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Bessel function

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 $$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + \left(x^2 - \alpha^2 \right)y = 0$$ for an arbitrary complex number $$\alpha$$, which represents the order of the Bessel function. Although $$\alpha$$ and $$-\alpha$$ produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of $$\alpha$$.

The most important cases are when $$\alpha$$ is an integer or half-integer. Bessel functions for integer $$\alpha$$ 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 $$\alpha$$ are obtained when solving the Helmholtz equation 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 ($α = n$); in spherical problems, one obtains half-integer orders ($α = n + 1⁄2$). 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
 * 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).

Definitions
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 $J_{α}$ and $Y_{α}$, respectively, rather than $I_{α}$ and $K_{α}$.

Bessel functions of the first kind: $H(1) α = J_{α} + iY_{α}$

Bessel functions of the first kind, denoted as $H(2) α = J_{α} − iY_{α}$, are solutions of Bessel's differential equation. For integer or positive $j_{n}$, Bessel functions of the first kind are finite at the origin ($h(1) n = j_{n} + iy_{n}$); while for negative non-integer $y_{n}$, Bessel functions of the first kind diverge as $N_{n}$ approaches zero. It is possible to define the function by $$x^\alpha$$ times a Maclaurin series (note that $n_{n}$ 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: $$ J_\alpha(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m!\, \Gamma(m+\alpha+1)} {\left(\frac{x}{2}\right)}^{2m + \alpha},$$ where $h(2) n = j_{n} − iy_{n}$ 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 $Y_{n}$ 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 $y_{n}$. (The series indicates that $J_{α}$ is the derivative of $J_{n}(z)$, much like $n = 0.5$ is the derivative of $−2 − 2i$; more generally, the derivative of $2 + 2i$ can be expressed in terms of $J_{α}(x)$ by the identities below.)

For non-integer $α$, the functions $α = 0, 1, 2$ and $J_{α}(x)$ are linearly independent, and are therefore the two solutions of the differential equation. On the other hand, for integer order $α$, the following relationship is valid (the gamma function has simple poles at each of the non-positive integers): $$J_{-n}(x) = (-1)^n J_n(x).$$

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 $x$, is possible using an integral representation: $$J_n(x) = \frac{1}{\pi} \int_0^\pi \cos (n \tau - x \sin \tau) \,d\tau = \frac{1}{2\pi} \int_{-\pi}^\pi e^{i(n \tau-x \sin \tau )} \,d\tau,$$ which is also called Hansen-Bessel formula.

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 $x = 0$: $$J_\alpha(x) = \frac{1}{\pi} \int_0^\pi \cos(\alpha\tau - x \sin\tau)\,d\tau - \frac{\sin(\alpha\pi)}{\pi} \int_0^\infty e^{-x \sinh t - \alpha t} \, dt. $$

Relation to hypergeometric series
The Bessel functions can be expressed in terms of the generalized hypergeometric series as $$J_\alpha(x) = \frac{\left(\frac{x}{2}\right)^\alpha}{\Gamma(\alpha+1)} \;_0F_1 \left(\alpha+1; -\frac{x^2}{4}\right).$$

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 $α$ and arbitrarily chosen parameter $α$, the Bessel function can be expressed as $$\frac{J_\alpha(x)}{\left( \frac{x}{2}\right)^\alpha} = \frac{e^{-t}}{\Gamma(\alpha+1)} \sum_{k=0}^\infty \frac{L_k^{(\alpha)}\left( \frac{x^2}{4 t}\right)}{\binom{k+\alpha}{k}} \frac{t^k}{k!}.$$

Bessel functions of the second kind: $Γ(z)$
The Bessel functions of the second kind, denoted by $−J_{1}(x)$, occasionally denoted instead by $J_{0}(x)$, are solutions of the Bessel differential equation that have a singularity at the origin ($−sin x$) and are multivalued. These are sometimes called Weber functions, as they were introduced by, and also Neumann functions after Carl Neumann.

For non-integer $x$, $cos x$ is related to $J_{n}(x)$ by $$Y_\alpha(x) = \frac{J_\alpha(x) \cos (\alpha\pi) - J_{-\alpha}(x)}{\sin (\alpha\pi)}.$$

In the case of integer order $α$, the function is defined by taking the limit as a non-integer $n$ tends to $n$: $$Y_n(x) = \lim_{\alpha \to n} Y_\alpha(x).$$

If $L_{k}$ is a nonnegative integer, we have the series $$Y_n(z) =-\frac{\left(\frac{z}{2}\right)^{-n}}{\pi}\sum_{k=0}^{n-1} \frac{(n-k-1)!}{k!}\left(\frac{z^2}{4}\right)^k +\frac{2}{\pi} J_n(z) \ln \frac{z}{2} -\frac{\left(\frac{z}{2}\right)^n}{\pi}\sum_{k=0}^\infty (\psi(k+1)+\psi(n+k+1)) \frac{\left(-\frac{z^2}{4}\right)^k}{k!(n+k)!}$$

where $$\psi(z)$$ is the digamma function, the logarithmic derivative of the gamma function.

There is also a corresponding integral formula (for $J_{n ± 1}(x)$): $$Y_n(x) = \frac{1}{\pi} \int_0^\pi \sin(x \sin\theta - n\theta) \, d\theta -\frac{1}{\pi} \int_0^\infty \left(e^{nt} + (-1)^n e^{-nt} \right) e^{-x \sinh t} \, dt.$$

In the case where $J_{α}(x)$, $$Y_{0}\left(x\right)=\frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos\left(x\cos\theta\right)\left(e+\ln\left(2x\sin^2\theta\right)\right)\, d\theta.$$

$J_{−α}(x)$ is necessary as the second linearly independent solution of the Bessel's equation when $t$ is an integer. But $Re(x) > 0$ has more meaning than that. It can be considered as a "natural" partner of $Y_{α}$. See also the subsection on Hankel functions below.

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

Both $Y_{n}(z)$ and $n = 0.5$ are holomorphic functions of $n$ on the complex plane cut along the negative real axis. When $α$ is an integer, the Bessel functions $n$ are entire functions of $n$. If $α$ is held fixed at a non-zero value, then the Bessel functions are entire functions of $α$.

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

Hankel functions: $−2 − 2i$, $2 + 2i$

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions of the first and second kind, $Y_{α}(x)$ and $N_{α}(x)$, defined as $$\begin{align} H_\alpha^{(1)}(x) &= J_\alpha(x) + iY_\alpha(x), \\[5pt] H_\alpha^{(2)}(x) &= J_\alpha(x) - iY_\alpha(x), \end{align}$$

where $α$ 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 $x = 0$. For real $$x>0$$ where $$J_\alpha(x)$$, $$Y_\alpha(x)$$ are real-valued, the Bessel functions of the first and second kind are the real and imaginary parts, respectively, of the first Hankel function and the real and negative imaginary parts of the second Hankel function. Thus, the above formulae are analogs of Euler's formula, substituting $Y_{α}(x)$, $J_{α}(x)$ for $$e^{\pm i x}$$ and $$J_\alpha(x)$$, $$Y_\alpha(x)$$ for $$\cos(x)$$, $$\sin(x)$$, as explicitly shown in the asymptotic expansion.

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

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

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

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

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

The Hankel functions admit the following integral representations for $α = 0, 1, 2$: $$\begin{align} H_\alpha^{(1)}(x) &= \frac{1}{\pi i}\int_{-\infty}^{+\infty + \pi i} e^{x\sinh t - \alpha t} \, dt, \\[5pt] H_\alpha^{(2)}(x) &= -\frac{1}{\pi i}\int_{-\infty}^{+\infty - \pi i} e^{x\sinh t - \alpha t} \, dt, \end{align}$$ where the integration limits indicate integration along a contour that can be chosen as follows: from $Re(x) > 0$ to 0 along the negative real axis, from 0 to $n = 0$ along the imaginary axis, and from $Y_{α}(x)$ to $Y_{α}(x)$ along a contour parallel to the real axis.

Modified Bessel functions: $J_{α}(x)$, $J_{α}(x)$
The Bessel functions are valid even for complex arguments $α$, 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 $$\begin{align} I_\alpha(x) &= i^{-\alpha} J_\alpha(ix) = \sum_{m=0}^\infty \frac{1}{m!\, \Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha}, \\[5pt] K_\alpha(x) &= \frac{\pi}{2} \frac{I_{-\alpha}(x) - I_\alpha(x)}{\sin \alpha \pi}, \end{align}$$ when $α$ is not an integer; when $i$ is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments $α$. The series expansion for $Y_{α}(x)$ is thus similar to that for $H(1) α$, but without the alternating $H(2) α$ factor.

$$K_{\alpha}$$ can be expressed in terms of Hankel functions: $$K_{\alpha}(x) = \begin{cases} \frac{\pi}{2} i^{\alpha+1} H_\alpha^{(1)}(ix) & -\pi < \arg x \leq \frac{\pi}{2} \\ \frac{\pi}{2} (-i)^{\alpha+1} H_\alpha^{(2)}(-ix) & -\frac{\pi}{2} < \arg x \leq \pi \end{cases}$$

Using these two formulae the result to $$J_{\alpha}^2(z)$$+$$Y_{\alpha}^2(z)$$, commonly known as Nicholson's integral or Nicholson's formula, can be obtained to give the following $$ J_{\alpha}^2(x)+Y_{\alpha}^2(x)=\frac{8}{\pi^2}\int_{0}^{\infty}\cosh(2\alpha t)K_0(2x\sinh t)\, dt, $$

given that the condition $H(1) n(x)$ 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 |$n = −0.5$| < $−2 − 2i$ and $2 + 2i$ but not when $H(2) n(x)$.

We can express the first and second Bessel functions in terms of the modified Bessel functions (these are valid if $n = −0.5$): $$\begin{align} J_\alpha(iz) &= e^{\frac{\alpha\pi i}{2}} I_\alpha(z), \\[1ex] Y_\alpha(iz) &= e^{\frac{(\alpha+1)\pi i}{2}}I_\alpha(z) - \tfrac{2}{\pi} e^{-\frac{\alpha\pi i}{2}}K_\alpha(z). \end{align}$$

$−2 − 2i$ and $2 + 2i$ are the two linearly independent solutions to the modified Bessel's equation: $$x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - \left(x^2 + \alpha^2 \right)y = 0.$$

Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, $α$ and $m$ are exponentially growing and decaying functions respectively. Like the ordinary Bessel function $x$, the function $α$ goes to zero at $H(1) α(x)$ for $H(2) α(x)$ and is finite at $e^{i f(x)}$ for $H(1) α(x)$. Analogously, $α$ diverges at $H(2) α(x)$ with the singularity being of logarithmic type for $x$, and $α = m + 1⁄2$ otherwise.

Two integral formulas for the modified Bessel functions are (for $Re(x) > 0$): $$\begin{align} I_\alpha(x) &= \frac{1}{\pi}\int_0^\pi e^{x\cos \theta} \cos \alpha\theta \,d\theta - \frac{\sin \alpha\pi}{\pi}\int_0^\infty e^{-x\cosh t - \alpha t} \,dt, \\[5pt] K_\alpha(x) &= \int_0^\infty e^{-x\cosh t} \cosh \alpha t \,dt. \end{align}$$

Bessel functions can be described as Fourier transforms of powers of quadratic functions. For example (for $−∞$): $$2\,K_0(\omega) = \int_{-\infty}^\infty \frac{e^{i\omega t}}{\sqrt{t^2+1}} \,dt.$$

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

Modified Bessel functions $±\pii$ and $+∞ ± \pii$ can be represented in terms of rapidly convergent integrals $$ \begin{align} K_{\frac{1}{3}}(\xi) &= \sqrt{3} \int_0^\infty \exp \left(- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}} \right) \,dx, \\[5pt] K_{\frac{2}{3}}(\xi) &= \frac{1}{\sqrt{3}} \int_0^\infty \frac{3+2x^2}{\sqrt{1+\frac{x^2}{3}}} \exp \left(- \xi \left(1+\frac{4x^2}{3}\right) \sqrt{1+\frac{x^2}{3}}\right) \,dx. \end{align}$$

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

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

 * Basset function after Alfred Barnard Basset
 * Modified Bessel function of the third kind
 * Modified Hankel function
 * Macdonald function after Hector Munro Macdonald

Spherical Bessel functions: $I_{α}$, $K_{α}$

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

The two linearly independent solutions to this equation are called the spherical Bessel functions $I_{α}$ and $K_{α}$, and are related to the ordinary Bessel functions $J_{α}$ and $I_{α}$ by $$\begin{align} j_n(x) &= \sqrt{\frac{\pi}{2x}} J_{n+\frac{1}{2}}(x), \\ y_n(x) &= \sqrt{\frac{\pi}{2x}} Y_{n+\frac{1}{2}}(x) = (-1)^{n+1} \sqrt{\frac{\pi}{2x}} J_{-n-\frac{1}{2}}(x). \end{align}$$

$K_{α}$ is also denoted $K_{0}$ or $j_{n}$; some authors call these functions the spherical Neumann functions.

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

The spherical Bessel functions can also be written as () $$\begin{align} j_n(x) &= (-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\sin x}{x}, \\ y_n(x) &= -(-x)^n \left(\frac{1}{x}\frac{d}{dx}\right)^n \frac{\cos x}{x}. \end{align}$$

The zeroth spherical Bessel function $I_{α}(x)$ is also known as the (unnormalized) sinc function. The first few spherical Bessel functions are: $$\begin{align} j_0(x) &= \frac{\sin x}{x}. \\ j_1(x) &= \frac{\sin x}{x^2} - \frac{\cos x}{x}, \\ j_2(x) &= \left(\frac{3}{x^2} - 1\right) \frac{\sin x}{x} - \frac{3\cos x}{x^2}, \\ j_3(x) &= \left(\frac{15}{x^3} - \frac{6}{x}\right) \frac{\sin x}{x} - \left(\frac{15}{x^2} - 1\right) \frac{\cos x}{x} \end{align}$$ and $$\begin{align} y_0(x) &= -j_{-1}(x) = -\frac{\cos x}{x}, \\ y_1(x) &= j_{-2}(x) = -\frac{\cos x}{x^2} - \frac{\sin x}{x}, \\ y_2(x) &= -j_{-3}(x) = \left(-\frac{3}{x^2} + 1\right) \frac{\cos x}{x} - \frac{3\sin x}{x^2}, \\ y_3(x) &= j_{-4}(x) = \left(-\frac{15}{x^3} + \frac{6}{x}\right) \frac{\cos x}{x} - \left(\frac{15}{x^2} - 1\right) \frac{\sin x}{x}. \end{align}$$

Generating function
The spherical Bessel functions have the generating functions $$\begin{align} \frac{1}{z} \cos \left(\sqrt{z^2 - 2zt}\right) &= \sum_{n=0}^\infty \frac{t^n}{n!} j_{n-1}(z), \\ \frac{1}{z} \sin \left(\sqrt{z^2 - 2zt}\right) &= \sum_{n=0}^\infty \frac{t^n}{n!} y_{n-1}(z). \end{align}$$

Finite series expansions
In contrast to the whole integer Bessel functions $J_{α}(x)$, the spherical Bessel functions $(−1)^{m}$ have a finite series expression: $$\begin{alignat}{2} j_n(x) &= \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}$$

Differential relations
In the following, $y_{n}$ is any of $J_{n}$, $Y_{n}$, $Re(x) > 0$, $Re(α)$ for $1⁄2$ $$\begin{align} \left(\frac{1}{z}\frac{d}{dz}\right)^m \left (z^{n+1} f_n(z)\right ) &= z^{n-m+1} f_{n-m}(z), \\ \left(\frac{1}{z}\frac{d}{dz}\right)^m \left (z^{-n} f_n(z)\right ) &= (-1)^m z^{-n-m} f_{n+m}(z). \end{align}$$

Spherical Hankel functions: $Re(x) ≥ 0$, $x = 0$

There are also spherical analogues of the Hankel functions: $$\begin{align} h_n^{(1)}(x) &= j_n(x) + i y_n(x), \\ h_n^{(2)}(x) &= j_n(x) - i y_n(x). \end{align}$$

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 $y_{n}$: $$h_n^{(1)}(x) = (-i)^{n+1} \frac{e^{ix}}{x} \sum_{m=0}^n \frac{i^m}{m!\,(2x)^m} \frac{(n+m)!}{(n-m)!},$$

and $−π < arg z ≤ π⁄2$ is the complex-conjugate of this (for real $n_{n}$). It follows, for example, that $I_{α}(x)$ and $K_{α}(x)$, 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: $x = 0$, $α > 0$, $x = 0$, $α = 0$
Riccati–Bessel functions only slightly differ from spherical Bessel functions: $$\begin{align} S_n(x)    &= x j_n(x) = \sqrt{\frac{\pi x}{2}} J_{n+\frac{1}{2}}(x) \\ C_n(x)    &= -x y_n(x) = -\sqrt{\frac{\pi x}{2}} Y_{n+\frac{1}{2}}(x) \\ \xi_n(x)  &= x h_n^{(1)}(x) = \sqrt{\frac{\pi x}{2}} H_{n+\frac{1}{2}}^{(1)}(x) = S_n(x) - iC_n(x) \\ \zeta_n(x) &= x h_n^{(2)}(x) = \sqrt{\frac{\pi x}{2}} H_{n+\frac{1}{2}}^{(2)}(x) = S_n(x) + iC_n(x) \end{align}$$ They satisfy the differential equation $$x^2 \frac{d^2 y}{dx^2} + \left (x^2 - n(n + 1)\right) y = 0.$$

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

Following Debye (1909), the notation $η_{n}$, $f_{n}$ is sometimes used instead of $j_{n}$, $y_{n}$.

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

When $n$ is a negative integer, we have $$J_\alpha(z) \sim \frac{(-1)^{\alpha}}{(-\alpha)!} \left( \frac{2}{z} \right)^\alpha.$$

For the Bessel function of the second kind we have three cases: $$Y_\alpha(z) \sim \begin{cases} \dfrac{2}{\pi} \left( \ln \left(\dfrac{z}{2} \right) + \gamma \right) & \text{if } \alpha = 0 \\[1ex] -\dfrac{\Gamma(\alpha)}{\pi} \left( \dfrac{2}{z} \right)^\alpha + \dfrac{1}{\Gamma(\alpha+1)} \left(\dfrac{z}{2} \right)^\alpha \cot(\alpha \pi) & \text{if } \alpha \text{ is a positive integer (one term dominates unless } \alpha \text{ is imaginary)}, \\[1ex] -\dfrac{(-1)^\alpha\Gamma(-\alpha)}{\pi} \left( \dfrac{z}{2} \right)^\alpha & \text{if } \alpha\text{ is a negative integer,} \end{cases}$$ where $x$ is the Euler–Mascheroni constant (0.5772...).

For large real arguments $x = 0$, one cannot write a true asymptotic form for Bessel functions of the first and second kind (unless $ψ_{n}$ 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 $1⁄2Γ(|α|)(2/x)^{|α|}$ one can write an equation containing a term of order $I_{α}(x)$: $$\begin{align} J_\alpha(z) &= \sqrt{\frac{2}{\pi z}}\left(\cos \left(z-\frac{\alpha\pi}{2} - \frac{\pi}{4}\right) + e^{\left|\operatorname{Im}(z)\right|}\mathcal{O}\left(|z|^{-1}\right)\right) && \text{for } \left|\arg z\right| < \pi, \\ Y_\alpha(z) &= \sqrt{\frac{2}{\pi z}}\left(\sin \left(z-\frac{\alpha\pi}{2} - \frac{\pi}{4}\right) + e^{\left|\operatorname{Im}(z)\right|}\mathcal{O}\left(|z|^{-1}\right)\right) && \text{for } \left|\arg z\right| < \pi. \end{align}$$

(For $α = 0, 1, 2, 3$ the last terms in these formulas drop out completely; see the spherical Bessel functions above.)

The asymptotic forms for the Hankel functions are: $$\begin{align} H_\alpha^{(1)}(z) &\sim \sqrt{\frac{2}{\pi z}}e^{i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } -\pi < \arg z < 2\pi, \\ H_\alpha^{(2)}(z) &\sim \sqrt{\frac{2}{\pi z}}e^{-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } -2\pi < \arg z < \pi. \end{align}$$

These can be extended to other values of $K_{α}(x)$ using equations relating $α = 0, 1, 2, 3$ and $K_{0}$ to $K_{1}$ and $K_{2}$.

It is interesting that although the Bessel function of the first kind is the average of the two Hankel functions, $I_{0}$ is not asymptotic to the average of these two asymptotic forms when $χ_{n}$ is negative (because one or the other will not be correct there, depending on the $I_{1}$ 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) $S_{n}$ so long as $I_{2}$ goes to infinity at a constant phase angle $Re(x) > 0$ (using the square root having positive real part): $$\begin{align} J_\alpha(z) &\sim \frac{1}{\sqrt{2\pi z}} e^{i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } -\pi < \arg z < 0, \\[1ex] J_\alpha(z) &\sim \frac{1}{\sqrt{2\pi z}} e^{-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } 0 < \arg z < \pi, \\[1ex] Y_\alpha(z) &\sim -i\frac{1}{\sqrt{2\pi z}} e^{i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } -\pi < \arg z < 0, \\[1ex] Y_\alpha(z) &\sim i\frac{1}{\sqrt{2\pi z}} e^{-i\left(z-\frac{\alpha\pi}{2}-\frac{\pi}{4}\right)} && \text{for } 0 < \arg z < \pi. \end{align}$$

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

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

When $Re(ω) > 0$, all the terms except the first vanish, and we have $$\begin{align} I_{{1}/{2}}(z) &= \sqrt{\frac{2}{\pi}} \frac{\sinh(z)}{\sqrt{z}} \sim \frac{e^z}{\sqrt{2\pi z}} && \text{for }\left|\arg z\right| < \tfrac{\pi}{2}, \\[1ex] K_{{1}/{2}}(z) &= \sqrt{\frac{\pi}{2}} \frac{e^{-z}}{\sqrt{z}}. \end{align}$$

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

Properties
For integer order $K_{0}$, $C_{n}$ is often defined via a Laurent series for a generating function: $$e^{\left(\frac{x}{2}\right)\left(t-\frac{1}{t}\right)} = \sum_{n=-\infty}^\infty J_n(x) t^n$$ an approach used by P. A. Hansen in 1843. (This can be generalized to non-integer order by contour integration or other methods.)

Infinite series of Bessel functions in the form $ \sum_{\nu=-\infty}^\infty J_{N\nu + p}(x)$ where  arise in many physical systems and defined in closed form by the Sung series. For example, when N = 3: $ \sum_{\nu=-\infty}^\infty J_{3\nu+p}(x) = \frac{1}{3}\left[1+2\cos{(x\sqrt{3}/2-2\pi p/3)}\right] $. More generally, the Sung series and the alternating Sung series are written as:

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: $$e^{iz \cos \phi} = \sum_{n=-\infty}^\infty i^n J_n(z) e^{in\phi}$$ and $$e^{\pm iz \sin \phi} = J_0(z)+2\sum_{n=1}^\infty J_{2n}(z) \cos(2n\phi) \pm 2i \sum_{n=0}^\infty J_{2n+1}(z)\sin((2n+1)\phi)$$ which is used to expand 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 $$f(z)=a_0^\nu J_\nu (z)+ 2 \cdot \sum_{k=1}^\infty a_k^\nu J_{\nu+k}(z)$$ is called Neumann expansion of $α$. The coefficients for $K_{1/3}$ have the explicit form $$a_k^0=\frac{1}{2 \pi i} \int_{|z|=c} f(z) O_k(z) \,dz$$ where $γ$ is Neumann's polynomial.

Selected functions admit the special representation $$f(z)=\sum_{k=0}^\infty a_k^\nu J_{\nu+2k}(z)$$ with $$a_k^\nu=2(\nu+2k) \int_0^\infty f(z) \frac{J_{\nu+2k}(z)}z \,dz$$ due to the orthogonality relation $$\int_0^\infty J_\alpha(z) J_\beta(z) \frac {dz} z= \frac 2 \pi \frac{\sin\left(\frac \pi 2 (\alpha-\beta) \right)}{\alpha^2 -\beta^2}$$

More generally, if $α$ has a branch-point near the origin of such a nature that $$f(z)= \sum_{k=0} a_k J_{\nu+k}(z)$$ then $$\mathcal{L}\left\{\sum_{k=0} a_k J_{\nu+k}\right\}(s)=\frac{1}{\sqrt{1+s^2}}\sum_{k=0}\frac{a_k}{\left(s+\sqrt{1+s^2} \right) ^{\nu+k}}$$ or $$\sum_{k=0} a_k \xi^{\nu+k}= \frac{1+\xi^2}{2\xi} \mathcal{L}\{f \} \left( \frac{1-\xi^2}{2\xi} \right)$$ where $$\mathcal{L}\{f \}$$ is the Laplace transform of $z$.

Another way to define the Bessel functions is the Poisson representation formula and the Mehler-Sonine formula: $$\begin{align} J_\nu(z) &= \frac{\left(\frac{z}{2}\right)^\nu}{\Gamma\left(\nu +\frac{1}{2}\right)\sqrt{\pi}} \int_{-1}^1 e^{izs}\left(1-s^2\right)^{\nu-\frac{1}{2}} \,ds \\[5px] &=\frac 2{{\left(\frac{z}{2}\right)}^\nu\cdot \sqrt{\pi} \cdot \Gamma\left(\frac{1}{2}-\nu\right)} \int_1^\infty \frac{\sin zu}{\left(u^2-1 \right )^{\nu+\frac 1 2}} \,du \end{align}$$ where $K_{2/3}$ and $j_{n}$. This formula is useful especially when working with Fourier transforms.

Because Bessel's equation becomes Hermitian (self-adjoint) if it is divided by $z$, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions. In particular, it follows that: $$\int_0^1 x J_\alpha\left(x u_{\alpha,m}\right) J_\alpha\left(x u_{\alpha,n}\right) \,dx = \frac{\delta_{m,n}}{2} \left[J_{\alpha+1} \left(u_{\alpha,m}\right)\right]^2 = \frac{\delta_{m,n}}{2} \left[J_{\alpha}'\left(u_{\alpha,m}\right)\right]^2$$ where $y_{n}$, $j_{n}(z)$ is the Kronecker delta, and $n = 0.5$ is the $J_{n}$th zero of $−2 − 2i$. 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 $2 + 2i$ for fixed $f$ and varying $O_{k}$.

An analogous relationship for the spherical Bessel functions follows immediately: $$\int_0^1 x^2 j_\alpha\left(x u_{\alpha,m}\right) j_\alpha\left(x u_{\alpha,n}\right) \,dx = \frac{\delta_{m,n}}{2} \left[j_{\alpha+1}\left(u_{\alpha,m}\right)\right]^2$$

If one defines a boxcar function of $f$ that depends on a small parameter $f$ as: $$f_\varepsilon(x)=\varepsilon \operatorname{rect}\left(\frac{x-1}\varepsilon\right)$$ (where $y_{n}(z)$ is the rectangle function) then the Hankel transform of it (of any given order $n = 0.5$), $−2 − 2i$, approaches $2 + 2i$ as $x$ approaches zero, for any given $m$. Conversely, the Hankel transform (of the same order) of $j_{n}(x)$ is $n = 0, 1, 2$: $$\int_0^\infty k J_\alpha(kx) g_\varepsilon(k) \,dk = f_\varepsilon(x)$$ which is zero everywhere except near 1. As $α$ approaches zero, the right-hand side approaches $y_{n}(x)$, where $m$ is the Dirac delta function. This admits the limit (in the distributional sense): $$\int_0^\infty k J_\alpha(kx) J_\alpha(k) \,dk = \delta(x-1)$$

A change of variables then yields the closure equation: $$\int_0^\infty x J_\alpha(ux) J_\alpha(vx) \,dx = \frac{1}{u} \delta(u - v)$$ for $n = 0, 1, 2$. The Hankel transform can express a fairly arbitrary function as an integral of Bessel functions of different scales. For the spherical Bessel functions the orthogonality relation is: $$\int_0^\infty x^2 j_\alpha(ux) j_\alpha(vx) \,dx = \frac{\pi}{2uv} \delta(u - v)$$ for $j_{0}(x)$.

Another important property of Bessel's equations, which follows from Abel's identity, involves the Wronskian of the solutions: $$A_\alpha(x) \frac{dB_\alpha}{dx} - \frac{dA_\alpha}{dx} B_\alpha(x) = \frac{C_\alpha}{x}$$ where $x$ and $ε$ are any two solutions of Bessel's equation, and $ε$ is a constant independent of $k$ (which depends on α and on the particular Bessel functions considered). In particular, $$J_\alpha(x) \frac{dY_\alpha}{dx} - \frac{dJ_\alpha}{dx} Y_\alpha(x) = \frac{2}{\pi x}$$ and $$I_\alpha(x) \frac{dK_\alpha}{dx} - \frac{dI_\alpha}{dx} K_\alpha(x) = -\frac{1}{x},$$ for $J_{n}(x), Y_{n}(x)$.

For $j_{n}(x), y_{n}(x)$, the even entire function of genus 1, $h(1) n$, has only real zeros. Let $$0<j_{\alpha,1}<j_{\alpha,2}<\cdots<j_{\alpha,n}<\cdots$$ be all its positive zeros, then $$J_{\alpha}(z)=\frac{\left(\frac{z}{2}\right)^\alpha}{\Gamma(\alpha+1)}\prod_{n=1}^{\infty}\left(1-\frac{z^2}{j_{\alpha,n}^2}\right)$$

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

Recurrence relations
The functions $ε$, $δ$, $h(2) n$, and $n = 0, ±1, ±2, ...$ all satisfy the recurrence relations $$\frac{2\alpha}{x} Z_\alpha(x) = Z_{\alpha-1}(x) + Z_{\alpha+1}(x)$$ and $$ 2\frac{dZ_\alpha (x)}{dx} = Z_{\alpha-1}(x) - Z_{\alpha+1}(x),$$ where $A_{α}$ denotes $B_{α}$, $C_{α}$, $h(1) n$, or $h(2) n$. 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 $$\begin{align} \left( \frac{1}{x} \frac{d}{dx} \right)^m \left[ x^\alpha Z_\alpha (x) \right] &= x^{\alpha - m} Z_{\alpha - m} (x), \\ \left( \frac{1}{x} \frac{d}{dx} \right)^m \left[ \frac{Z_\alpha (x)}{x^\alpha} \right] &= (-1)^m \frac{Z_{\alpha + m} (x)}{x^{\alpha + m}}. \end{align}$$

Modified Bessel functions follow similar relations: $$e^{\left(\frac{x}{2}\right)\left(t+\frac{1}{t}\right)} = \sum_{n=-\infty}^\infty I_n(x) t^n$$ and $$e^{z \cos \theta} = I_0(z) + 2\sum_{n=1}^\infty I_n(z) \cos n\theta$$ and $$ \frac{1}{2\pi} \int_0^{2\pi} e^{z \cos (m\theta) + y \cos \theta} d\theta = I_0(z)I_0(y) + 2\sum_{n=1}^\infty I_n(z)I_{mn}(y).$$

The recurrence relation reads $$\begin{align} C_{\alpha-1}(x) - C_{\alpha+1}(x) &= \frac{2\alpha}{x} C_\alpha(x), \\[1ex] C_{\alpha-1}(x) + C_{\alpha+1}(x) &= 2\frac{d}{dx}C_\alpha(x), \end{align}$$ where $x$ denotes $J_{α}$ or $h(1) n(x)$. These recurrence relations are useful for discrete diffusion problems.

Transcendence
In 1929, Carl Ludwig Siegel proved that $n = -0.5$, $−2 − 2i$, and the logarithmic derivative $2 + 2i$ are transcendental numbers when ν is rational and x is algebraic and nonzero. The same proof also implies that $h(2) n(x)$ is transcendental under the same assumptions.

Multiplication theorem
The Bessel functions obey a multiplication theorem $$\lambda^{-\nu} J_\nu(\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{\left(1 - \lambda^2\right)z}{2}\right)^n J_{\nu+n}(z),$$ where $Y_{α}$ and $Z$ may be taken as arbitrary complex numbers. For $n = −0.5$, the above expression also holds if $J$ is replaced by $Y$. The analogous identities for modified Bessel functions and $−2 − 2i$ are $$\lambda^{-\nu} I_\nu(\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{\left(\lambda^2 - 1\right)z}{2}\right)^n I_{\nu+n}(z)$$ and $$\lambda^{-\nu} K_\nu(\lambda z) = \sum_{n=0}^\infty \frac{(-1)^n}{n!} \left(\frac{\left(\lambda^2 - 1\right)z}{2}\right)^n K_{\nu+n}(z).$$

Bourget's hypothesis
Bessel himself originally proved that for nonnegative integers $C_{α}$, the equation $2 + 2i$ has an infinite number of solutions in $I_{α}$. When the functions $h(2) n$ are plotted on the same graph, though, none of the zeros seem to coincide for different values of $λ$ except for the zero at $j_{0}(x) = sin x⁄x$. 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 $y_{0}(x) = −cos x⁄x$ and $S_{n}$, the functions $C_{n}$ and $ξ_{n}$ have no common zeros other than the one at $ζ_{n}$. The hypothesis was proved by Carl Ludwig Siegel in 1929.

Transcendence
Siegel proved in 1929 that when ν is rational, all nonzero roots of $z ≫ |α^{2} − 1⁄4|$ and $arg z$ are transcendental, as are all the roots of $|z|^{−1}$. It is also known that all roots of the higher derivatives $$J_\nu^{(n)}(x)$$ for $α = 1⁄2$ are transcendental, except for the special values $$J_1^{(3)}(\pm\sqrt3) = 0$$ and $$J_0^{(4)}(\pm\sqrt3) = 0$$.

Numerical approaches
For numerical studies about the zeros of the Bessel function, see, and.

Numerical values
The first zero in J0 (i.e, j0,1, j0,2 and j0,3) occurs at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.

title: Bessel function


text:
Backpacking

Backpacking may refer to:

 * Backpacking (travel), low-cost, independent, international travel
 * Backpacking (hiking), trekking and camping overnight in the wilderness
 * Ultralight backpacking, a style of wilderness backpacking with an emphasis on carrying as little as possible

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Backpacking


text:
Brahui language

Brahui (also known as Brahvi or Brohi) is a Dravidian language spoken by the Brahui people who are mainly found in the central Balochistan Province of Pakistan, with smaller communities of speakers scattered in parts of Iranian Baluchestan, Afghanistan, and Turkmenistan (around Merv) and by expatriate Brahui communities in Iraq, Qatar, and the United Arab Emirates. It is isolated from the nearest Dravidian-speaking neighbouring population of South India by a distance of more than 1500 km. The Kalat, Khuzdar, Mastung, Quetta, Bolan, Nasirabad, Nushki, and Kharan districts of Balochistan Province are predominantly Brahui-speaking.

Brahui is the only Dravidian language written in Perso-Arabic script, and is also the only Dravidian language written in Latin script.

Distribution
Brahui is spoken in the central part of Pakistani Balochistan, mainly in Kalat, Khuzdar and Mastung districts, but also in smaller numbers in neighboring districts, as well as in Afghanistan which borders Pakistani Balochistan; however, many members of the ethnic group no longer speak Brahui. There are also an unknown (but very small) number of expatriate Brahuis in the Arab States of the Persian Gulf, and Turkmenistan.

History
There is no consensus as to whether Brahui is a relatively recent language introduced into Balochistan or the remnant of a formerly more widespread Dravidian language family. According to Josef Elfenbein (1989), the most common theory is that the Brahui were part of a Dravidian migration into north-western parts of the Indian subcontinent in the 3rd millennium BC, but unlike other Dravidians who migrated to the south, they remained in Sarawan and Jahlawan since before 2000 BC. However, some other scholars see it as a recent migrant language to its present region. They postulate that Brahui could only have migrated to Balochistan from central India after 1000 AD. This is contradicted by genetic evidence that shows the Brahui population to be indistinguishable from neighbouring Balochi speakers, and genetically distant from central Dravidian speakers. The main Iranian contributor to Brahui vocabulary, Balochi, is a Northwestern Iranian language, and moved to the area from the west only around 1000 AD. One scholar places the migration as late as the 13th or 14th century. The Brahui lexicon is believed to be of: 35% Perso-Arabic origin, 20% Balochi origin, 20% Indo-Aryan origin, 15% Dravidian origin, and 10% unknown origin.

Franklin Southworth (2012) proposes that Brahui is not a Dravidian language, but can be linked with the remaining Dravidian languages and Elamite to form the "Zagrosian family," which originated in Southwest Asia (southern Iran) and was widely distributed in South Asia and parts of eastern West Asia before the Indo-Aryan migration.

Dialects
There are no important dialectal differences. Jhalawani (southern, centered on Khuzdar) and Sarawani (northern, centered on Kalat) dialects are distinguished by the pronunciation of *h, which is retained only in the north (Elfenbein 1997). Brahui has been influenced by the Iranian languages spoken in the area, including Persian, Balochi and Pashto.

Phonology
Brahui vowels show a partial length distinction between long and diphthongs  and short. Brahui does not have short /e, o/ due to influence from neighbouring Indo-Aryan and Iranic languages, the PD short *e was replaced by a, ē and i, and ∗o by ō, u and a in root syllables.

Brahui consonants show patterns of retroflexion but lack the aspiration distinctions found in surrounding languages and include several fricatives such as the voiceless lateral fricative, a sound not otherwise found in the region. Consonants are also very similar to those of Balochi, but Brahui has more fricatives and nasals (Elfenbein 1993).

 * of north corresponds to a glottal stop of south initially and intervocalically. Before a C in word-final position it is lost. Non-phonemic glottal stop before word-initial vowels, e.g. hust (N), ʔust (S) 'heart'.
 * and vary freely in many cases; contrast is limited to two or three items. Conditions for the emergence of  are not clear.
 * does not occur word-initially. →  before  in northern Brahui (Elfenbein 1998: 394), e.g. xūrt → xūṛt 'tiny'.
 * The consonants freely alternate with aspirated counterparts in the northeast. Aspirated stops word-initially occur in loanwords in the south, where they freely vary with unaspirated stops.
 * occurs before velar stops.
 * Brahui preserves the PD laryngeal * as in some words e.g. PD. *caH- ~ *ceH- > Br. kah-.

Stress
Stress in Brahui follows a quantity-based pattern, occurring either on the first long vowel or diphthong, or on the first syllable if all vowels are short.

Perso-Arabic script

Brahui is the only Dravidian language which is not known to have been written in a Brahmi-based script; instead, it has been written in the Arabic script since the second half of the 20th century. In Pakistan, an Urdu based Nastaʿlīq script is used in writing:

Latin script
More recently, a Roman-based orthography named Brolikva (an abbreviation of Brahui Roman Likvar) was developed by the Brahui Language Board of the University of Balochistan in Quetta and adopted by the newspaper Talár.

Below is the new promoted Bráhuí Báşágal Brolikva orthography:

The letters with diacritics are the long vowels, post-alveolar and retroflex consonants, the voiced velar fricative and the voiceless lateral fricative.

English
All human beings are born free and equal in dignity and rights. They are endowed with reason and conscience and should act towards one another in a spirit of brotherhood.

Latin script
Muccá insáńk ájo o izzat ná rid aŧ barebar vadí massuno. Ofte puhí o dalíl raseńgáne. andáde ofte asi elo ton ílumí e vaddifoí e.

Endangerment
According to a 2009 UNESCO report, Brahui is one of the 27 languages of Pakistan that are facing the danger of extinction. It was classified as "unsafe", the least endangered level out of the five levels of concern (Unsafe, Definitely Endangered, Severely Endangered, Critically Endangered and Extinct). This status has since been renamed to "vulnerable".

Publications
Talár is the first daily newspaper in the Brahui language. It uses the new Roman orthography and is "an attempt to standardize and develop [the] Brahui language to meet the requirements of modern political, social and scientific discourse."

title: Brahui language


text:
Berkeley DB

Berkeley DB (BDB) is an embedded database software library for key/value data, historically significant in open-source software. Berkeley DB is written in C with API bindings for many other programming languages. BDB stores arbitrary key/data pairs as byte arrays and supports multiple data items for a single key. Berkeley DB is not a relational database, although it has database features including database transactions, multiversion concurrency control and write-ahead logging. BDB runs on a wide variety of operating systems, including most Unix-like and Windows systems, and real-time operating systems.

BDB was commercially supported and developed by Sleepycat Software from 1996 to 2006. Sleepycat Software was acquired by Oracle Corporation in February 2006, who continued to develop and sell the C Berkeley DB library. In 2013 Oracle re-licensed BDB under the AGPL license and released new versions until May 2020. Bloomberg L.P. continues to develop a fork of the 2013 version of BDB within their Comdb2 database, under the original Sleepycat permissive license.

Origin
Berkeley DB originated at the University of California, Berkeley as part of BSD, Berkeley's version of the Unix operating system. After 4.3BSD (1986), the BSD developers attempted to remove or replace all code originating in the original AT&T Unix from which BSD was derived. In doing so, they needed to rewrite the Unix database package. Seltzer and Yigit created a new database, unencumbered by any AT&T patents: an on-disk hash table that outperformed the existing dbm libraries. Berkeley DB itself was first released in 1991 and later included with 4.4BSD. In 1996 Netscape requested that the authors of Berkeley DB improve and extend the library, then at version 1.86, to suit Netscape's requirements for an LDAP server and for use in the Netscape browser. That request led to the creation of Sleepycat Software. This company was acquired by Oracle Corporation in February 2006.

Berkeley DB 1.x releases focused on managing key/value data storage and are referred to as "Data Store" (DS). The 2.x releases added a locking system enabling concurrent access to data. This is what is known as "Concurrent Data Store" (CDS). The 3.x releases added a logging system for transactions and recovery, called "Transactional Data Store" (TDS). The 4.x releases added the ability to replicate log records and create a distributed highly available single-master multi-replica database. This is called the "High Availability" (HA) feature set. Berkeley DB's evolution has sometimes led to minor API changes or log format changes, but very rarely have database formats changed. Berkeley DB HA supports online upgrades from one version to the next by maintaining the ability to read and apply the prior release's log records.

Starting with the 6.0.21 (Oracle 12c) release, all Berkeley DB products are licensed under the GNU AGPL. Previously, Berkeley DB was redistributed under the 4-clause BSD license (before version 2.0), and the Sleepycat Public License, which is an OSI-approved open-source license as well as an FSF-approved free software license. The product ships with complete source code, build script, test suite, and documentation. The comprehensive feature along with the licensing terms have led to its use in a multitude of free and open-source software. Those who do not wish to abide by the terms of the GNU AGPL, or use an older version with the Sleepycat Public License, have the option of purchasing another proprietary license for redistribution from Oracle Corporation. This technique is called dual licensing.

Berkeley DB includes compatibility interfaces for some historic Unix database libraries: dbm, ndbm and hsearch (a System V and POSIX library for creating in-memory hash tables).

Architecture
Berkeley DB has an architecture notably simpler than relational database management systems. Like SQLite and LMDB, it is not based on a server/client model, and does not provide support for network access – programs access the database using in-process API calls. Oracle added support for SQL in 11g R2 release based on the popular SQLite API by including a version of SQLite in Berkeley DB (it uses Berkeley DB for storage).

A program accessing the database is free to decide how the data is to be stored in a record. Berkeley DB puts no constraints on the record's data. The record and its key can both be up to four gigabytes long.

Berkeley DB supports database features such as ACID transactions, fine-grained locking, hot backups and replication.

Oracle Corporation use of name "Berkeley DB"
The name "Berkeley DB" is used by Oracle Corporation for three different products, only one of which is BDB:

 * 1) Berkeley DB, the C database library that is the subject of this article
 * 2) Berkeley DB Java Edition, a pure Java library whose design is modelled after the C library but is otherwise unrelated
 * 3) Berkeley DB XML, a C++ program that supports XQuery, and which includes a legacy version of the C database library

Open-source programs still using Berkeley DB
BDB was once very widespread, but usage dropped steeply from 2013 (see licensing section). Notable software that still uses Berkeley DB for data storage include:
 * Bogofilter – A free/open-source spam filter that saves its wordlists using Berkeley DB by default.
 * Citadel/UX – A collaborative software (messaging and groupware) that is directly descended from the Citadel family of programs, which became popular in the 1980s and 1990s as a bulletin board system platform.
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title: Berkeley DB


text:
Talk:Basque language

Steppe
Has anyone ever tried to look for clues that could point to Basque being of Early Bronze-age European steppe (R1b-M269, WSH) origin?
 * Hello. Wikipedia article talk pages are reserved for discussion of ways to improve the article, not for general conversation about their subject. Please see WP:NOTFORUM. Largoplazo (talk) 09:44, 2 August 2022 (UTC)

Hand of Irulegi - Earliest Basque inscription
An interesting discovery was made in January 2022, but only reported today, stating that the earliest inscription in Basque has been dated to the first century B.C., and speculates on potential connections to the Iberian language. Culloty82 (talk) 20:13, 14 November 2022 (UTC) Culloty82 (talk) 20:13, 14 November 2022 (UTC)

 * Hm yes, it's an important find but I think, given we can't read most of it and only have a single word that admittedly looks a lot like a Basque word, it's a bit premature to call this a Basque text. The state we're at, for all we know it may be a Basque loanword in Iberian (think of how the word benedictus has made it into languages like Irish in the form of beannacht in the wake of Christianity. Akerbeltz (talk) 22:46, 14 November 2022 (UTC)

Neolithic
Can any linguist bet his head that the ancestor of Basque language was present in Western Europe before Late Neolithic? "Late Neolithic and later are IE times" is not a linguistic arguement. — Preceding unsigned comment added by 85.249.44.2 (talk) 06:53, 30 December 2022 (UTC)
 * Is this related to anything that is currently mentioned in the article or that needs to be added to it? Otherwise, please see WP:NOTFORUM. –Austronesier (talk) 20:27, 31 December 2022 (UTC)

Video is messed up
The "Euskara iparraldean" video in the "Demographics" section loses sound about a minute in. It might be a poor recording or it might be fixable, but that's beyond my humble skill. Anyone who knows anything about these sort of things should take a look at it. Gamle Kvitrafn (talk) 18:47, 8 April 2023 (UTC)

 * It plays fine for me all the way through, maybe you were on a slow connection and had a buffering issue? Akerbeltz (talk) 11:16, 1 June 2023 (UTC)
 * It seems ok to me. In the last minute the sound does drop down somewhat and sounds like it's off in the distance compared to the first part of the tape - but still totally audible. It may be that if the sound on the device you're using is not strong that bit may be problematic. DeCausa (talk) 11:45, 1 June 2023 (UTC)

Map not supported by source?
The map made and added by user User:Enekorga is very helpful indeed for an overview of where Basque is spoken and where the strongholds of the language are. For all the municipalities on the Spanish side, everything seems entirely accurate. Enekorga provides the sources, and they detail the how many speak Basque per municipality. Big thanks to Enekorga for creating a map to show all data! My concern is with Iparralde, the French side. The source given was a telephone survey that did give indications of language use for Lapurdi, Behe Nafarroa and Zuberroa - but despite going through it in detail, I can find no data per municipality nor even any indication that such was collected. As the map is so useful, I hope we can keep it. That would require a source of the data from Iparralde actually containing the data used. If Enekorga or anyone else know of such a source and can add it, that's great. In the absence of such a source, the map might be seen to violate WP:OR. Rather than tagging it, not to mention removing it (neither of which I want), I hope we can simply find and add the proper source. (Apart from Enekorga, also tagging User:Akerbeltz, User:Austronesier, User:Ksarasola and User:DeCausa (active contributors) as it's my wish to find a source to keep the very useful map in the article). Jeppiz (talk) 12:36, 10 June 2023 (UTC)

 * My guess is that Enekorga has access to the raw data. I have a copy of the 2nd Linguistic Survey of the Basque Country (I think the last one where they published both a short and a long version) and it has down-to-the-district data. It may just be that Enekorga wasn't entirely sure how to reference a data source that isn't published? Not sure I would know how to reference that myself :) Akerbeltz (talk) 16:32, 10 June 2023 (UTC)
 * Surely, an unpublished source shouldn't be referenced. It won't be WP:RS. DeCausa (talk) 18:17, 10 June 2023 (UTC)
 * Akerbeltz, it's possible Enekorga has raw data, but as DeCausa already wrote, that would not meet WP:RS. The data should be verifiable, and at the moment it is not. I really hope Enekorga (or anyone else) can provide the data so we can keep the map. There's no hurry, but in the absence of any source/data in the next few days, the correct procedure will be remove the map pending data to support it. Jeppiz (talk) 20:13, 10 June 2023 (UTC)
 * PS I left a comment on Enekorga's page on Basque wikipedia about this discussion Jeppiz (talk) 20:34, 10 June 2023 (UTC)
 * Hi everybody,
 * the data for Iparralde does come from the cited source. The precise document I used was actually online, but I guess it must not be accessible anymore. Still, there is this tool created by Udalbiltza called Euskal Herriari Begira which takes back the same data from the Office public de la langue basque and is actually available online as of today, in case that would suit you.
 * The website does not show exclusively that data, there is a lot more about various topics. The one I am referring to is named Euskararen ezagutza tasa (%) and is found under Tokiko nortasunaren bizitasuna. However, I must warn you that the scale used for the displayed map is somewhat different to the one used on my map, since I decided to use the same scale of the precedent map. Still, the numeric data, which I think is the thing that you are more interested in, can also be found on the website.
 * I hope this helps. Enekorga (talk) 18:41, 12 June 2023 (UTC)
 * Milesker ainitz, that is very helpful. Thanks again for your great contributions to Wikipedia! Jeppiz (talk) 00:44, 13 June 2023 (UTC)

Shameless editors
You don't care about the truth. A native Gipuzkera Basque speaker, added a LINK with evidence proving tt is extremely common in both Gipuzkoa and Bizkaia and that the DD sound is very much used there and you choose to silence an actual euskalduna. SHAME ON YOU. 79.116.130.131 (talk) 20:30, 11 June 2023 (UTC)

 * here the tt sound all over Bizkaia and Gipuzkoa https://www.euskaltzaindia.eus/index.php?&option=com_ehha&view=frontpage&Itemid=466&lang=eu&mapa=2481 79.116.130.131 (talk) 20:34, 11 June 2023 (UTC)
 * Map not supported by source? how dare you??? That map was created by the Euskaltzaindia! 79.116.130.131 (talk) 20:39, 11 June 2023 (UTC)

 * On the Internet, nobody knows you're a dog. Have you heard that meme before? In Wikipedia we have a very important policy called No original research. That means we must not take account of the personal knowledge of anyone who tries to edit Wikipedia because it is impossible to know whether or not they have real expertise. Instead, we only accept verifiable information. This is explained in this policy: WP:V. The most important aspect of this is that what is written here should be supported by reliable sources which are defined here: WP:RS and these should be cited. You are trying to replace information that is cited to reliable sources with your own knowledge and alternative sources. That is why you have been reverted. If you want to change what is written in the article (and you can do that) you need to persuade other people editing this page that:
 * existing information and the cited sources are incorrect
 * you have reliable sources that make it clear it is incorrect
 * the sources you cite are more reliable than the existing sources
 * This is a collaborative project an changes are only made with consensus agreement. The onus is on the person wanting to make the change to persuade everyone else. You will find that editors here are happy to change incorrect information if you present the evidence on this talk page. What they will not do is change the article just because you say it should be changed. I hope this leads you to understand Wikipedia better. Please therefore compare the sources you are putting forward and explain why they should be followed rather than the existing sources. Please avoid arguments that are based on your personal knowledge. Please also do not insult other people editing here. We have a very strict policy of No personal attacks and if you continue to attack other editors you will be blocked. DeCausa (talk) 20:49, 11 June 2023 (UTC)
 * It's a euskaltzandia page, with a map in which the word "aita" pronounce either atta or atte is ALL OVER GIPUZKOA AND BIZKAIA. the nerve you have! 79.116.130.131 (talk) 20:54, 11 June 2023 (UTC)
 * I will always defend my language, you're the ones acting like you know better and ignoring information, do as you wish, you've proven how little you care about the Basque language already. Your ego is more importan than actual facts. I'd certainty be ashamed. 79.116.130.131 (talk) 20:56, 11 June 2023 (UTC)
 * The tldr version of my previous post is no one cares what you think unless you are prepared to understand and comply with Wikipedia policy. DeCausa (talk) 20:58, 11 June 2023 (UTC)
 * It's not about what I think. I most definitely provided evidence that you're willingly choosing to ignore. 79.116.130.131 (talk) 21:02, 11 June 2023 (UTC)
 * Imagine the frustration you'd feel if someone were spreading misinformation about your language, and completely ignoring evidence you're presenting. Evidence from an official academic language regulatory institution. Of course I'm upset because it's absolutely outrageous. 79.116.130.131 (talk) 21:00, 11 June 2023 (UTC)
 * Stop lying about it too! my argument is based on the Euskaltzaindia! Not simply "personal experience"! 79.116.130.131 (talk) 21:05, 11 June 2023 (UTC)
 * You're blocking a legit correction backed up by the Euskaltzaindia and still think you are in the right. That level of ego is beyond my comprehension. https://www.euskaltzaindia.eus/index.php?&option=com_ehha&view=frontpage&Itemid=466&lang=eu&mapa=2481 2A0C:5A80:1307:EA00:191:668C:FF76:59AA (talk) 21:09, 24 November 2023 (UTC)

Mesedez, can we calm this down? :) So let's see what the article says  In the Gipuzkoan and Biscayan dialects tt is often pronounced the same as tx, that is, as [tʃ], and dd does not exist.

So to begin with, it does not state that tt does not exist, it merely states that where tt may be expected, it is often realized as tx. It also states that (according to de Rijk), dd does not exist in Gipuzkoan and Biscayan.

Formalities first, the Euskaltzaindia uses non-standard IPA dd instead of [c] and tt instead of [ɟ] (NB this is not criticism, I'm merely clarifying so we're all on the same sheet, because [atta] in pure IPA means something else). Also, to locate items with a particular letter sequence, wildcards can be used in the Zer field e.g. *dd* to find words containing dd.

Let's do the easy one first: dd. This is not easy to pin down in the Atlas. We might expect it in onddo 'mushroom' but here we find the Euskaltzaindia data backs up de Rijk, onddo is realised as ondo in G and B. I can't find any other items with dd which would contradict that but I haven't looked at all.

For tt, the picture is more complex, there are clearly things going on that govern where tt appears as [c] in G and B which are beyond my ken. Yes, there is the aita map but this data shows that tt is realized as [ɟ] in aita in parts of G and B, but it does NOT state that tt is never realized as tx.

The main problem is that by and large, tt and dd are the result of palatalising t and d and while in some words these forms are the citation forms (e.g. Maddalen, onddo), such words are rare (especially with tt) and as a result it's really hard to pin down where an organic (for the want of a better word) tt is realised as [ɟ] and there it is tx.

Where we're talking active palatalization i.e. a word with underlying t being realised as tt or tx, the picture is very mixed. The inflected verb ditu in G and B largely is realized as dittu but note also the tx in part of B. At the same time, tortola > ttorttola does NOT appear to appear in G and B at all, nor with tanta/ttantta/txanta Ignoring the fact that what I just did may be argued as being OR, I think it's safe to say that the source is clearly correct regarding dd in G and B. Regarding tt, while the picture is much more complicated, it would be difficult to argue using the Atlas data that "tt realized as tx" is incorrect in a lot of cases. But I stress again that de Rijk doesn't say tt is absent in G and B, just that it CAN often be heard as tx. Akerbeltz (talk) 12:37, 13 June 2023 (UTC)

 * Ttantto is extremely common in gipuzkera Basque and so is atta, attona, ze ittezu (egiten duzu). No shame at all... 2A0C:5A80:1307:EA00:191:668C:FF76:59AA (talk) 21:06, 24 November 2023 (UTC)

2021 Sociolinguistic Survey results
I cannot read Basque, but it appears 680,629 (36.2%) of population in the Basque Autonomous Community claim to speak Basque, with a further 18% stating they are passive speakers:

https://www.argia.eus/fitxategiak/ckfinder/files/VII_INKESTA_SOZIOLINGUISTIKOA_laburpena.pdf Culloty82 (talk) 12:41, 9 November 2023 (UTC)

Tt sound
As a native gipuzkera speaker, the VAST majority of gipuzkera subdialects do have a tt sound very clearly distinguished from tx and dd also does exist. It's an absolute disgrace you will not allow me and many others to correct that. Shame on all of you. 2A0C:5A80:1307:EA00:191:668C:FF76:59AA (talk) 21:04, 24 November 2023 (UTC)

 * It's an absolute disgrace that Wikipedia isn't left to being a collection of claims that thousands of anonymous users make, all accepted just because they say so? If you're correct and it's documented in a reliable source, then you need only provide such a source. If there isn't any, then it's going to have to remain one of the many true things that aren't covered here because the alternative would be also to allow millions of falsehoods. Largoplazo (talk) 23:53, 24 November 2023 (UTC)

Steppe hypothesis
We now know that the Basques have one of the highest Steppe migrant Y-DNA - almost 100%. Did anyone address the Basque language in this regard? Did anyone try to analize it on the possibility of it being a Steppe language, searching for Steppe traits in it, like Indo-European has been studied? --95.24.65.132 (talk) 15:26, 30 November 2023 (UTC)

 * Do you have any source backing that up? Who is "we" exactly? -- VerusPhoebusApollo (talk) 14:52, 26 April 2024 (UTC)

2021 language survey data
Data from the most recent language survey can be found here, if anyone is motivated to add it in: https://basquetribune.com/the-basque-language-gains-speakers-but-no-surge-in-usage/ 2600:1702:6D0:5160:ACFA:FB1C:2E00:E86F (talk) 01:39, 30 December 2023 (UTC)

First mention of the noun Euskara?
It seems that the first mention of Euskara is possibly from Marzuq ibn Uskara, a basque from Medieval Iberia, can anyone check this out? https://puebladecastro.blogspot.com/2018/10/archuza-de-bahlul-ibn-marzuq-el.html Yefa36 (talk) 07:40, 20 March 2024 (UTC)
 * It's claimed in that WP:BLOG that this work (without page number) is the source for that statement: Cañada Juste, Alberto (1980), Los Banu Qasi (714-924), Príncipe de Viana, Año Nº 41, Nº 158-159, págs. 5-96, ISSN 0032-8472. DeCausa (talk) 08:10, 20 March 2024 (UTC)

IE theory variant
The article begins non-scientifically asserting IE coming from Anatolia at some point in Bronze Age, something totally unproven. Kostandisloukos (talk) 09:42, 30 April 2024 (UTC)

 * There is no mention of Anatolia anywhere on the page. Try your luck arguing that IE didn't move into Europe from further east on the Indo-European languages page and should you succeed there (you won't), come back here and we'll look into it. Akerbeltz (talk) 15:27, 30 April 2024 (UTC)
 * (I had already removed the funky passages in question.) Remsense  诉  15:28, 30 April 2024 (UTC)

title: Talk:Basque language


text:
Talk:Books of the Bible

WikiProject
Based on a suggestion in Pages needing attention, I have started the skeleton of a WikiProject to try to cut down on the overlap between the various presentations of the canon. I think that a lot of people working here will want input on this. Feel free! Mpolo 13:24, Sep 14, 2004 (UTC)

Major and Minor. How about Former and Latter
Though there is no distinction between Major nad Minor prophets in the Tanakh, the "N" (i.e. Neviim) are divided into two categories. Rishonim and Acharonim (Priori and Posteriori). Joshua, Judges, Samuel 1-2, Kings 1-2 are considered Neviim Rishonim, whereas the rest are considered Latter Prophets. Perhaps that ought to be included. I understand that would be hard to fit into the graph, but at least in text...

Other Bibles
The OT list could also include lists for the Armenian Apostolic, Ethiopian and Syriac Bibles. Likewise the NT list could add the Syriac NT. There are 2 Catholic orders: Vulgate (Maccabees after Malachi) & Septuagint (Maccabees after Malachi).

Collated Jewish books in the chart
I collated Jewish books in the chart. Previously the Ketuvim books were separated out from the rest of the chart and added to the bottom of the chart as a separate list. This made it appear (without close inspection and mental collation) that there was a massive difference between the Jewish Tanakh and the Christian Old Testament, which, of course, there is not. The Hebrew Bible and the Protestant Old Testament are even identical in content. This is made much more plain by the collation job done. I marked the Ketuvim books in bold, and I footnoted each one so that there would be no confusion with the Nevi'im books. I think this brings much more clarity to the chart, which before seemed rather confusing. -- Guðsþegn (talk) 23:36, 13 October 2012 (UTC)

Christian Books
Can someone please add a Christian (not Catholic) table I'm having trouble making one. 173.20.102.126 (talk) 15:41, 25 June 2014 (UTC)

Re: reversion of changes to book order
(Copied from my user talk page: I thought it might be more useful here.) Evensteven (talk) 17:55, 29 October 2015 (UTC)

You stated that the order of the books of the Bible (specifically the New Testament) varies based on the denomination. This may be true, but the statement in the article that the order you have is accepted by most Protestants, Catholics, and Greek Orthodox is not. You can see the Catholic order here. Just on the bookshelf of my office, the same order is used in the NIV, NRSV, CEB, KJV, NKJV, TEV, CEV, RSV, and New Oxford Annotated Bible. All of these versions are used regularly across Protestant denominations. In fact, this is the order found in both my Greek texts: "A Reader's Greek New Testament: 2nd Edition" by Goodrich and Lukaszewski and "The Greek New Testament: 4th Revised Edition" edited by Aland et. al. Every Bible Commentary in my office, including the New Interpreter's Bible and The Interpreter's Bible has them in this order; however, to be fair, they do rely on the NIV and NRSV (former) and RSV and KJV (latter). You can also find this order in Box 1.4 on page 10 of "The New Testament (4th ed)" by Bart D. Ehrman.

I would request one of two things happen. 1) The correct order as cited above be placed in the article or 2) Both your order and the order I cited above be placed in the article with the statement that this order is the one used by most Protestant denominations as well as Roman Catholics. I have been unable to confirm that the Greek Orthodox use this order.

In the interest of learning, would you tell me which denominations use the order you have, and where this can be found? I was unable to find it in any google search.

Pastor Melissa (talk) 16:20, 29 October 2015 (UTC)Pastor Melissa

 * You appear to be new to Wikipedia. Welcome! I assume you are IP 50.195.146.93 who made the edits I recently reverted.

 * I wasn't calling into question the accuracy of your changes. I was questioning the editing process. Edit summaries are fine for simple matters, but it's clear that the book order in the Bible tends to be more complex, and moreover, could be subject to partisan rivalries among editors. The article has been stable for some time, indicating its original state has been generally acceptable to the community of editors who watch it, representing an editing consensus (click or right-click on the blue word to bring up the editorial consensus policy). That consensus can change, but my action in reverting you was designed to draw attention to your edits so that sufficient attention could be drawn, and its desirability be weighed also by experienced editors here. The edit summaries could not explain your reasons or approach sufficiently for due consideration to be given (there just isn't room there). Now you have responded, quite perfectly I might add, and everyone can see your thoughts, which also appear to me to be quite reasonable.

 * The various Biblical versions don't just differ in order, of course, but in content (differences in accepted Biblical canon), and in grouping. Protestants eliminate several books (sometimes differently from each other too), often calling them Apochrypha, while Catholics and Orthodox take them as canonical (the Orthodox accept a couple more than Catholic). Western translations tend to stick with content and order as in the Vulgate, whereas Orthodox used the Greek of the Septuagint (I assume that's the basis for your "Greek" versions). Different orderings might be useful to different people for different reasons, depending on how far they are going into historical development of the canon and collections of original texts. In addition, while Protestant-oriented ordering may be natural to many westerners who speak English, there are also a great many English speakers worldwide of Catholic and Orthodox backgrounds, and all such use English Wikipedia.

 * I don't really oppose your changes. I hope I merely brought up a few things you may not yet have considered in approaching Wikipedia, asking that you give them what you consider to be proper weight. Then, if you still wish your changes to be made, it's ok with me if you want to revert my reversion. And now all the other editors here can also weigh in with any comments they want to bring, and respond to changes with a good idea of what's going on. Please be aware that we on Wikipedia often get editing from IP addresses (essentially anonymous) who leave behind a very few edits, all too often without merit or appreciation of things they haven't thought about. While yours did not have the earmarks of the less knowledgeable of those, I hope you now have also gained some insight into how valuable editorial review can be, and why we may pause to consider before going ahead with changes. Welcome again, and I hope we see you again in the Christianity pages. Evensteven (talk) 17:44, 29 October 2015 (UTC)

 * The section I actually wished to change was the New Testament ordering. The Hebrew Bible/Old Testament list on the page seemed to do a good job of showing the variety of possible orders. As such, no, the Greek texts I used are not based on the Septuagint. They are compiled from a variety of manuscripts and are the two texts used most often in Koine Greek classes in undergraduate and graduate studies.  The manuscripts most used in the one I prefer, edited by Aland et al, comes from papyri, uncials, miniscules, lectionaries, and several more.  I have never seen arguments before about the ordering of the New Testament, so discovering this problem on Wikipedia surprised me.  Admittedly, I do live in the US, but I have not been able to find any non-western sources that confirm a different order of the New Testament canon either.

 * Thank you for the insight into how editorial changes should be made. In the future, should I encounter something I would like to add or edit, should I come to a page like this before making a change? Pastor Melissa (talk) 18:25, 29 October 2015 (UTC)Pastor Melissa

 * Yes, this is the place for anything that needs the consideration of the editing community, and that isn't readily addressed in simple fashion by edit comments themselves. But in many small matters, the edit comment alone is sufficient. So it's often a judgment call where to go first, and often the talk page is used once it's clear that there's an actual question that needs resolving.

 * As for arguments on Wikipedia, I can't say I've seen one specifically about Bible book order, either Testament, but stuff that is associated with differences among the churches can be sensitive, and I was seeking only to get explicit confirmation that all sides were fine. But as for your changes being to the New Testament ordering, indeed I see it now, but somehow I was thinking OT at the time I reverted you. That was completely my own blunder, for which I apologize. If I had only seen that aright the first time, I would have let your edit pass without all this commentary, and I would not have considered the matter to be sensitive. For indeed, I know of no real differences in NT canon, or ordering. I've only ever heard of infrequent criticisms of an early Reformation leader or two regarding bits from a couple of the general epistles, but nothing that altered it in major ways. Anyway, I hope I haven't influenced you to become too shy. After all, two more WP policies talk about editing boldly, and how to respond when that doesn't work out the first time, so no one really needs to walk on eggshells either. Don't worry, you're doing great! Evensteven (talk) 19:07, 29 October 2015 (UTC)

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Abbreviations of the names of the books
Hi there,

i came here looking for those, and searching for "abbreviation" or "short" didn't make any hits. You know, "Jn" for "John", etc. --Jerome Potts (talk) 02:18, 2 June 2018 (UTC)

 * https://www.logos.com/bible-book-abbreviations Editor2020 (talk) 01:17, 1 July 2020 (UTC)

The ancestors or God
What revelation was meant Tobe reading the book of God thus the debate of the Xhosa religion say they pray on ancestors Simbulele luhle simjo (talk) 05:12, 30 June 2020 (UTC)
 * Who is Tobe? Editor2020 (talk) 01:15, 1 July 2020 (UTC)

title: Talk:Books of the Bible