# Bessel functions

2020 Mathematics Subject Classification: *Primary:* 33C10 [MSN][ZBL]

Some authors use this term for all the cylinder functions. In this entry the term is used for the cylinder functions of the first kind (which are usually called *Bessel functions of the first kind* by those authors which use the term Bessel functions for all cylinder functions). For the Bessel functions of the second kind,
denoted by $Y_\nu$ (more rarely by $N_\nu$) and also called Neumann functions or Weber functions, see Cylinder functions and Neumann function. For the Bessel functions of the third kind see
Cylinder functions and Hankel functions.

### Definition

The Bessel function of order $\nu \in \mathbb C$ can be defined, when $\nu$ is *not* a negative integer, via the series
\begin{equation}\label{e:series}
J_\nu (z) := \sum_{k=0}^\infty \frac{(-1)^k}{\Gamma (k+1) \Gamma (k+\nu+1)} \left(\frac{z}{2}\right)^{\nu + 2k} = \left(\frac{z}{2}\right)^\nu \sum_{k=0}^\infty \frac{(-1)^k}{\Gamma (k+1) \Gamma (k+\nu+1)} \left(\frac{z}{2}\right)^{2k}\, ,
\end{equation}
where $\Gamma$ is the Gamma-function. The series in the second identity converges on the entire complex plane (when $\nu$ is not a negative integer) and hence the indeterminacy (or multi-valued nature) in the analytic function $J_\nu$ is reduced to that of $z^\nu$ when $\nu\not \in \mathbb N$. When $\nu = n \in \mathbb N$, the Bessel function is therefore entire and the Taylor series in \eqref{e:series} takes the simple form
\[
J_n (z) = \left(\frac{z}{2}\right)^n \sum_{k=0}^\infty \frac{(-1)^k}{k!(n+k)!} \left(\frac{z}{2}\right)^{2k}\, .
\]
When $\nu = -n$ with $n\in \mathbb N$, the formulas in \eqref{e:series} are not "strictly speaking" well-defined because the $\Gamma$ function has poles in all negative integers. However, given the Laurent series of $\Gamma$ at such poles and the fact that the function appears in the denominators of fractions, the formula \eqref{e:series} can be intepreted, for $\nu = -n$ as
\[
J_{-n} (z) = \left(\frac{z}{2}\right)^{-n} \sum_{k=0}^\infty \frac{(-1)^{n+k}}{k!(n+k)!} \left(\frac{z}{2}\right)^{2k+2n} = (-1)^n J_n (z)\, .
\]

### Real and integer order

If the argument is real and the order $\nu$ is integer, the Bessel function is real, and its graph has the form of a damped vibration (Fig. 1). If the order is even, the Bessel function is even, if odd, it is odd. If $\nu$ is real and the argument is real, it is a common convention to take the determination of $z^\nu$ which takes real values for positive real values of $z$. Thus the Bessel function $J_\nu$ is real on the positive real axis when $\nu\in \mathbb R$.

Fig. 1. Graphs of the functions $y = J_0 (x)$ and $y= J_1 (x)$ for positive real values of $x$.

The behaviour of a Bessel function $J_n$ for $n$ integer in a neighbourhood of zero is given by the first term of the series \eqref{e:series}; for large $z$, $|{\arg z}|<\pi$, the asymptotic representation \[ J_n (z) = \sqrt{\frac{2}{\pi z}} \cos \left(z - \frac{\pi}{2} n - \frac{\pi}{4}\right) +O\left(\frac{1}{z}\right) \] (which holds also for real or complex $\nu$).

The zeros of a Bessel function $J_\nu$ with $\nu \in \mathbb R$ (i.e. the roots of the equation $J_\nu (x) =0$) are simple, and the zeros of $J_\nu$ are situated between the zeros of $J_{\nu+1}$.

#### Half integer order

Bessel functions of "half-integral" order $\nu = n+ \frac{1}{2}$ with $n\in \mathbb Z$ can be expressed by trigonometric functions; in particular we have \[ J_{1/2} (x) = \sqrt{\frac{2}{\pi x}} \sin x \qquad J_{-1/2} (x) = \sqrt{\frac{2}{\pi x}} \cos x\, . \] More in general there are polynomials $P_n$ and $Q_n$, for $n\in \mathbb N$, with \begin{eqnarray*} && \deg P_n = \deg Q_n = n\\ && P_n (-x) = (-1)^n P_n (x) \qquad Q_n (-x) = (-1)^n Q_n (x) \end{eqnarray*} such that \begin{eqnarray} &&J_{n+1/2} (x) = \sqrt{\frac{2}{\pi x}} \left(P_n \left(\frac{1}{x}\right) \sin x - Q_{n-1} \left(\frac{1}{x}\right) \cos x \right)\\ &&J_{-n-1/2} (x) = (-1)^n \sqrt{\frac{2}{\pi x}} \left(P_n \left(\frac{1}{x}\right) \cos x + Q_{n-1} \left(\frac{1}{x}\right) \sin x \right)\, . \end{eqnarray}

### Fourier-Bessel series

Let $\nu$ be real larger than $-\frac{1}{2}$ and denote by $\mu_n^\nu$ the positive zeros of $J_\nu$. Then the functions $x \mapsto b_n (x) := J_\nu (\mu_n^\nu x)$, $n\in \mathbb N\setminus \{0\}$ form an orthogonal system with weight $x$ in the interval $[0,1]$. Under certain conditions the following expansion, called Fourier-Bessel series, is then valid: \[ f(x) = \sum_{n=1}^\infty c_n b_n (x) \] where \[ c_n = \frac{2}{(J_{\nu+1} (\mu_n^\nu))} \int_0^1 x\, f (x)\, b_n (x)\, dx\, . \] On the infinite half-line $[0, \infty[$ this expansion is replaced by the Fourier–Bessel integral: \[ f(x) = \int_0^\infty c (\lambda)\, J_\nu (\lambda x)\, d\lambda \] where \[ c (\lambda) = \int J_\nu (\lambda x)\, f(x)\, x\, dx\, . \]

### Notable formulas

The following formulas play an important role in the theory of Bessel functions and their applications:

1) The integral representation, which for $n$ integer takes the form \[ J_n (z) = \frac{1}{\pi} \int_0^\pi \cos\, (z \sin \phi - n \phi)\, d \phi \] and was the starting point of Bessel himself in his original investigations. For $\nu$ complex and ${\rm Re}\, z > 0$ the identity can be extended as \[ J_\nu (z) = \frac{1}{\pi} \int_0^\pi \cos\, (z \sin \phi - \nu \phi)\, d \phi - \frac{\sin (\pi \nu)}{\pi} \int_0^\infty e^{-z \sinh t - \nu t}\, dt\, . \]

2) The generating function \[ e^{z\, (\xi - \xi^{-1})/2} = \sum_{k=-\infty}^\infty J_k (z)\, \xi^k\, . \]

3) The addition theorem for integer order $n\in \mathbb Z$ \[ J_n (z_1+z_2) = \sum_{k=-\infty}^\infty J_k (z_1)\, J_{n-k} (z_2)\, . \]

4) The recurrence formulas \begin{eqnarray*} && J_{\nu-1} (z) + J_{\nu+1} (z) = \frac{2\nu}{z} J_\nu (z)\\ && J'_\nu (z) = \frac{1}{2} (J_{\nu-1} (z) - J_{\nu+1} (z))\, . \end{eqnarray*}

### References

[Co] | H. Cohen, "Number theory volume II: analytic and modern tools", Springer (2007) |

[GR] | I.S., Gradshteyn, I.M. Ryzhik, "Table of integrals, series and products", Academic Press (2000) |

[Wa] | G.N. Watson, "A Treatise on the Theory of Bessel Functions", Cambridge University Press (1922) |

**How to Cite This Entry:**

Bessel functions.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Bessel_functions&oldid=53295