Bessel Functions
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Bessel Functions

Bessel functions are the radial part of the modes of vibration of a circular drum.

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation

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

for an arbitrary complex number ?, the order of the Bessel function. Although ? and -? produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of ?.

The most important cases are when ? is an integer or half-integer. Bessel functions for integer ? are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer ? are obtained when the Helmholtz equation is solved in spherical coordinates.

## Applications of Bessel functions

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 (? = n); in spherical problems, one obtains half-integer orders (? = n + ). For example:

Bessel functions also appear in other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).

## Definitions

Because this is a second-order linear differential equation, there must be two linearly independent solutions. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections.

Type First kind Second kind
Bessel functions J? Y?
Modified Bessel functions I? K?
Hankel functions H(1)
?
= J? + iY?
H(2)
?
= J? - iY?
Spherical Bessel functions jn yn
Spherical Hankel functions h(1)
n
= jn + iyn
h(2)
n
= jn - iyn

Bessel functions of the second kind and the spherical Bessel functions of the second kind are sometimes denoted by Nn and nn, respectively, rather than Yn and yn.[1][2]

### Bessel functions of the first kind: J?

Bessel functions of the first kind, denoted as J?(x), are solutions of Bessel's differential equation. For integer or positive ?, Bessel functions of the first kind are finite at the origin (x = 0); while for negative non-integer ?, Bessel functions of the first kind diverge as x approaches zero. It is possible to define the function by its series expansion around x = 0, which can be found by applying the Frobenius method to Bessel's equation:[3]

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

where ?(z) is the gamma function, a shifted generalization of the factorial function to non-integer values. The Bessel function of the first kind is an entire function if ? is an integer, otherwise it is a multivalued function with singularity at zero. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to ${\displaystyle x^{-{\frac {1}{2}}}}$ (see also their asymptotic forms below), although their roots are not generally periodic, except asymptotically for large x. (The series indicates that -J1(x) is the derivative of J0(x), much like -sin x is the derivative of cos x; more generally, the derivative of Jn(x) can be expressed in terms of Jn ± 1(x) by the identities below.)

Plot of Bessel function of the first kind, J?(x), for integer orders ? = 0, 1, 2

For non-integer ?, the functions J?(x) and J-?(x) are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order n, the following relationship is valid (the gamma function has simple poles at each of the non-positive integers):[4]

${\displaystyle J_{-n}(x)=(-1)^{n}J_{n}(x).}$

This means that the two solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below.

#### Bessel's integrals

Another definition of the Bessel function, for integer values of n, is possible using an integral representation:[5]

${\displaystyle J_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(n\tau -x\sin \tau )\,d\tau .}$

Another integral representation is:[5]

${\displaystyle J_{n}(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{i(x\sin \tau -n\tau )}\,d\tau .}$

This was the approach that Bessel used, and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by one of Schläfli's integrals, for Re(x) > 0:[5][6][7][8][9]

${\displaystyle J_{\alpha }(x)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\alpha \tau -x\sin \tau )\,d\tau -{\frac {\sin \alpha \pi }{\pi }}\int _{0}^{\infty }e^{-x\sinh t-\alpha t}\,dt.}$

#### Relation to hypergeometric series

The Bessel functions can be expressed in terms of the generalized hypergeometric series as[10]

${\displaystyle J_{\alpha }(x)={\frac {\left({\frac {x}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}\;_{0}F_{1}\left(\alpha +1;-{\frac {x^{2}}{4}}\right).}$

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

#### Relation to Laguerre polynomials

In terms of the Laguerre polynomials Lk and arbitrarily chosen parameter t, the Bessel function can be expressed as[11]

${\displaystyle {\frac {J_{\alpha }(x)}{\left({\frac {x}{2}}\right)^{\alpha }}}={\frac {e^{-t}}{\Gamma (\alpha +1)}}\sum _{k=0}^{\infty }{\frac {L_{k}^{(\alpha )}\left({\frac {x^{2}}{4t}}\right)}{\binom {k+\alpha }{k}}}{\frac {t^{k}}{k!}}.}$

### Bessel functions of the second kind: Y?

The Bessel functions of the second kind, denoted by Y?(x), occasionally denoted instead by N?(x), are solutions of the Bessel differential equation that have a singularity at the origin (x = 0) and are multivalued. These are sometimes called Weber functions, as they were introduced by H. M. Weber (1873), and also Neumann functions after Carl Neumann.[12]

Plot of Bessel function of the second kind, Y?(x), for integer orders ? = 0, 1, 2

For non-integer ?, Y?(x) is related to J?(x) by

${\displaystyle Y_{\alpha }(x)={\frac {J_{\alpha }(x)\cos(\alpha \pi )-J_{-\alpha }(x)}{\sin(\alpha \pi )}}.}$

In the case of integer order n, the function is defined by taking the limit as a non-integer ? tends to n:

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

If n is a nonnegative integer, we have the series[13]

${\displaystyle Y_{n}(z)=-{\frac {\left({\frac {z}{2}}\right)^{-n}}{\pi }}\sum _{k=0}^{n-1}{\frac {(n-k-1)!}{k!}}\left({\frac {z^{2}}{4}}\right)^{k}+{\frac {2}{\pi }}J_{n}(z)\ln {\frac {z}{2}}-{\frac {\left({\frac {z}{2}}\right)^{n}}{\pi }}\sum _{k=0}^{\infty }(\psi (k+1)+\psi (n+k+1)){\frac {\left(-{\frac {z^{2}}{4}}\right)^{k}}{k!(n+k)!}}}$

where ${\displaystyle \psi (z)}$ is the digamma function, the logarithmic derivative of the gamma function.[14]

There is also a corresponding integral formula (for Re(x) > 0):[15]

${\displaystyle Y_{n}(x)={\frac {1}{\pi }}\int _{0}^{\pi }\sin(x\sin \theta -n\theta )\,d\theta -{\frac {1}{\pi }}\int _{0}^{\infty }\left(e^{nt}+(-1)^{n}e^{-nt}\right)e^{-x\sinh t}\,dt.}$

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

When ? is an integer, moreover, as was similarly the case for the functions of the first kind, the following relationship is valid:

${\displaystyle Y_{-n}(x)=(-1)^{n}Y_{n}(x).}$

Both J?(x) and Y?(x) are holomorphic functions of x on the complex plane cut along the negative real axis. When ? is an integer, the Bessel functions J are entire functions of x. If x is held fixed at a non-zero value, then the Bessel functions are entire functions of ?.

The Bessel functions of the second kind when ? is an integer is an example of the second kind of solution in Fuchs's theorem.

### Hankel functions: H(1)?, H(2)?

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, H(1)
?
(x)
and H(2)
?
(x)
, defined as[16]

{\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(x)&=J_{\alpha }(x)+iY_{\alpha }(x),\\H_{\alpha }^{(2)}(x)&=J_{\alpha }(x)-iY_{\alpha }(x),\end{aligned}}}

where i is the imaginary unit. These linear combinations are also known as Bessel functions of the third kind; they are two linearly independent solutions of Bessel's differential equation. They are named after Hermann Hankel.

These forms of linear combination satisfy numerous simple-looking properties, like asymptotic formulae or integral representations. Here, "simple" means an appearance of a factor of the form eif(x). For real ${\displaystyle x>0}$ where ${\displaystyle J_{\alpha }(x)}$, ${\displaystyle Y_{\alpha }(x)}$ are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of the first Hankel function and the real and negative imaginary parts of the second Hankel function. Thus, the above formulae are analogs of Euler's formula, substituting H(1)
?
(x)
, H(2)
?
(x)
for ${\displaystyle e^{\pm ix}}$ and ${\displaystyle J_{\alpha }(x)}$, ${\displaystyle Y_{\alpha }(x)}$ for ${\displaystyle \cos(x)}$, ${\displaystyle \sin(x)}$, as explicitly shown in the asymptotic expansion.

The Hankel functions are used to express outward- and inward-propagating cylindrical-wave solutions of the cylindrical wave equation, respectively (or vice versa, depending on the sign convention for the frequency).

Using the previous relationships, they can be expressed as

{\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(x)&={\frac {J_{-\alpha }(x)-e^{-\alpha \pi i}J_{\alpha }(x)}{i\sin \alpha \pi }},\\H_{\alpha }^{(2)}(x)&={\frac {J_{-\alpha }(x)-e^{\alpha \pi i}J_{\alpha }(x)}{-i\sin \alpha \pi }}.\end{aligned}}}

If ? is an integer, the limit has to be calculated. The following relationships are valid, whether ? is an integer or not:[17]

{\displaystyle {\begin{aligned}H_{-\alpha }^{(1)}(x)&=e^{\alpha \pi i}H_{\alpha }^{(1)}(x),\\H_{-\alpha }^{(2)}(x)&=e^{-\alpha \pi i}H_{\alpha }^{(2)}(x).\end{aligned}}}

In particular, if ? = m + with m a nonnegative integer, the above relations imply directly that

{\displaystyle {\begin{aligned}J_{-(m+{\frac {1}{2}})}(x)&=(-1)^{m+1}Y_{m+{\frac {1}{2}}}(x),\\Y_{-(m+{\frac {1}{2}})}(x)&=(-1)^{m}J_{m+{\frac {1}{2}}}(x).\end{aligned}}}

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

The Hankel functions admit the following integral representations for Re(x) > 0:[18]

{\displaystyle {\begin{aligned}H_{\alpha }^{(1)}(x)&={\frac {1}{\pi i}}\int _{-\infty }^{+\infty +i\pi }e^{x\sinh t-\alpha t}\,dt,\\H_{\alpha }^{(2)}(x)&=-{\frac {1}{\pi i}}\int _{-\infty }^{+\infty -i\pi }e^{x\sinh t-\alpha t}\,dt,\end{aligned}}}

where the integration limits indicate integration along a contour that can be chosen as follows: from -? to 0 along the negative real axis, from 0 to ±i? along the imaginary axis, and from ±i? to +? ± i? along a contour parallel to the real axis.[15]

### Modified Bessel functions: I?, K?

The Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as[19]

{\displaystyle {\begin{aligned}I_{\alpha }(x)&=i^{-\alpha }J_{\alpha }(ix)=\sum _{m=0}^{\infty }{\frac {1}{m!\,\Gamma (m+\alpha +1)}}\left({\frac {x}{2}}\right)^{2m+\alpha },\\K_{\alpha }(x)&={\frac {\pi }{2}}{\frac {I_{-\alpha }(x)-I_{\alpha }(x)}{\sin \alpha \pi }},\end{aligned}}}

when ? is not an integer; when ? is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments x. The series expansion for I?(x) is thus similar to that for J?(x), but without the alternating (-1)m factor.

${\displaystyle K_{\alpha }}$ can be expressed in terms of Hankel functions:

${\displaystyle K_{\alpha }={\begin{cases}{\frac {\pi }{2}}i^{\alpha +1}H_{\alpha }^{(1)}(ix)&-\pi <\arg x\leq {\frac {\pi }{2}}\\{\frac {\pi }{2}}(-i)^{\alpha +1}H_{\alpha }^{(2)}(-ix)&-{\frac {\pi }{2}}<\arg x\leq \pi \end{cases}}}$

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if -? < arg z ):[20]

{\displaystyle {\begin{aligned}J_{\alpha }(iz)&=e^{\frac {\alpha i\pi }{2}}I_{\alpha }(z),\\Y_{\alpha }(iz)&=e^{\frac {(\alpha +1)i\pi }{2}}I_{\alpha }(z)-{\frac {2}{\pi }}e^{-{\frac {\alpha i\pi }{2}}}K_{\alpha }(z).\end{aligned}}}

I?(x) and K?(x) are the two linearly independent solutions to the modified Bessel's equation:[21]

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

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, I? and K? are exponentially growing and decaying functions respectively. Like the ordinary Bessel function J?, the function I? goes to zero at x = 0 for ? > 0 and is finite at x = 0 for ? = 0. Analogously, K? diverges at x = 0 with the singularity being of logarithmic type for K0, and ½?(|?|)(2/x)|?| otherwise.[22]

 Modified Bessel functions of the first kind, I?(x), for ? = 0, 1, 2, 3 Modified Bessel functions of the second kind, K?(x), for ? = 0, 1, 2, 3

Two integral formulas for the modified Bessel functions are (for Re(x) > 0):[23]

{\displaystyle {\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,\\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:

${\displaystyle 2\,K_{0}(\omega )=\int _{-\infty }^{\infty }{\frac {e^{i\omega t}}{\sqrt {t^{2}+1}}}\,dt.}$

It can be proven by showing equality to the above integral definition for K0. This is done by integrating a closed curve in the first quadrant of the complex plane.

Modified Bessel functions K1/3 and K2/3 can be represented in terms of rapidly convergent integrals[24]

{\displaystyle {\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,\\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 of the second kind has also been called by the following names (now rare):

### Spherical Bessel functions: jn, yn

Spherical Bessel functions of the first kind, jn(x), for n = 0, 1, 2
Spherical Bessel functions of the second kind, yn(x), for n = 0, 1, 2

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

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

The two linearly independent solutions to this equation are called the spherical Bessel functions jn and yn, and are related to the ordinary Bessel functions Jn and Yn by[26]

{\displaystyle {\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}}}

yn is also denoted nn or ?n; some authors call these functions the spherical Neumann functions.

The spherical Bessel functions can also be written as (Rayleigh's formulas)[27]

{\displaystyle {\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 first spherical Bessel function j0(x) is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are:[28]

{\displaystyle {\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[29]

{\displaystyle {\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}}}

#### Generating function

The spherical Bessel functions have the generating functions[30]

{\displaystyle {\begin{aligned}{\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{aligned}}}

#### Differential relations

In the following, fn is any of jn, yn, h(1)
n
, h(2)
n
for n = 0, ±1, ±2, ...[31]

{\displaystyle {\begin{aligned}\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{aligned}}}

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

There are also spherical analogues of the Hankel functions:

{\displaystyle {\begin{aligned}h_{n}^{(1)}(x)&=j_{n}(x)+iy_{n}(x),\\h_{n}^{(2)}(x)&=j_{n}(x)-iy_{n}(x).\end{aligned}}}

In fact, there are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, and therefore for the spherical Bessel functions. In particular, for non-negative integers n:

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

and h(2)
n
is the complex-conjugate of this (for real x). It follows, for example, that j0(x) = and y0(x) = -, and so on.

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

### Riccati-Bessel functions: Sn, Cn, ?n, ?n

Riccati-Bessel functions only slightly differ from spherical Bessel functions:

{\displaystyle {\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}}}

They satisfy the differential equation

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

For example, this kind of differential equation appears in quantum mechanics while solving the radial component of the Schrödinger's equation with hypothetical cylindrical infinite potential barrier.[32] 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)[33] for recent developments and references.

Following Debye (1909), the notation ?n, ?n is sometimes used instead of Sn, Cn.

## Asymptotic forms

The Bessel functions have the following asymptotic forms. For small arguments 0 < z? + 1, one obtains, when ? is not a negative integer:[3]

${\displaystyle J_{\alpha }(z)\sim {\frac {1}{\Gamma (\alpha +1)}}\left({\frac {z}{2}}\right)^{\alpha }.}$

When ? is a negative integer, we have

${\displaystyle J_{\alpha }(z)\sim {\frac {(-1)^{\alpha }}{(-\alpha )!}}\left({\frac {2}{z}}\right)^{\alpha }.}$

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

${\displaystyle Y_{\alpha }(z)\sim {\begin{cases}{\dfrac {2}{\pi }}\left(\ln \left({\dfrac {z}{2}}\right)+\gamma \right)&{\text{if }}\alpha =0\\-{\dfrac {\Gamma (\alpha )}{\pi }}\left({\dfrac {2}{z}}\right)^{\alpha }+{\dfrac {1}{\Gamma (\alpha +1)}}\left({\dfrac {z}{2}}\right)^{\alpha }\cot(\alpha \pi )&{\text{if }}\alpha {\text{ is not a non-positive integer (one term dominates unless }}\alpha {\text{ is imaginary)}},\\-{\dfrac {(-1)^{\alpha }\Gamma (-\alpha )}{\pi }}\left({\dfrac {z}{2}}\right)^{\alpha }&{\text{if }}\alpha {\text{ is a negative integer,}}\end{cases}}}$

where ? is the Euler-Mascheroni constant (0.5772...).

For large real arguments z >> ||, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless ? is half-integer) because they have zeros all the way out to infinity, which would have to be matched exactly by any asymptotic expansion. However, for a given value of arg z one can write an equation containing a term of order ||-1:[34]

{\displaystyle {\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|}\mathrm {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|}\mathrm {O} \left(|z|^{-1}\right)\right)&&{\text{for }}\left|\arg z\right|<\pi .\end{aligned}}}

(For ? = the last terms in these formulas drop out completely; see the spherical Bessel functions above.) Even though these equations are true, better approximations may be available for complex z. For example, J0(z) when z is near the negative real line is approximated better by

${\displaystyle J_{0}(z)\approx {\sqrt {\frac {-2}{\pi z}}}\cos \left(z+{\frac {\pi }{4}}\right)}$

than by

${\displaystyle J_{0}(z)\approx {\sqrt {\frac {2}{\pi z}}}\cos \left(z-{\frac {\pi }{4}}\right).}$

The asymptotic forms for the Hankel functions are:

{\displaystyle {\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 of arg z using equations relating H(1)
?
(zeim?)
and H(2)
?
(zeim?)
to H(1)
?
(z)
and H(2)
?
(z)
.[35]

It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, J?(z) is not asymptotic to the average of these two asymptotic forms when z is negative (because one or the other will not be correct there, depending on the arg z used). But the asymptotic forms for the Hankel functions permit us to write asymptotic forms for the Bessel functions of first and second kinds for complex (non-real) z so long as || goes to infinity at a constant phase angle arg z (using the square root having positive real part):

{\displaystyle {\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,\\J_{\alpha }(z)&\sim {\frac {1}{\sqrt {2\pi z}}}e^{-i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}0<\arg z<\pi ,\\Y_{\alpha }(z)&\sim -i{\frac {1}{\sqrt {2\pi z}}}e^{i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}-\pi <\arg z<0,\\Y_{\alpha }(z)&\sim -i{\frac {1}{\sqrt {2\pi z}}}e^{-i\left(z-{\frac {\alpha \pi }{2}}-{\frac {\pi }{4}}\right)}&&{\text{for }}0<\arg z<\pi .\end{aligned}}}

For the modified Bessel functions, Hankel developed asymptotic (large argument) expansions as well:[36][37]

{\displaystyle {\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}}}

When ? = , all the terms except the first vanish, and we have

{\displaystyle {\begin{aligned}I_{\frac {1}{2}}(z)&={\sqrt {\frac {2}{\pi z}}}\sinh(z)\sim {\frac {e^{z}}{\sqrt {2\pi z}}}&&{\text{for }}\left|\arg z\right|<{\tfrac {\pi }{2}},\\K_{\frac {1}{2}}(z)&={\sqrt {\frac {\pi }{2z}}}e^{-z}.\end{aligned}}}

For small arguments 0 < || √? + 1, we have

{\displaystyle {\begin{aligned}I_{\alpha }(z)&\sim {\frac {1}{\Gamma (\alpha +1)}}\left({\frac {z}{2}}\right)^{\alpha },\\K_{\alpha }(z)&\sim {\begin{cases}-\ln \left({\dfrac {z}{2}}\right)-\gamma &{\text{if }}\alpha =0\\{\frac {\Gamma (\alpha )}{2}}\left({\dfrac {2}{z}}\right)^{\alpha }&{\text{if }}\alpha >0\end{cases}}\end{aligned}}}

## Full domain approximations with elementary functions

A very good approximation (error below ${\displaystyle 0.3\%}$ of the maximum value 1)[] of the Bessel function ${\displaystyle J_{0}}$ for an arbitrary value of the argument x may be obtained with the elementary functions by joining the trigonometric approximation working for smaller values of x with the expression containing attenuated cosine function valid for large arguments with a usage of the smooth transition function ${\displaystyle {\frac {1}{1+(x/7)^{20}}}}$ i.e.

${\displaystyle J_{0}(x)\approx \left[{\frac {1}{6}}+{\frac {1}{3}}\cos {\frac {x}{2}}+{\frac {1}{3}}\cos {\frac {{\sqrt {3}}x}{2}}+{\frac {1}{6}}\cos x\right]{\frac {1}{1+(x/7)^{20}}}+{\sqrt {\frac {2}{\pi |x|}}}\cos \left[x-{\frac {\pi }{4}}\operatorname {sgn}(x)\right]\left[1-{\frac {1}{1+(x/7)^{20}}}\right].}$

## Properties

For integer order ? = n, Jn is often defined via a Laurent series for a generating function:

${\displaystyle e^{\left({\frac {x}{2}}\right)\left(t-{\frac {1}{t}}\right)}=\sum _{n=-\infty }^{\infty }J_{n}(x)t^{n}}$

an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.) Another important relation for integer orders is the Jacobi-Anger expansion:

${\displaystyle e^{iz\cos \phi }=\sum _{n=-\infty }^{\infty }i^{n}J_{n}(z)e^{in\phi }}$

and

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

which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone-modulated FM signal.

More generally, a series

${\displaystyle f(z)=a_{0}^{\nu }J_{\nu }(z)+2\cdot \sum _{k=1}^{\infty }a_{k}^{\nu }J_{\nu +k}(z)}$

is called Neumann expansion of f. The coefficients for ? = 0 have the explicit form

${\displaystyle a_{k}^{0}={\frac {1}{2\pi i}}\int _{|z|=c}f(z)O_{k}(z)\,dz}$

where Ok is Neumann's polynomial.[38]

Selected functions admit the special representation

${\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}^{\nu }J_{\nu +2k}(z)}$

with

${\displaystyle a_{k}^{\nu }=2(\nu +2k)\int _{0}^{\infty }f(z){\frac {J_{\nu +2k}(z)}{z}}\,dz}$

due to the orthogonality relation

${\displaystyle \int _{0}^{\infty }J_{\alpha }(z)J_{\beta }(z){\frac {dz}{z}}={\frac {2}{\pi }}{\frac {\sin \left({\frac {\pi }{2}}(\alpha -\beta )\right)}{\alpha ^{2}-\beta ^{2}}}}$

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

${\displaystyle f(z)=\sum _{k=0}a_{k}J_{\nu +k}(z)}$

then

${\displaystyle {\mathcal {L}}\left\{\sum _{k=0}a_{k}J_{\nu +k}\right\}(s)={\frac {1}{\sqrt {1+s^{2}}}}\sum _{k=0}{\frac {a_{k}}{\left(s+{\sqrt {1+s^{2}}}\right)^{\nu +k}}}}$

or

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

where ${\displaystyle {\mathcal {L}}\{f\}}$ is the Laplace transform of f.[39]

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

{\displaystyle {\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}}}

where ? > - and z ? C.[40] This formula is useful especially when working with Fourier transforms.

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

${\displaystyle \int _{0}^{1}xJ_{\alpha }\left(xu_{\alpha ,m}\right)J_{\alpha }\left(xu_{\alpha ,n}\right)\,dx={\frac {\delta _{m,n}}{2}}\left[J_{\alpha +1}\left(u_{\alpha ,m}\right)\right]^{2}={\frac {\delta _{m,n}}{2}}\left[J_{\alpha }'\left(u_{\alpha ,m}\right)\right]^{2}}$

where ? > -1, ?m,n is the Kronecker delta, and u?,m is the mth zero of J?(x). This orthogonality relation can then be used to extract the coefficients in the Fourier-Bessel series, where a function is expanded in the basis of the functions J?(x u?,m) for fixed ? and varying m.

An analogous relationship for the spherical Bessel functions follows immediately:

${\displaystyle \int _{0}^{1}x^{2}j_{\alpha }\left(xu_{\alpha ,m}\right)j_{\alpha }\left(xu_{\alpha ,n}\right)\,dx={\frac {\delta _{m,n}}{2}}\left[j_{\alpha +1}\left(u_{\alpha ,m}\right)\right]^{2}}$

If one defines a boxcar function of x that depends on a small parameter ? as:

${\displaystyle f_{\varepsilon }(x)=\varepsilon \operatorname {rect} \left({\frac {x-1}{\varepsilon }}\right)}$

(where rect is the rectangle function) then the Hankel transform of it (of any given order ? > -), g?(k), approaches J?(k) as ? approaches zero, for any given k. Conversely, the Hankel transform (of the same order) of g?(k) is f?(x):

${\displaystyle \int _{0}^{\infty }kJ_{\alpha }(kx)g_{\varepsilon }(k)\,dk=f_{\varepsilon }(x)}$

which is zero everywhere except near 1. As ? approaches zero, the right-hand side approaches ?(x - 1), where ? is the Dirac delta function. This admits the limit (in the distributional sense):

${\displaystyle \int _{0}^{\infty }kJ_{\alpha }(kx)J_{\alpha }(k)\,dk=\delta (x-1)}$

A change of variables then yields the closure equation:[41]

${\displaystyle \int _{0}^{\infty }xJ_{\alpha }(ux)J_{\alpha }(vx)\,dx={\frac {1}{u}}\delta (u-v)}$

for ? > -. The Hankel transform can express a fairly arbitrary function[clarification needed]as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is:

${\displaystyle \int _{0}^{\infty }x^{2}j_{\alpha }(ux)j_{\alpha }(vx)\,dx={\frac {\pi }{2u^{2}}}\delta (u-v)}$

for ? > -1.

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

${\displaystyle A_{\alpha }(x){\frac {dB_{\alpha }}{dx}}-{\frac {dA_{\alpha }}{dx}}B_{\alpha }(x)={\frac {C_{\alpha }}{x}}}$

where A? and B? are any two solutions of Bessel's equation, and C? is a constant independent of x (which depends on ? and on the particular Bessel functions considered). In particular,

${\displaystyle J_{\alpha }(x){\frac {dY_{\alpha }}{dx}}-{\frac {dJ_{\alpha }}{dx}}Y_{\alpha }(x)={\frac {2}{\pi x}}}$

and

${\displaystyle I_{\alpha }(x){\frac {dK_{\alpha }}{dx}}-{\frac {dI_{\alpha }}{dx}}K_{\alpha }(x)=-{\frac {1}{x}},}$

for ? > -1.

For ? > -1, the even entire function of genus 1, x-?J?(x), has only real zeros. Let

${\displaystyle 0

be all its positive zeros, then

${\displaystyle J_{\alpha }(z)={\frac {\left({\frac {z}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{j_{\alpha ,n}^{2}}}\right)}$

(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

The functions J?, Y?, H(1)
?
, and H(2)
?
all satisfy the recurrence relations[42]

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

and

${\displaystyle 2{\frac {dZ_{\alpha }(x)}{dx}}=Z_{\alpha -1}(x)-Z_{\alpha +1}(x),}$

where Z denotes J, Y, H(1), or H(2). These two identities are often combined, e.g. added or subtracted, to yield various other relations. In this way, for example, one can compute Bessel functions of higher orders (or higher derivatives) given the values at lower orders (or lower derivatives). In particular, it follows that[43]

{\displaystyle {\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}}}

Modified Bessel functions follow similar relations:

${\displaystyle e^{\left({\frac {x}{2}}\right)\left(t+{\frac {1}{t}}\right)}=\sum _{n=-\infty }^{\infty }I_{n}(x)t^{n}}$

and

${\displaystyle e^{z\cos \theta }=I_{0}(z)+2\sum _{n=1}^{\infty }I_{n}(z)\cos n\theta .}$

and

${\displaystyle {\frac {1}{2\pi }}\int _{0}^{2\pi }e^{z\cos(m\theta )+y\cos \theta }=I_{0}(z)I_{0}(y)+2\sum _{n=1}^{\infty }I_{n}(z)I_{mn}(y).}$

{\displaystyle {\begin{aligned}C_{\alpha -1}(x)-C_{\alpha +1}(x)&={\frac {2\alpha }{x}}C_{\alpha }(x),\\C_{\alpha -1}(x)+C_{\alpha +1}(x)&=2{\frac {dC_{\alpha }}{dx}},\end{aligned}}}

where C? denotes I? or e?i?K?. These recurrence relations are useful for discrete diffusion problems.

## Multiplication theorem

The Bessel functions obey a multiplication theorem

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

where ? and ? may be taken as arbitrary complex numbers.[44][45] For || < 1,[44] the above expression also holds if J is replaced by Y. The analogous identities for modified Bessel functions and || < 1 are

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

and

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

## Zeros of the Bessel function

### Bourget's hypothesis

Bessel himself originally proved that for nonnegative integers n, the equation Jn(x) = 0 has an infinite number of solutions in x.[46] When the functions Jn(x) are plotted on the same graph, though, none of the zeros seem to coincide for different values of n except for the zero at x = 0. This phenomenon is known as Bourget's hypothesis after the 19th-century French mathematician who studied Bessel functions. Specifically it states that for any integers n >= 0 and m >= 1, the functions Jn(x) and Jn + m(x) have no common zeros other than the one at x = 0. The hypothesis was proved by Carl Ludwig Siegel in 1929.[47]

### Numerical approaches

For numerical studies about the zeros of the Bessel function, see Gil, Segura & Temme (2007), Kravanja et al. (1998) and Moler (2004).

## Notes

1. ^ Weisstein, Eric W. "Spherical Bessel Function of the Second Kind". MathWorld.
2. ^
3. ^ a b Abramowitz and Stegun, p. 360, 9.1.10.
4. ^ Abramowitz and Stegun, p. 358, 9.1.5.
5. ^ a b c Temme, 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.
6. ^ Watson, p. 176
7. ^ "Archived copy". Archived from the original on 2010-09-23. Retrieved .CS1 maint: archived copy as title (link)
8. ^ "Integral representations of the Bessel function". www.nbi.dk. Retrieved 2018.
9. ^ Arfken & Weber, exercise 11.1.17.
10. ^ Abramowitz and Stegun, p. 362, 9.1.69.
11. ^ Szeg?, Gábor (1975). Orthogonal Polynomials (4th ed.). Providence, RI: AMS.
12. ^ http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap4.pdf
13. ^ NIST Digital Library of Mathematical Functions, (10.8.1). Accessed on line Oct. 25, 2016.
14. ^
15. ^ a b Watson, p. 178.
16. ^ Abramowitz and Stegun, p. 358, 9.1.3, 9.1.4.
17. ^ Abramowitz and Stegun, p. 358, 9.1.6.
18. ^ Abramowitz and Stegun, p. 360, 9.1.25.
19. ^ Abramowitz and Stegun, p. 375, 9.6.2, 9.6.10, 9.6.11.
20. ^ Abramowitz and Stegun, p. 375, 9.6.3, 9.6.5.
21. ^ Abramowitz and Stegun, p. 374, 9.6.1.
22. ^ Greiner, Walter; Reinhardt, Joachim (2009). Quantum Electrodynamics. Springer. p. 72. ISBN 978-3-540-87561-1.
23. ^ Watson, p. 181.
24. ^ 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 to I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Fizmatgiz, Moscow, 1963; Academic Press, New York, 1980).
25. ^ 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.
26. ^ Abramowitz and Stegun, p. 437, 10.1.1.
27. ^ Abramowitz and Stegun, p. 439, 10.1.25, 10.1.26.
28. ^ Abramowitz and Stegun, p. 438, 10.1.11.
29. ^ Abramowitz and Stegun, p. 438, 10.1.12.
30. ^ Abramowitz and Stegun, p. 439, 10.1.39.
31. ^ Abramowitz and Stegun, p. 439, 10.1.23, 10.1.24.
32. ^ Griffiths. Introduction to Quantum Mechanics, 2nd edition, p. 154.
33. ^ Du, Hong (2004). "Mie-scattering calculation". Applied Optics. 43 (9): 1951-1956. Bibcode:2004ApOpt..43.1951D. doi:10.1364/ao.43.001951. PMID 15065726.
34. ^ Abramowitz and Stegun, p. 364, 9.2.1.
35. ^
36. ^ Abramowitz and Stegun, p. 377, 9.7.1.
37. ^ Abramowitz and Stegun, p. 378, 9.7.2.
38. ^ Abramowitz and Stegun, p. 363, 9.1.82 ff.
39. ^ Watson, G. N. (25 August 1995). A Treatise on the Theory of Bessel Functions. Cambridge University Press. ISBN 9780521483919. Retrieved 2018 – via Google Books.
40. ^ 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.
41. ^ Arfken & Weber, section 11.2
42. ^ Abramowitz and Stegun, p. 361, 9.1.27.
43. ^ Abramowitz and Stegun, p. 361, 9.1.30.
44. ^ a b Abramowitz and Stegun, p. 363, 9.1.74.
45. ^ Truesdell, C. (1950). "On the Addition and Multiplication Theorems for the Special Functions" (PDF). 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.
46. ^ Bessel, F. (1824) "Untersuchung des Theils der planetarischen Störungen", Berlin Abhandlungen, article 14.
47. ^ Watson, pp. 484-485.