric functions

Category:Fourier analysis