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Legendre polynomials

In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications.

Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions.

Definition by construction as an orthogonal system
In this approach, the polynomials are defined as an orthogonal system with respect to the weight function $$w(x) = 1$$ over the interval $$ [-1,1]$$. That is, $$P_n(x)$$ is a polynomial of degree $$n$$, such that $$\int_{-1}^1 P_m(x) P_n(x) \,dx = 0 \quad \text{if } n \ne m.$$

With the additional standardization condition $$P_n(1) = 1$$, all the polynomials can be uniquely determined. We then start the construction process: $$P_0(x) = 1$$ is the only correctly standardized polynomial of degree 0. $$P_1(x)$$ must be orthogonal to $$P_0$$, leading to $$P_1(x) = x$$, and $$P_2(x)$$ is determined by demanding orthogonality to $$P_0$$ and $$P_1$$, and so on. $$P_n$$ is fixed by demanding orthogonality to all $$P_m$$ with $$ m < n $$. This gives $$ n $$ conditions, which, along with the standardization $$ P_n(1) = 1$$ fixes all $$ n+1$$ coefficients in $$ P_n(x)$$. With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of $$x$$ given below.

This definition of the $$P_n$$'s is the simplest one. It does not appeal to the theory of differential equations. Second, the completeness of the polynomials follows immediately from the completeness of the powers 1, $$ x, x^2, x^3, \ldots$$. Finally, by defining them via orthogonality with respect to the most obvious weight function on a finite interval, it sets up the Legendre polynomials as one of the three classical orthogonal polynomial systems. The other two are the Laguerre polynomials, which are orthogonal over the half line $$[0,\infty)$$, and the Hermite polynomials, orthogonal over the full line $$(-\infty,\infty)$$, with weight functions that are the most natural analytic functions that ensure convergence of all integrals.

Definition via generating function
The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of $$t$$ of the generating function

The coefficient of $$t^n$$ is a polynomial in $$ x $$ of degree $$n$$ with $$|x| \leq 1$$. Expanding up to $$t^1$$ gives $$P_0(x) = 1 \,,\quad P_1(x) = x.$$ Expansion to higher orders gets increasingly cumbersome, but is possible to do systematically, and again leads to one of the explicit forms given below.

It is possible to obtain the higher $$P_n$$'s without resorting to direct expansion of the Taylor series, however. Equation $$ is differentiated with respect to $$ on both sides and rearranged to obtain $$\frac{x-t}{\sqrt{1-2xt+t^2}} = \left(1-2xt+t^2\right) \sum_{n=1}^\infty n P_n(x) t^{n-1} \,.$$ Replacing the quotient of the square root with its definition in Eq. $t$, and equating the coefficients of powers of $t$ in the resulting expansion gives Bonnet’s recursion formula $$ (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)\,.$$ This relation, along with the first two polynomials $P_{0}$ and $P_{1}$, allows all the rest to be generated recursively.

The generating function approach is directly connected to the multipole expansion in electrostatics, as explained below, and is how the polynomials were first defined by Legendre in 1782.

Definition via differential equation
A third definition is in terms of solutions to Legendre's differential equation:

This differential equation has regular singular points at $x = ±1$ so if a solution is sought using the standard Frobenius or power series method, a series about the origin will only converge for $|x| < 1$ in general. When $n$ is an integer, the solution $P_{n}(x)$ that is regular at $x = 1$ is also regular at $x = −1$, and the series for this solution terminates (i.e. it is a polynomial). The orthogonality and completeness of these solutions is best seen from the viewpoint of Sturm–Liouville theory. We rewrite the differential equation as an eigenvalue problem, $$\frac{d}{dx} \left( \left(1-x^2\right) \frac{d}{dx} \right) P(x) = -\lambda P(x) \,,$$ with the eigenvalue $$\lambda$$ in lieu of $$ n(n+1)$$. If we demand that the solution be regular at $$x = \pm 1$$, the differential operator on the left is Hermitian. The eigenvalues are found to be of the form $n(n + 1)$, with $$n = 0, 1, 2, \ldots$$ and the eigenfunctions are the $$P_n(x)$$. The orthogonality and completeness of this set of solutions follows at once from the larger framework of Sturm–Liouville theory.

The differential equation admits another, non-polynomial solution, the Legendre functions of the second kind $$Q_n$$. A two-parameter generalization of (Eq. $$) is called Legendre's general differential equation, solved by the Associated Legendre polynomials. Legendre functions are solutions of Legendre's differential equation (generalized or not) with non-integer parameters.

In physical settings, Legendre's differential equation arises naturally whenever one solves Laplace's equation (and related partial differential equations) by separation of variables in spherical coordinates. From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the spherical harmonics, of which the Legendre polynomials are (up to a multiplicative constant) the subset that is left invariant by rotations about the polar axis. The polynomials appear as $$P_n(\cos\theta)$$ where $$\theta$$ is the polar angle. This approach to the Legendre polynomials provides a deep connection to rotational symmetry. Many of their properties which are found laboriously through the methods of analysis — for example the addition theorem — are more easily found using the methods of symmetry and group theory, and acquire profound physical and geometrical meaning.

Orthogonality and completeness
The standardization $$P_n(1) = 1$$ fixes the normalization of the Legendre polynomials (with respect to the $L^{2}$ norm on the interval $−1 ≤ x ≤ 1$). Since they are also orthogonal with respect to the same norm, the two statements can be combined into the single equation, $$\int_{-1}^1 P_m(x) P_n(x)\,dx = \frac{2}{2n + 1} \delta_{mn},$$ (where $δ_{mn}$ denotes the Kronecker delta, equal to 1 if $m = n$ and to 0 otherwise). This normalization is most readily found by employing Rodrigues' formula, given below.

That the polynomials are complete means the following. Given any piecewise continuous function $$ f(x) $$ with finitely many discontinuities in the interval $$, the sequence of sums $$ f_n(x) = \sum_{\ell=0}^n a_\ell P_\ell(x)$$ converges in the mean to $$ f(x) $$ as $$ n \to \infty $$, provided we take $$ a_\ell = \frac{2\ell + 1}{2} \int_{-1}^1 f(x) P_\ell(x)\,dx.$$

This completeness property underlies all the expansions discussed in this article, and is often stated in the form $$\sum_{\ell=0}^\infty \frac{2\ell + 1}{2} P_\ell(x)P_\ell(y) = \delta(x-y), $$ with $−1 ≤ x ≤ 1$ and $−1 ≤ y ≤ 1$.

Rodrigues' formula and other explicit formulas
An especially compact expression for the Legendre polynomials is given by Rodrigues' formula: $$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 -1)^n \,.$$

This formula enables derivation of a large number of properties of the $$P_n$$'s. Among these are explicit representations such as $$\begin{align} P_n(x) & = [t^n] \frac{\left((t+x)^2 - 1\right)^n}{2^n} = [t^n] \frac{\left(t+x+1\right)^n \left(t+x-1\right)^n}{2^n}, \\[1ex] P_n(x)&= \frac{1}{2^n} \sum_{k=0}^n \binom{n}{k}^{\!2} (x-1)^{n-k}(x+1)^k, \\[1ex] P_n(x)&= \sum_{k=0}^n \binom{n}{k} \binom{n+k}{k} \left( \frac{x-1}{2} \right)^{\!k}, \\[1ex] P_n(x)&= \frac{1}{2^n}\sum_{k=0}^{\left\lfloor n/2 \right\rfloor} \left(-1\right)^k \binom{n}{k}\binom{2n-2k}n x^{n-2k},\\[1ex] P_n(x)&= 2^n \sum_{k=0}^n x^k \binom{n}{k} \binom{\frac{n+k-1}{2}}{n}. \end{align}$$ Expressing the polynomial as a power series, $P_n(x) = \sum a_k x^k $, the coefficients of powers of $$x$$ can also be calculated using a general formula:$$a_{k+2} = - \frac{(l-k)(l+k+1)}{(k+2)(k+1)}a_k. $$The Legendre polynomial is determined by the values used for the two constants $a_0 $ and $a_1 $, where $a_0=0 $  if $$n$$ is odd and $a_1=0 $  if $$n$$ is even.

In the fourth representation, $$\lfloor n/2 \rfloor$$ stands for the largest integer less than or equal to $$n/2$$. The last representation, which is also immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the generalized form of the binomial coefficient.

The first few Legendre polynomials are:

The graphs of these polynomials (up to $n = 5$) are shown below:

Expanding a 1/r potential
The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre as the coefficients in the expansion of the Newtonian potential $$\frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma),$$ where $r$ and $r′$ are the lengths of the vectors $x$ and $x′$ respectively and $γ$ is the angle between those two vectors. The series converges when $r > r′$. The expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution.

Legendre polynomials occur in the solution of Laplace's equation of the static potential, $∇^{2} Φ(x) = 0$, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle). Where $ẑ$ is the axis of symmetry and $θ$ is the angle between the position of the observer and the $ẑ$ axis (the zenith angle), the solution for the potential will be $$\Phi(r,\theta) = \sum_{\ell=0}^\infty \left( A_\ell r^\ell + B_\ell r^{-(\ell+1)} \right) P_\ell(\cos\theta) \,.$$

$A_{l}$ and $B_{l}$ are to be determined according to the boundary condition of each problem.

They also appear when solving the Schrödinger equation in three dimensions for a central force.

Legendre polynomials in multipole expansions
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently): $$\frac{1}{\sqrt{1 + \eta^2 - 2\eta x}} = \sum_{k=0}^\infty \eta^k P_k(x),$$ which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials.

As an example, the electric potential $Φ(r,θ)$ (in spherical coordinates) due to a point charge located on the $z$-axis at $z = a$ (see diagram right) varies as $$\Phi (r, \theta ) \propto \frac{1}{R} = \frac{1}{\sqrt{r^2 + a^2 - 2ar \cos\theta}}.$$

If the radius $r$ of the observation point $P$ is greater than $a$, the potential may be expanded in the Legendre polynomials $$\Phi(r, \theta) \propto \frac{1}{r} \sum_{k=0}^\infty \left( \frac{a}{r} \right)^k P_k(\cos \theta),$$ where we have defined $η = a⁄r < 1$ and $x = cos θ$. This expansion is used to develop the normal multipole expansion.

Conversely, if the radius $r$ of the observation point $P$ is smaller than $a$, the potential may still be expanded in the Legendre polynomials as above, but with $a$ and $r$ exchanged. This expansion is the basis of interior multipole expansion.

Legendre polynomials in trigonometry
The trigonometric functions $cos nθ$, also denoted as the Chebyshev polynomials $T_{n}(cos θ) ≡ cos nθ$, can also be multipole expanded by the Legendre polynomials $P_{n}(cos θ)$. The first several orders are as follows: $$\begin{alignat}{2} T_0(\cos\theta)&=1          &&=P_0(\cos\theta),\\[4pt] T_1(\cos\theta)&=\cos \theta&&=P_1(\cos\theta),\\[4pt] T_2(\cos\theta)&=\cos 2\theta&&=\tfrac{1}{3}\bigl(4P_2(\cos\theta)-P_0(\cos\theta)\bigr),\\[4pt] T_3(\cos\theta)&=\cos 3\theta&&=\tfrac{1}{5}\bigl(8P_3(\cos\theta)-3P_1(\cos\theta)\bigr),\\[4pt] T_4(\cos\theta)&=\cos 4\theta&&=\tfrac{1}{105}\bigl(192P_4(\cos\theta)-80P_2(\cos\theta)-7P_0(\cos\theta)\bigr),\\[4pt] T_5(\cos\theta)&=\cos 5\theta&&=\tfrac{1}{63}\bigl(128P_5(\cos\theta)-56P_3(\cos\theta)-9P_1(\cos\theta)\bigr),\\[4pt] T_6(\cos\theta)&=\cos 6\theta&&=\tfrac{1}{1155}\bigl(2560P_6(\cos\theta)-1152P_4(\cos\theta)-220P_2(\cos\theta)-33P_0(\cos\theta)\bigr). \end{alignat}$$

Another property is the expression for $sin (n + 1)θ$, which is $$\frac{\sin (n+1)\theta}{\sin\theta}=\sum_{\ell=0}^n P_\ell(\cos\theta) P_{n-\ell}(\cos\theta).$$

Legendre polynomials in recurrent neural networks
A recurrent neural network that contains a $d$-dimensional memory vector, $$\mathbf{m} \in \R^d$$, can be optimized such that its neural activities obey the linear time-invariant system given by the following state-space representation: $$\theta \dot{\mathbf{m}}(t) = A\mathbf{m}(t) + Bu(t),$$ $$\begin{align} A &= \left[ a \right]_{ij} \in \R^{d \times d} \text{,} \quad && a_{ij} = \left(2i + 1\right) \begin{cases} -1 & i < j \\ (-1)^{i-j+1} & i \ge j \end{cases},\\ B &= \left[ b \right]_i \in \R^{d \times 1} \text{,} \quad && b_i = (2i + 1) (-1)^i. \end{align}$$

In this case, the sliding window of $$u$$ across the past $$\theta$$ units of time is best approximated by a linear combination of the first $$d$$ shifted Legendre polynomials, weighted together by the elements of $$\mathbf{m}$$ at time $$t$$: $$u(t - \theta') \approx \sum_{\ell=0}^{d-1} \widetilde{P}_\ell \left(\frac{\theta'}{\theta} \right) \, m_{\ell}(t), \quad 0 \le \theta' \le \theta .$$

When combined with deep learning methods, these networks can be trained to outperform long short-term memory units and related architectures, while using fewer computational resources.

Additional properties of Legendre polynomials
Legendre polynomials have definite parity. That is, they are even or odd, according to $$P_n(-x) = (-1)^n P_n(x) \,.$$

Another useful property is $$\int_{-1}^1 P_n(x)\,dx = 0 \text{ for } n\ge1,$$ which follows from considering the orthogonality relation with $$P_0(x) = 1$$. It is convenient when a Legendre series $\sum_i a_i P_i$ is used to approximate a function or experimental data: the average of the series over the interval $$ is simply given by the leading expansion coefficient $$a_0$$.

Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but the actual norm is not 1) by being scaled so that $$P_n(1) = 1 \,.$$

The derivative at the end point is given by $$P_n'(1) = \frac{n(n+1)}{2} \,. $$

The Askey–Gasper inequality for Legendre polynomials reads $$\sum_{j=0}^n P_j(x) \ge 0 \quad \text{for }\quad x\ge -1 \,.$$

The Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using $$P_\ell \left(r \cdot r'\right) = \frac{4\pi}{2\ell + 1} \sum_{m=-\ell}^\ell Y_{\ell m}(\theta,\varphi) Y_{\ell m}^*(\theta',\varphi')\,,$$ where the unit vectors $r$ and $r′$ have spherical coordinates $(θ, φ)$ and $(θ′, φ′)$, respectively.

The product of two Legendre polynomials $$ \sum_{p=0}^\infty t^{p}P_p(\cos\theta_1)P_p(\cos\theta_2)=\frac2\pi\frac{\mathbf K\left( 2\sqrt{\frac{t\sin\theta_1\sin\theta_2}{t^2-2t\cos\left( \theta_1+\theta_2 \right)+1}} \right)}{\sqrt{t^2-2t\cos\left( \theta_1+\theta_2 \right)+1}}\,,$$ where $$K(\cdot)$$ is the complete elliptic integral of the first kind.

Recurrence relations
As discussed above, the Legendre polynomials obey the three-term recurrence relation known as Bonnet's recursion formula given by $$ (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x)$$ and $$ \frac{x^2-1}{n} \frac{d}{dx} P_n(x) = xP_n(x) - P_{n-1}(x) $$ or, with the alternative expression, which also holds at the endpoints $$ \frac{d}{dx} P_{n+1}(x) = (n+1)P_n(x) + x \frac{d}{dx}P_{n}(x) \,.$$

Useful for the integration of Legendre polynomials is $$(2n+1) P_n(x) = \frac{d}{dx} \bigl( P_{n+1}(x) - P_{n-1}(x) \bigr) \,.$$

From the above one can see also that $$\frac{d}{dx} P_{n+1}(x) = (2n+1) P_n(x) + \bigl(2(n-2)+1\bigr) P_{n-2}(x) + \bigl(2(n-4)+1\bigr) P_{n-4}(x) + \cdots$$ or equivalently $$\frac{d}{dx} P_{n+1}(x) = \frac{2 P_n(x)}{\left\| P_n \right\|^2} + \frac{2 P_{n-2}(x)}{\left\| P_{n-2} \right\|^2} + \cdots$$ where $\|P_{n}\|$ is the norm over the interval $−1 ≤ x ≤ 1$ $$\| P_n \| = \sqrt{\int_{-1}^1 \bigl(P_n(x)\bigr)^2 \,dx} = \sqrt{\frac{2}{2 n + 1}} \,.$$

Asymptotics
Asymptotically, for $$\ell \to \infty$$, the Legendre polynomials can be written as $$\begin{align} P_\ell (\cos \theta) &= \sqrt{\frac{\theta}{\sin\left(\theta\right)}} \, J_0{\left(\left(\ell+\tfrac{1}{2}\right)\theta\right)} + \mathcal{O}\left(\ell^{-1}\right) \\[1ex] &= \sqrt{\frac{2}{\pi \ell\sin\left(\theta\right)}}\cos\left(\left(\ell + \tfrac{1}{2} \right)\theta - \tfrac{\pi}{4}\right) + \mathcal{O}\left(\ell^{-3/2}\right), \quad \theta \in (0,\pi), \end{align}$$ and for arguments of magnitude greater than 1 $$\begin{align} P_\ell \left(\cosh\xi\right) &= \sqrt{\frac{\xi}{\sinh\xi}} I_0\left(\left(\ell+\frac{1}{2}\right)\xi\right)\left(1+\mathcal{O}\left(\ell^{-1}\right)\right)\,,\\ P_\ell \left(\frac{1}{\sqrt{1-e^2}}\right) &= \frac{1}{\sqrt{2\pi\ell e}} \frac{(1+e)^\frac{\ell+1}{2}}{(1-e)^\frac{\ell}{2}} + \mathcal{O}\left(\ell^{-1}\right) \end{align}$$ where $J_{0}$ and $I_{0}$ are Bessel functions.

Zeros
All $$ n$$ zeros of $$P_n(x)$$ are real, distinct from each other, and lie in the interval $$(-1,1)$$. Furthermore, if we regard them as dividing the interval $$[-1,1]$$ into $$ n+1 $$ subintervals, each subinterval will contain exactly one zero of $$P_{n+1}$$. This is known as the interlacing property. Because of the parity property it is evident that if $$x_k$$ is a zero of $$P_n(x)$$, so is $$-x_k$$. These zeros play an important role in numerical integration based on Gaussian quadrature. The specific quadrature based on the $$P_n$$'s is known as Gauss-Legendre quadrature.

From this property and the facts that $$ P_n(\pm 1) \ne 0 $$, it follows that $$ P_n(x) $$ has $$ n-1 $$ local minima and maxima in $$ (-1,1) $$. Equivalently, $$ dP_n(x)/dx $$ has $$ n -1 $$ zeros in $$ (-1,1) $$.

Pointwise evaluations
The parity and normalization implicate the values at the boundaries $$ x=\pm 1 $$ to be $$ P_n(1) = 1 \,, \quad P_n(-1) = (-1)^n $$ At the origin $$ x=0 $$ one can show that the values are given by $$ P_{2n}(0) = \frac{(-1)^{n}}{4^n} \binom{2n}{n} = \frac{(-1)^{n}}{2^{2n}} \frac{(2n)!}{\left(n!\right)^2} = (-1)^n\frac{(2n-1)!!}{(2n)!!} $$$$ P_{2n+1}(0) = 0 $$

Shifted Legendre polynomials
The shifted Legendre polynomials are defined as $$\widetilde{P}_n(x) = P_n(2x-1) \,.$$ Here the "shifting" function $x ↦ 2x − 1$ is an affine transformation that bijectively maps the interval $[−1, 1]$ to the interval $[−1, 1]$, implying that the polynomials $P̃_{n}(x)$ are orthogonal on $[0, 1]$: $$\int_0^1 \widetilde{P}_m(x) \widetilde{P}_n(x)\,dx = \frac{1}{2n + 1} \delta_{mn} \,.$$

An explicit expression for the shifted Legendre polynomials is given by $$\widetilde{P}_n(x) = (-1)^n \sum_{k=0}^n \binom{n}{k} \binom{n+k}{k} (-x)^k \,.$$

The analogue of Rodrigues' formula for the shifted Legendre polynomials is $$\widetilde{P}_n(x) = \frac{1}{n!} \frac{d^n}{dx^n} \left(x^2 -x \right)^n \,.$$

The first few shifted Legendre polynomials are:

Legendre rational functions
The Legendre rational functions are a sequence of orthogonal functions on [0, ∞). They are obtained by composing the Cayley transform with Legendre polynomials.

A rational Legendre function of degree n is defined as: $$R_n(x) = \frac{\sqrt{2}}{x+1}\,P_n\left(\frac{x-1}{x+1}\right)\,.$$

They are eigenfunctions of the singular Sturm–Liouville problem: $$\left(x+1\right) \frac{d}{dx} \left(x \frac{d}{dx} \left[\left(x+1\right) v(x)\right]\right) + \lambda v(x) = 0$$ with eigenvalues $$\lambda_n=n(n+1)\,.$$

title: Legendre polynomials


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The Wild, the Innocent & the E Street Shuffle

The Wild, the Innocent & the E Street Shuffle is the second studio album by the American rock singer-songwriter Bruce Springsteen. It was recorded by Springsteen with the E Street Band at 914 Sound Studios in Blauvelt, New York, and released on November 5, 1973, by Columbia Records. It includes the song "Rosalita (Come Out Tonight)", the band's most-used set-closing song through 1985.

As with Springsteen's debut album released earlier in the year, Greetings from Asbury Park, N.J., The Wild, the Innocent & the E Street Shuffle was well-received critically but had little commercial success at the time. Locally, though, the album sold well, was played regularly on Northeast album-oriented rock stations and made Springsteen a local phenomenon. Once Springsteen achieved nationwide popularity with his subsequent album Born to Run (1975), several selections from this album became popular FM radio airplay and concert favorites. On November 7, 2009, Springsteen and the E Street Band played the album in its entirety for the first time during a concert at Madison Square Garden. In Rolling Stone's 2020 edition of its "The 500 Greatest Albums of All Time" list, the album was ranked at number 345.

Recording
Recording began on May 14, 1973, with the day spent on "Circus Song", which would be finished on June 28, and re-titled "Wild Billy's Circus Story". Two days later, "The Fever" was recorded in one take, then discarded and not included on the album. Sessions did not resume until June 22, but all backing tracks and most of the album would be recorded by the end of the week. September 23, 1973, was the final day of sessions, with final touches applied to "Kitty's Back", the last verse of "4th of July, Asbury Park (Sandy)" overdubbed with new lyrics, "Rosalita (Come Out Tonight)" and a brand new song, "Incident on 57th Street", both recorded from scratch and completed. Along with these tracks, the album would be composed of "New York City Serenade", completed August 7, and "The E Street Shuffle", recorded on June 28 in one session.

Musical style
According to biographer Peter Ames Carlin, Springsteen had developed a "renewed passion for full-band rock 'n' roll" with The Wild, the Innocent & the E Street Shuffle. Sputnikmusic critic Adam Thomas later wrote that the album departed from the folk influences of Springsteen's 1973 debut album Greetings from Asbury Park, N.J. and was instead characterized by "a grand fusion of nostalgic rock 'n' roll and soulful R&B".

Artwork and name
The back photo for the album featured six members of Springsteen's backing E Street Band standing in a doorway of an antique store on Sairs Ave in the West End section of Long Branch, New Jersey. The building was across the street from West End Elementary School, and for years was Tommy Reed's bicycle repair shop and penny candy store; it has since been demolished and its former location is occupied by a parking lot.

The album was named in part after the 1959 Audie Murphy--Sandra Dee film, The Wild and the Innocent.

Release
There is disagreement on The Wild, the Innocent & the E Street Shuffle's release date between official sources and third-party sources, including various Springsteen biographies covering the era. Springsteen's official website provides a release date of September 11, 1973; however, it is known that Springsteen was still in the studio recording for the album past this date. Indeed, September 23 was the final recording date, and after mixing and sequencing tweaks, the album was released on November 5, 1973. However, there was very little press, no advertisements in the trade papers and no release party, possibly because of Springsteen's deteriorating relationship with Columbia Records, and the departures of John Hammond and Clive Davis, the men who had signed him as a Columbia artist. This situation would continue, and would almost result in the termination of Springsteen's contract in 1974.

Reception
The Wild, the Innocent & the E Street Shuffle sold poorly when it was first released in 1973 but received acclaim from critics. Rolling Stone magazine's Ken Emerson said that its lengthy, vividly written songs make for a more challenging and romantic album than Greetings from Asbury Park, N.J., but still retain that album's ebullient music because of Springsteen and the E Street Band's masterful playing. In a less enthusiastic review for Creem, Robert Christgau wrote that it does not cohere as a whole, although its livelier songs make it "the kind of album that will be fun to go back to" if Springsteen improves upon it. According to him in Christgau's Record Guide: Rock Albums of the Seventies (1981), Springsteen eschewed the limiting folk conventions of his first album for a vibrant, quirky style of rock and roll that balances his celebrations of wild youth with a mature embrace of city life: "This guy may not be God yet, but he has his sleeveless undershirt in the ring."

The Wild, the Innocent and the E Street Shuffle made its first appearance on the British albums chart on June 15, 1985. In the wake of Springsteen's Born in the U.S.A. Tour arriving in Britain, the record hit number 33 and remained in the Top 100 for 12 weeks. The Rolling Stone Album Guide (1992) later called the album a "masterpiece", "cinematic in its sweep" and densely poetic with "vignettes of urban dreams and adolescent restlessness". Goldmine magazine's Rush Evans said it was not only a five-star album but also Springsteen's "most overlooked album ... to those who know its seven richly vivid songs, it is recognized as an innovative masterpiece." Reviewing it for AllMusic, William Ruhlmann felt the record epitomized Springsteen's romanticized songwriting and diversity as a composer, making it his best work and "one of the greatest albums in the history of rock & roll". In 2003, Rolling Stone ranked the record at number 132 on the magazine's list of the 500 greatest albums of all time, and 133 in a 2012 revised list. Writing in Hot Press, Pat Carty declared "if I want to beam like an idiot and remember why I fell in love with music in the first place, this is the Springsteen record I put on". Chicago Tribune critic Greg Kot was less enthusiastic and remarked only on how the record was highlighted by the beautiful three-song suite that ended it.

Rejected songs
On November 4 and 5, 1997, eight tracks not included on The Wild, the Innocent and the E Street Shuffle were mixed and evaluated for inclusion on the Tracks box set, to be released in 1998. They included "Zero and Blind Terry", "Thundercrack", "Seaside Bar Song", "Santa Ana", "Bishop Danced", "Evacuation of the West" aka "No More Kings In Texas", "Phantoms" and "Fire on the Wing". The first five, which included a live version of "Bishop Danced", recorded on January 31, 1973, at Max's Kansas City, New York City, were included; "Phantoms", "Fire on the Wing" and "No More Kings/Evacuation" remain officially unreleased.

"The Fever", recorded in a single take on May 16, 1973, was also rejected and disliked by Springsteen. His manager at the time, Mike Appel, without informing Springsteen or Columbia Records, sent a cassette of the song to approximately forty disc jockeys across the US, including Ed Sciaky in Philadelphia, Kid Leo in Cleveland, and other DJs in Boston, New York, Houston, and Dallas. The song was then included on an endless stream of bootlegs, and was well known to Springsteen fans as one of his best songs. When it failed to appear on Tracks, along with "The Promise", another highly regarded unreleased song, a backlash occurred among fans on the Internet. In response, a second release, 18 Tracks, was hastily assembled and released on April 13, 1999, with both songs included.

Personnel
The E Street Band
 * Bruce Springsteen – lead vocals (all tracks), guitar (all tracks), recorder (track 2), maracas (track 1), harmonica (track 4), mandolin (track 4)
 * Clarence Clemons – tenor saxophone (tracks 1, 2, 3, 6, 7), backing vocals (tracks 1, 2, 3, 6, 7)
 * Danny Federici – backing vocals (tracks 1, 2, 4, 6, 7), accordion (tracks 2, 4) organ (tracks 3, 5), second piano (track 5)
 * Garry Tallent – bass guitar (all tracks except 4), tuba (tracks 1, 4), backing vocals (track 3)
 * David Sancious – piano (all tracks except 1, 4), clavinet (track 1), soprano saxophone (track 1), electric piano (tracks 2, 3), backing vocals (track 1), organ solo (track 3), Mellotron (track 7), string arrangement (track 7)
 * "Mad Dog" Vini Lopez – drums, percussion (all tracks), backing vocals (tracks 1, 3, 7), cornet (track 1)

Additional personnel

 * Richard Blackwell – percussion (tracks 1, 7)
 * Albany "Al" Tellone – baritone saxophone (track 1)
 * Suki Lahav – choir vocals (tracks 2, 5)

Technical
 * Mike Appel & Jim Cretecos – record producers (for Laurel Canyon Ltd.)
 * Louis Lahav – engineer
 * Teresa Alfieri & John Berg – design
 * David Gahr – photography

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


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Andhaka

In Hindu literature, Andhaka (Sanskrit: अन्धक, IAST: Andhaka; lit. "He who darkens") refers to a malevolent asura, whose pride is vanquished by Shiva and Parvati.

His story finds mention in various Hindu texts, including the Matsya Purāṇa, the Kūrma Purāṇa, the Liṅga Purāṇa, the Padma Purana, and the Shiva Purana. He is believed to have one thousand heads, and one thousand arms, and also having two thousand eyes. In another version, he has two thousand arms, and two thousand legs. In some versions of his story, Andhaka is described as a son of Shiva and Parvati.

Shiva Purana
In the Shiva Purana, when Shiva was meditating on Mount Mandara, Parvati was in a playful mood and covered Shiva's eyes. This caused the whole universe to become covered in darkness. The sweat that oozed out of Parvati's hands, due to touching Shiva's powerful third eye, fell to the ground and created a horrible looking and blind boy. Parvati was terrified on seeing him, but Shiva rebuked her, claiming that since he was born due to their physical contact, he was their child.

When the demon king Hiraṇyākṣa performed penance to please Shiva in order to beget a child, Shiva gifted the boy to him and named him Andhaka due to his blindness. After Hiraṇyākṣa's death by Vishnu, Andhaka became the new king, but was not regarded as an Asura since he was born of the Devas. Disowned by majority of his clan, he performed a severe penance to please Brahmā. Brahmā thus appeared to him and offered him a boon. Andhaka requested that Brahmā give and make divine vision and immortality. Brahmā granted Andhaka these wishes, but warned him that he could still be killed by Shiva. Andhaka went back to his kingdom and subdued all his opponents as well as the Devas.

Andhaka asked his minister if there was anyone who could match him in strength, majesty and riches. The minister informed him that he did not have the company of a beautiful woman. The world's most beautiful woman, he explained, was Parvati, who belonged to a matted ascetic who lived on Mount Kailāśa. Andhaka was advised that if he wished to be truly matchless, he should possess her. Andhaka sent a messenger to Shiva with the demand to hand over his wife. Andhaka attacked Shiva with his greatest warriors, but they were defeated by Shiva's army.

One day, when Shiva was performing meditation in forest, Andhaka thought of attacking Mandar mountain. Terrible war took place between gods and danavs. Andhaka's trusted general Bali single-handedly defeated all the gods and swallowed them. Shiva fired powerful weapons at Bali, which forced him to disgorge all the gods. In retaliation, Shiva swallowed Śukra, the guru of the Asuras. Andhaka then launched an attack on Indra. Shiva intervened to save Indra and assaulted the demon with his trident. However, whenever Andhaka's blood fell on the ground, a copy of him would spawn. Vishnu created several Mātṛkās to drink the blood of the demon every time he was hurt, preventing the spread of new demons. Finally, Shiva decapitated Andhaka. According to Shiva Purana, in a twist of events, since Andhaka had chanted Shiva's name in repentance after which he was told of his biological parentage and he was made a Gaṇa chief.

Alternate account
There is another version in the Purāṇa in which Andhaka's birth story and disownment by his clan remains the same. His kingdom was taken over by his cousins, including Prahlāda, when he went to the forest to perform penance to Brahmā. During his penance he abstained from both water and food while severing his own limbs as an act of desperation to please Brahmā.

His actions proved successful. A pleased Brahmā appeared before him and offered him a boon of his choice. Andhaka asked Brahma to repair his vision and also asked him to make him immortal. Brahmā replied that he could not make him immortal since all that take birth must die, although he could choose the condition of his death. Andhaka asked that he could only be killed if he lusted after a woman who was like a mother. Brahma agreed and granted all his boons.

When Andhaka returned to his kingdom, his cousins became fearful of his new power and returned his kingdom as well as their own. After becoming the lord of all Asuras, Andhaka fought with the Devas along with his army and conquered heaven. He then proceeded to conquer the Nāgas, Gandharvas, Rākṣasas, Yakṣas and the humans. He thus became the lord of the three worlds.

He was a cruel ruler and disrespected the Vedas, the Brahmins and the Devas. Once while on a journey, Andhaka happened to visit Mount Mandara. Charmed by the beauty of the mountain, he decided to stay there and ordered his three generals Duryodhana, Vighasa and Hasti to search for a place suitable to stay.

While searching, his generals found a cave where a hermit was meditating and saw a beautiful woman along with him. They told their master about this, who ordered them to bring the woman to him. The hermit happened to be Shiva and the woman his wife Parvati. When they told the ascetic to hand over the woman to their master, Shiva refused, stating that if their master wanted her, he should take her himself.

When his generals informed Andhaka about this, he became furious and proceeded to fight Shiva. With his army of Asuras, Andhaka fought Nandi and Shiva's Gaṇas, but were defeated and forced to flee. Andhaka soon returned to battle, which lasted for seven hundred years.

Vishnu, Brahmā and the Devas joined the battle against Andhaka and his army. Andhaka's general Vighasa swallowed all the gods, to which Shiva retaliated by charging with his bull and plowing into the demon. Śukra, the preceptor of Asura, brought the dead Asuras back to life by using his medicinal herb, the Mṛtasañjīvanī. Shiva ordered the Gaṇas to capture Śukra. When they brought him to Shiva, he swallowed the demon guru.

Shiva attacked Andhaka with his trident, but every drop of Andhaka's blood that fell to the ground spawned another demon like him. Shiva requested the goddess chandi to drink the blood while he took care of the rest of Andhaka's duplicates. After vanquishing the remaining Asura, Shiva impaled Andhaka with his trident and lifted him thereupon, where he remained for a very long time. Once he had realised that the woman who he was lusting for was his own mother and her husband was his own father, he felt deeply grieved and ashamed of his actions. He sincerely prayed to Shiva for forgiveness. Shiva, the most humble one, forgave him and made him the chief of the Gaṇas.

Vāmana Purāṇa
In the Vāmana Purāṇa, the story of Andhaka's birth remains same. Andhaka was the son of Hiraṇyākṣa and cousin of Prahlāda. Andhaka and Prahlāda, along with their army defeated the Devas, making Prahlāda the king of the three worlds. Sometime later, Prahlāda battled Vishnu, but lost the battle due to a curse the sages have placed upon him. After his loss, he appointed Andhaka as king and successfully acquires Vishnu's forgiveness. Upon returning, Andhaka tries to make him king again, but is met with refusal. At one point, Mahiṣāsura and Tārakāsura are killed by the gods, a feat which Andhaka thought was impossible.

After this incident, Andhaka came to desire a beautiful wife and was informed about Parvati, said to be the most beautiful woman in the world. Unaware that Parvati is his mother, Andhaka makes his way to Shiva's abode. There he tried to abduct Parvati, but she assumed a hundred forms and knocked him unconscious in battle. Andhaka fled back to Pātāla with the intention to recuperate and make another attempt at procuring Parvati.

Prahlāda tried to dissuade him. He revealed to Andhaka his true origins about being born from the sweat of Shiva when Parvati covered his eyes. His explanation falls on deaf ears and he fails to persuade the Asura. Andhaka attacked Shiva and the other deities with his army, most of whom were obliterated. Andhaka then disguised himself as Shiva to fool Parvati, but she managed to recognised him and hid among her servants. Unable to find her, Andhaka returns to the battle.

Kārttikeya and Gaṇeśa, accompanied by the Gaṇas, destroyed Andhaka's chariot. Shiva engaged him in battle and pierced his heart, but Andhaka was able to recover and strike Shiva with his mace. The blood that fell on the ground from the wound gave rise to the eight forms of Bhairava.

Shiva impaled Andhaka on his trident and lifted him upon it. The sweat that emanated from Shiva gave rise to a girl and a boy of the colour of charcoal, who proceed to consume Andhaka's blood before it falls onto the ground. Shiva names the girl Cārcikā and the boy Maṅgala. He holds Andhaka impaled on his trident for thousands of years, reducing his body to a mere skeletal appearance. Andhaka begged for forgiveness and prostrated infront of Shiva as his father and Parvati as his mother. He is forgiven and made a Gaṇa chief. Shiva takes him to the Mandara mountain where Parvati also blesses the same boon to him and he later becomes famous by the name of Bhṛṅgi.

Kūrma Purāṇa
In the Kūrma Purāṇa too, Andhaka is the natural son of Hiraṇyākṣa and becomes the king of the Asuras after the death of Prahlāda. Like other variants, he lusts after Parvati and goes to Mount Mandara to abduct her when Shiva is away. Shiva had entrusted Nandi to guard his household and the gods including Vishnu, to attend to and protect Parvati. When Andhaka arrives, Nandi battles with him and pierces him with a trident. Upon spilling of his blood, a thousand more Asura like him spawn and defeat Nandi and the Devas. Nandi prays to Vishnu for aid.

When Vishnu arrives he creates several Mātṛkā goddesses who vanquish the demons. When Shiva returns, Andhaka returns in another attempt to abduct Parvati. Shiva and Vishnu wage battle against Andhaka and his army. Shiva succeeds in impaling Andhaka on his trident and begins to dance. By his mere touch, Andhaka's sins are burned away and he begs for forgiveness. He is later named a Gaṇa chief. Shiva also makes Andhaka a handsome man who then prostrates before Parvati in repentance. Parvati forgives him and accepts him as her son.

Matsya Purāṇa
In the Matsya Purāṇa, Andhaka, the son of Hiraṇyākṣa, tries to abduct Parvati from Shiva. A battle ensues in the Mahākāla forest and the blood that flows from Andhaka gives rise to a thousand more demons. Shiva creates numerous divine mothers, the Mātṛkā, who drink the demon's blood every time it issues forth. After they become satiated, Vāsudeva creates the goddess Śuṣkarevatī who drinks the blood of all the demons and kills them. When Shiva is about to blow the final strike, Andhaka surrenders and begs for forgiveness by praying to him. Pleased by his devotion, Shiva makes him a Gaṇa.

Harivaṃśa Purana
According to the Harivaṃśa Purāṇa, Andhaka was a Daitya and the son of Diti and the sage Kaśyapa. After the defeat of Daityas by Vishnu, Diti pleaded with Kaśyapa to give her such a son who couldn't be defeated by the gods. Kaśyapa granted her boon and told her that only Shiva had the power to destroy her son, as no one could match his power. Kaśyapa then touched her belly and a child was born with a thousand eyes and limbs. Although he wasn't blind, Andhaka walked like a blind person, thus bearing the name. Over time, Andhaka became arrogant since he cannot be harmed by anyone.

At one time, Andhaka went to the court of Indra, abducted all the Apsaras, and defeated all of the gods in battle. He also hindered the performance of yajñas by the gods. Being so tormented, the gods approached Nārada for assistance. Nārada visited Andhaka to counsel with him. Andhaka was intrigued by the fragrant garland of Mandara flowers worn by Nārada and inquired as to the source of the flowers. Nārada tells him that the flowers come from the Mandara forest, protected by guards so that no one may enter without Shiva's permission.

Andhaka conversed with Mount Mandara, informing the mountain of his invulnerability, and asked about the whereabouts of the forest. The mountain refuses to answer and disappears. Andhaka erupts in anger and uproots the entire mountain and with the help of his Asura army. They manage to grind the mountain down, destroying all of its natural beauty. Once Shiva become informed of this, he blesses the mountain with a boon which restored it to its original splendor. The mountain summits began to kill the Asuras who had attempted to destroy them. Upon seeing this, Andhaka calls out the owner of the mountain, wishing to burn the entire Mandara mountain. In response, Shiva, with his mace, mounts his bull and charges Andhaka. When Shiva releases his spear, it strikes the demon in the chest and reduces him to ash.

Liṅga Purāṇa
In the Liṅga Purāṇa, Andhaka's story is related as a variation of accounts of Harvamsa and Kurma Purana. Andhaka was the son of Hiraṇyākṣa and was given immunity to death by Brahmā after he performed a severe penance. He traversed the three worlds and conquered the heavens. The Devas retreated to Mount Mandara, where the demon followed them. Shiva, on the request of the Devas, challenges Andhaka, destroys his army and impales him on his trident. However, the divine sight of Shiva burns away all of Andhaka's sins, prompting the latter to eulogizing him. Shiva was pleased with his devotion and forgave him, making him the chief of all his Gaṇas.

It is here that the reader is informed that Andhaka's son is Āḍi.

Alternate account
Another version of the tale states that Andhaka fought with the Gaṇas and Shiva's sons, but that the gods lost the battle. Shiva sends Vīrabhadra to battle it out with Andhaka, but every time Vīrabhadra kills Andhaka, another form arises from his blood that falls upon the earth. Parvati becomes furious and assumes the form of Kālī. She destroys each and every copy of Andhaka, except for the real one, who is saved by Shiva and given a new life.

In the Rāmāyaṇa and Mahābhārata
In the Rāmāyaṇa, the story of Kālī killing Andhaka is briefly noted in Chapter 30 of the Araṇya Kāṇḍa, at the moment when Khara, the younger brother of Rāvaṇa is killed by Rāma. The scripture reads that Andhaka was killed by Shiva's third eye in the forest of the sage Śveta.

In the Mahābhārata, Andhaka is killed by Kālī, though not by his third eye as in the Rāmāyaṇa.

title: Andhaka


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Angiras

Angiras or Angira (अङ्गिरा,, ) was a Vedic rishi (sage) of Hinduism. He is described in the Rigveda as a teacher of divine knowledge, a mediator between men and gods, as well as stated in other hymns to be the first of Agni-devas (fire gods). In some texts, he is considered to be one of the seven great sages or Saptarishis, but in others he is mentioned but not counted in the list of seven great sages. In some manuscripts of Atharvaveda, the text is attributed to "Atharvangirasah", which is a compound of sage Atharvan and Angira. The student family of Angira are called "Angira", and they are credited to be the authors of some hymns in the first, second, fifth, eighth, ninth, and tenth books of the Rigveda. By the time of the composition of the Rigveda, the Angirases were an old Rishi clan, and were stated to have participated in several events.

Texts
Many hymns of the Rigveda credit the Angirases as their authors, mainly in Mandalas I and VIII. Various Angirasa sub-clans, including the Śunahotras, the Gautamas, and the Bhāradvājas composed Mandalas II, IV, and VI respectively.

Other than crediting authorship, the Vedic texts mention sage Angiras in various roles such as a fire priest or a singer. For example, the allegorical hymn 3.31 of the Rigveda calls him a singer:

"The most inspired one came, assuming a friendly attitude, The rock made ripe (its) fruit for the one who performs the kind deed, The young hero attained (his aim) with the youths, assuming a warlike attitude, And here right away, the singing Angiras appeared."

- Rigveda 3.31.7

According to Max Muller – a professor of Sanskrit and Indology at the Oxford University, the sage Angiras in Vedic literature is different from the plural term Angirasa, and these terms refer to different people. The Angiras rishi is different from the group of sorcerers in Atharvaveda also named Angirasa, and according to Muller, the Vedic rishi is also different from a class of divine beings who too are called Angirasa in the Vedic texts and described as "sprung from coals (angara)".

In Buddhist Pali canonical texts such as Digha Nikaya, Tevijja Sutta describes a discussion between the Buddha and Vedic scholars of his time. The Buddha names ten rishis, calls them "early sages" and makers of ancient verses that have been collected and chanted in his era, and among those ten rishis is Angiras.

Life in Puranas
The name Angirasas is applied generically to several Puranic individuals. Further, the Vedic sage Angiras appears in medieval Hindu texts with contradictory roles as well as many different versions of his birth, marriage and biography. In some, he is described to be the son of Brahma, in others he is one of many Prajapatis. Depending on the legend, he has one, two or four wives. In one myth, his wife is stated to be Surūpa and his sons are Utathya, Samvartana and Brahaspati. Other accounts say that he married Smriti (memory), the daughter of Daksha and later married Svadha (oblation). Yet other Puranic accounts state, he married Shubha and they had seven daughters named after aspects of "fire" and a son named Brihaspati. In some legends, sage Brihaspati is his son.

According to one legend, Angirasa turned his senses inwards and meditated on Para Brahman, the creator of the creator, for several years. The great Tejas he got by birth had multiplied infinitely by his penance. He attained many divine qualities, powers, and riches, and control over many worlds. But he was oblivious of all the worldly attainments and did not stop his penance. Due to this penance he became one with the Para-Brahman and thus attained the state of “Brahmarshi”. He had visions of many Vedic Mantras and brought them to this earthly world. He is credited as being the source of great number of Vedic Hymns and mantras and also believed to have introduced fire-worship along with sage Bhrigu.

He is one of Saptarishis in the Puranas.

Ghora of the Angiras family is identified by some scholars as Neminatha, the twenty-second tirthankara in Jainism.

title: Angiras


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Angiris

In Hindu mythology the Angiris (or Angiras) are a group of celestial beings who are descendants of the Fire God Agni and the Goddess Agneya, responsible for watching over humans performing Yagna (sacrifices) and protecting the sacrificial fires

title: Angiris


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Annapurna

Annapurna ( अन्नपूर्ण) is a mountain situated in the Annapurna mountain range of Gandaki Province, north-central Nepal. It is the 10th highest mountain in the world at 8091 m above sea level and is well known for the difficulty and danger involved in its ascent.

Maurice Herzog led a French expedition to its summit through the north face in 1950, making it the first eight-thousander to be successfully climbed. The entire massif and surrounding area are protected within the 7629 km2 Annapurna Conservation Area, the first and largest conservation area in Nepal. The Annapurna Conservation Area is home to several world-class treks, including Annapurna Sanctuary and Annapurna Circuit.

For decades, Annapurna I held the highest fatality-to-summit rate of all principal eight-thousander summits; it has, however, seen great climbing successes in recent years, with the fatality rate falling from 32% to under 20% from 2012 to 2022. This figure places it just under the most recent fatality rate estimates for K2, at about 24%. The mountain still poses grave threats to climbers through avalanche danger, unpredictable weather and the extremely steep and committing nature of its climbing routes, in particular its 3000 m south face, renowned as one of the most difficult climbs in the world. It is also a dangerous peak for trekkers, as in the case of a 2014 snowstorm near it and Dhaulagiri which claimed at least 43 lives. As of 2022, 365 people had reached the summit of Annapurna I, while 72 had died in the attempt.

Etymology
The mountain is named after Annapurna, the Hindu goddess of food and nourishment, who is said to reside there. The name Annapurna is derived from the Sanskrit-language words purna ("filled") and anna ("food"), and can be translated as "everlasting food". Many streams descending from the slopes of the Annapurna Massif provide water for the agricultural fields and pastures located at lower elevations.

Climbing expeditions

Annapurna I was the first 8000 m peak to be climbed. Maurice Herzog and Louis Lachenal, of the French Annapurna expedition led by Herzog (including Lionel Terray, Gaston Rébuffat, Marcel Ichac, Jean Couzy, Marcel Schatz, Jacques Oudot, Francis de Noyelle), reached the summit on 3 June 1950. Ichac made a documentary of the expedition, called Victoire sur l'Annapurna. Its summit was the highest summit attained till that time, but not the highest climb; higher non-summit points – at least 8500 m – had already been attained on Mount Everest in the 1920s.

The south face of Annapurna was first climbed in 1970 by Don Whillans and Dougal Haston also without using supplementary oxygen, members of a British expedition led by Chris Bonington that included Ian Clough, who was killed by a falling serac during the descent. They were, however, beaten to the second ascent of Annapurna by a matter of days by a British Army expedition led by Colonel Henry Day.

In 1978, the American Women's Himalayan Expedition, a team led by Arlene Blum, became the first United States team to climb Annapurna I. The first summit team, composed of Vera Komarkova and Irene Miller, and Sherpas Mingma Tsering and Chewang Ringjing, reached the top at 3:30 pm on 15 October 1978. The second summit team, Alison Chadwick-Onyszkiewicz and Vera Watson, died during this climb.

In 1981 Polish expedition Zakopane Alpine Club set a new route on Annapurna I Central (8051 m). Maciej Berbeka and Bogusław Probulski reached the summit on 23 May 1981. The route called Zakopiańczyków Way was recognized as the best achievement of the Himalayan season in 1981.

On 3 February 1987, Polish climbers Jerzy Kukuczka and Artur Hajzer made the first winter ascent of Annapurna I.

The first solo ascent of the south face was made in October 2007 by Slovenian climber Tomaž Humar;   he climbed to the Roc Noir and then to Annapurna East (8,047m).

On 8 and 9 October 2013 Swiss climber Ueli Steck soloed the Lafaille route on the main and highest part of the face; this was his third attempt on the route and has been called "one of the most impressive Himalayan climbs in history", with Steck taking 28 hours to make the trip from Base Camp to summit and back again.

Flights
Several airlines offer sightseeing flights over Annapurna.

Fatality rate
Along with K2 and Nanga Parbat, Annapurna I has consistently ranked as one of the most dangerous of the principal eight-thousander summits. Climbers killed on the peak include Britons Ian Clough in 1970 and Alex MacIntyre in 1982, Frenchman Pierre Béghin in 1992, Kazakh Russian Anatoli Boukreev in 1997, Spaniard Iñaki Ochoa in 2008, Korean Park Young-seok in 2011 and the Finn Samuli Mansikka in 2015.

title: Annapurna


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Anumati

Anumati may refer to :
 * Anumati (lunar phase), a phase of the moon
 * Anumati (deity), in Hinduism
 * Anumati (film), a 2013 Marathi film

title: Anumati


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Avatar

Avatar (अवतार, ; ) is a concept within Hinduism that in Sanskrit literally means descent. It signifies the material appearance or incarnation of a powerful deity, or spirit on Earth. The relative verb to "alight, to make one's appearance" is sometimes used to refer to any guru or revered human being.

The word avatar does not appear in the Vedic literature; however, it appears in developed forms in post-Vedic literature, and as a noun particularly in the Puranic literature after the 6th century CE. Despite that, the concept of an avatar is compatible with the content of the Vedic literature like the Upanishads as it is symbolic imagery of the Saguna Brahman concept in the philosophy of Hinduism. The Rigveda describes Indra as endowed with a mysterious power of assuming any form at will. The Bhagavad Gita expounds the doctrine of Avatara but with terms other than avatar.

Theologically, the term is most often associated with the Hindu god Vishnu, though the idea has been applied to other deities. Varying lists of avatars of Vishnu appear in Hindu scriptures, including the ten Dashavatara of the Garuda Purana and the twenty-two avatars in the Bhagavata Purana, though the latter adds that the incarnations of Vishnu are innumerable. The avatars of Vishnu are important in Vaishnavism theology. In the goddess-based Shaktism tradition of Hinduism, avatars of the Devi in different appearances such as Tripura Sundari, Durga, Chandi, Chamunda, Mahakali, and Kali are commonly found. While avatars of other deities such as Ganesha and Shiva are also mentioned in medieval Hindu texts, this is minor and occasional. The incarnation doctrine is one of the important differences between Vaishnavism and Shaivism traditions of Hinduism.

Incarnation concepts that are in some aspects similar to avatar are also found in Buddhism, Christianity, and other religions.

The scriptures of Sikhism include the names of numerous Hindu gods and goddesses, but it rejected the doctrine of savior incarnation and endorsed the view of Hindu Bhakti movement saints such as Namdev, that formless eternal god is within the human heart, and man is his own savior.

Etymology and meaning
The Sanskrit noun is derived from the Sanskrit prefix  down and the root  to cross over. These roots trace back, states Monier-Williams, to -taritum, -tarati, -rītum.

Avatar means descent, alight, to make one's appearance, and refers to the embodiment of the essence of a superhuman being or a deity in another form. The word also implies "to overcome, to remove, to bring down, to cross something". In Hindu traditions, the "crossing or coming down" is symbolism, states Daniel Bassuk, of the divine descent from "eternity into the temporal realm, from unconditioned to the conditioned, from infinitude to finitude". An avatar, states Justin Edwards Abbott, is a saguna (with form, attributes) embodiment of the nirguna Brahman or Atman (soul). Avatar, according to Bhaktisiddhanta Sarasvati actually means divine descent in his commentaries of The Shrimad Bhagavatam and The Bramha Samhita (mentioned in Brahmavaivarta Purana).

Neither the Vedas nor the Principal Upanishads ever mention the word avatar as a noun. The verb roots and form, such as, appear in ancient post-Vedic Hindu texts, but as "action of descending", but not as an incarnated person (avatara). The related verb is, states Paul Hacker, used with double meaning, one as action of the divine descending, another as "laying down the burden of man" suffering from the forces of evil.

The term is most commonly found in the context of the Hindu god Vishnu. The earliest mention of Vishnu manifested in a human form to establish Dharma on Earth, uses other terms such as the word sambhavāmi in verse 4.6 and the word tanu in verse 9.11 of the Bhagavad Gita, as well as other words such as akriti and rupa elsewhere. It is in medieval era texts, those composed after the sixth century CE, that the noun version of avatar appears, where it means embodiment of a deity. The idea proliferates thereafter, in the Puranic stories for many deities, and with ideas such as ansha-avatar or partial embodiments.

The term avatar, in colloquial use, is also an epithet or a word of reverence for any extraordinary human being who is revered for his or her ideas. In some contexts, the term avatara just means a landing place, site of sacred pilgrimage, or just achieve one's goals after effort, or retranslation of a text in another language. The term avatar is not unique to Hinduism even though the term originated with Hinduism. It is found in the Trikaya doctrine of Mahayana Buddhism, in descriptions for the Dalai Lama in Tibetan Buddhism, and many ancient cultures.

Avatar versus incarnation
The manifest embodiment is sometimes referred to as an incarnation. The translation of avatar as "incarnation" has been questioned by Christian theologians, who state that an incarnation is in flesh and imperfect, while avatar is mythical and perfect. The theological concept of Christ as an incarnation, as found in Christology, presents the Christian concept of incarnation. The term avatar in Hinduism refers to act of various gods taking form to perform a particular task which in most of the times is bringing dharma back. The concept of avatar is widely accepted all over the India. Sheth disagrees and states that this claim is an incorrect understanding of the Hindu concept of avatar. Avatars are embodiments of spiritual perfection, driven by noble goals, in Hindu traditions such as Vaishnavism. The concept of the avatar in Hinduism is not incompatible with natural conception through a sexual act, which is again different from the Christian concept of the Virgin Birth.

As a loanword
Following 19th Century Western interest in Indian culture and Hinduism, the word "Avatar" was taken as loanword into English and other Western languages, where it is used in various contexts and meanings, often considerably different from its original meaning in Hinduism - see Avatar (disambiguation).

Avatars of Vishnu
The concept of avatar within Hinduism is most often associated with Vishnu, the preserver or sustainer aspect of God within the Hindu Trinity or Trimurti of Brahma, Vishnu and Shiva. Vishnu's avatars descend to empower the good and fight evil, thereby restoring Dharma. Traditional Hindus see themselves not as "Hindu", but as Vaishnava (Worshippers of Vishnu), Shaiva (Worshippers of Shiva), or Shakta (Worshipper of the Shakti). Each of the deities has its own iconography and mythology, but common to all is the fact that the divine reality has an explicit form, a form that the worshipper can behold. An oft-quoted passage from the Bhagavad Gita describes the typical role of an avatar of Vishnu:

The Vishnu avatars appear in Hindu mythology whenever the cosmos is in crisis, typically because evil has grown stronger and has thrown the cosmos out of its balance. The avatar then appears in a material form, to destroy evil and its sources, and restore the cosmic balance between the ever-present forces of good and evil.

The most known and celebrated avatars of Vishnu, within the Vaishnavism traditions of Hinduism, are Krishna, Rama, Narayana and Vasudeva. These names have extensive literature associated with them, each has its own characteristics, legends and associated arts. The Mahabharata, for example, includes Krishna, while the Ramayana includes Rama.

Dashavatara
The Bhagavata Purana describes Vishnu's avatars as innumerable, though ten of his incarnations, the Dashavatara, are celebrated therein as his major appearances. The ten major Vishnu avatars are mentioned in the Agni Purana, the Garuda Purana and the Bhagavata Purana.

The ten best known avatars of Vishnu are collectively known as the Dashavatara (a Sanskrit compound meaning "ten avatars"). Five different lists are included in the Bhagavata Purana, where the difference is in the sequence of the names. Freda Matchett states that this re-sequencing by the composers may be intentional, so as to avoid implying priority or placing something definitive and limited to the abstract.

Longer alternatives
The Bhagavata Purana also goes on to give an alternate list, wherein it numerically lists out 23 Vishnu avatars in chapter 1.3.

 * 1) Four Kumaras (Catuḥsana): the four sons of Brahma who exemplify the path of devotion.
 * 2) Varaha: The boar avatar. He rescues Bhumi, the goddess of the earth, when the asura Hiranyaksha abducts her, restoring her rightful place in the universe.
 * 3) Narada: the divine-sage who travels the worlds as a devotee of Vishnu.
 * 4) Nara-Narayana: the twin-sages.
 * 5) Kapila: a renowned sage spoken of in the Mahabharata, son of Kardama and Devahuti. He is sometimes identified with the founder of the Samkhya school of philosophy.
 * 6) Dattatreya: the combined avatar of the Hindu trinity of Brahma, Vishnu, and Shiva.
 * 7) Yajna: the embodiment of sacrifices.
 * 8) Rishabha: the father of emperor Bharata.
 * 9) Prithu: the sovereign-king who milked the earth as a cow to obtain the world's grain and vegetation.
 * 10) Matsya: The fish avatar. He saves Manu and the seven sages from the cosmic flood, and in some traditions, saves the Vedas from an asura called Hayagriva.
 * 11) Kurma: The tortoise/turtle avatar. He supports the mountain named Mandara while the devas and the asuras churn the ocean of milk to produce the nectar of immortality.
 * 12) Dhanvantari: the father of Ayurvedic medicine and a physician to the devas.
 * 13) Mohini: the enchantress who beguiles the asuras into offering her the elixir of eternal life.
 * 14) Narasimha: The lion avatar. He saves his devotee Prahlada and frees the three worlds from the tyranny of an asura named Hiranyakashipu.
 * 15) Vamana: The dwarf avatar. He vanquishes the asura king Mahabali to the netherworld after taking three strides upon the universe, restoring the rule of Indra.
 * 16) Parashurama: The warrior-sage avatar. He destroys the oppressive kings of the military class and creates a new social order.
 * 17) Rama: The prince avatar. He rescues his wife Sita when she is abducted by the rakshasa king Ravana, restoring just rule to the world.
 * 18) Vyasa: the compiler of the Vedas and writer of the scriptures (Puranas) and the epic Mahabharata.
 * 19)  Krishna: The eighth avatar of Vishnu who incarnates to re-establish righteousness in the world. He slays Kamsa, the tyrant of Mathura and his uncle, and participates in the Kurukshetra War as the charioteer of Arjuna.
 * 20) Gautama Buddha: The historical Buddha, who incarnates to delude the asuras from the path of the Vedas, ensuring the victory of the devas. In some traditions, he is referred to as an avatar of Vishnu.
 * 21) Kalki: The prophesied tenth avatar of Vishnu. He incarnates to bring an end to the present age of corruption called the Kali Yuga, re-establishing the four classes and law to the world.

Avatars like Hayagriva, Hamsa ,and Garuda are also mentioned in the Pancharatra ,making a total of forty-six avatars. However, despite these lists, the commonly accepted number of ten avatars for Vishnu was fixed well before the 10th century CE. Madhvacharya also regards Gautama Buddha as an avatar of Vishnu.

Manava Purana

Manava Purana is one of Upapuranas. It lists 42 avatars of Vishnu.

 * 1) Adipurusha
 * 2) Four Kumaras (Catuḥsana)
 * 3) Narada
 * 4) Dattatreya
 * 5) Kapila
 * 6) Nara- Narayana
 * 7) Yajna
 * 8) Vibhu
 * 9) Satyasena
 * 10) Hari
 * 11) Vaikunta
 * 12) Ajita
 * 13) Sharvabhouma
 * 14) Vrishbha
 * 15) Visvaksena
 * 16) Dharmasetu
 * 17) Sudhama
 * 18) Yogeshwara
 * 19) Brihadbhanu
 * 20) Shaligram
 * 21) Hayagriva
 * 22) Hamsa
 * 23) Vyasa
 * 24) Matsya
 * 25) Kurma
 * 26) Dhanvantri
 * 27) Mohini
 * 28) Prithu
 * 29) Vrishbha deva
 * 30) Varaha
 * 31) Narasimha
 * 32) Vamana
 * 33) Parashurama
 * 34) Rama
 * 35) Krishna
 * 36) Buddha
 * 37) Vikhanasa
 * 38) Venkateswara
 * 39) Chaitanya Mahaprabhu
 * 40) Dhyaneshwar
 * 41) Kalki

Types
The avatar concept was further developed and refined in later Hindu texts. One approach was to identify full avatars and partial avatars. Krishna, Rama, and Narasimha were full avatars (purna avatars), while others were partial avatars (ansha avatars). Some declared, states Noel Sheth, that every living creature is an avatar of Vishnu. The Pancharatra text of Vaishnavism declares that Vishnu's avatars include those that are direct and complete, indirect and endowed , cosmic and salvific , inner and inspirational , consecrated and in the form of image.

Yet another classification, developed in Krishna schools, centers around Guna-avatars, Purusha-avatars and Lila-avatars, with their subtypes. The Guna-avatar classification of avatars is based on the Guṇas concept of the Samkhya school of Hindu philosophy, that is Rajas (Brahma), Sattva (Vishnu), and Tamas (Shiva). These personalities of the Trimurti are referred to as Guna avatars. The Purushavatara are three. The first evolves all matter (Prakriti), the second is the soul present in each individual creature, the third is the interconnected oneness or Brahman that connects all souls. The Lilavataras are partial or full manifestations of Vishnu, where either some powers (Shakti) or material parts of him exist.

Vishnu is Purushavatara. The Matsya, Kurma, and Vamana avatars of Vishnu are Lilavataras. A Purnarupa in this classification, is when Vishnu manifests completely along with his qualities and powers. In Bengal Vaishnavism, Krishna is the Purnarupa. In Shaivism, Bhairava is the purnarupa of Shiva.

In Sikhism
24 avatars of Vishnu are mentioned in Bachitar Natak's composition in Dasam Granth, the second scripture of Sikhism written by Guru Gobind Singh:

• # Mach (Matsya)

• # Kach (Kurma)

• # Nara (Nara in Nara-Narayana)

• # Narayan (Narayana in Nara-Narayana)

• # Maha Mohini (Mohini)

• # Bairaha (Varaha)

• # Nar Singha (Narasimha)

• # Baman (Vamana)

• # Parshuram (Parashurama)

• # Bramma (Brahma)

• # Balram (Balarama)

• # Jalandhar (Jalandhara)

• # Bishan (Vishnu)

• # Sheshayi (Shesha)

• # Arihant Dev (Arihanta)

• # Manu Raja (Manu)

• # Dhanvantari (Dhanvantari)

• # Suraj (Surya)

• # Chandar (Chandra)

• # Ram (Rama)

• # Kishan (Krishna)

• # Nar (Arjuna)

• # Rudra (Shiv)

• # Kalki (Kalki)

The Guru Granth Sahib reverentially includes the names of numerous Hindu deities, including Vishnu avatars such as Krishna, Hari, and Rama, as well those of Devi as Durga.

Dasam Granth has three major compositions, one each dedicated to avatars of Vishnu (Chaubis avatar) and Brahma. However, Sikhism rejects the doctrine of savior incarnation, and only accepts the abstract nirguna formless god. The Sikh Gurus endorsed the view of Hindu Bhakti movement saints such as Namdev (≈1270 – 1350 CE) that formless eternal god is within the human heart and man is his own savior.

In Isma'ilism
The Gupti Ismailis, who observe pious circumspection as Hindus, uphold that the first Shi‘i Imam, ‘Ali b. Abi Talib, as well as his descendants through the line of Isma‘il, are collectively Kalki, the tenth and final avatāra of Vishnu. According to this interpretation, these figures represent the continuity of divine guidance to humankind. In the view of some Guptis, this is corroborated by the Quranic verse 14:4 which mentions the idea that God had sent a messenger to every land. They understand the avatāras to be these messengers sent by God to their people in the Indian subcontinent.

Avatars of Shiva

Although Puranic scriptures contain occasional references to avatars of Shiva, the avatar doctrine is neither universally accepted nor commonly adopted in Shaivism. The views on the doctrine of incarnation has been one of the significant doctrinal differences between Vaishnavism and Shaivism, in addition to their differences on the role of householder life versus monastic life for spiritual release. Shaivism is a transcendental theology, where man, with the help of his Guru, is his own savior.

The Linga Purana lists twenty-eight avatars of Shiva. In the Shiva Purana there is a distinctly Saivite version of a traditional avatar myth: Shiva brings forth Virabhadra, one of his terrifying forms, in order to calm Narasimha, an avatar of Vishnu. When that fails, Shiva manifests as the human-lion-bird Sharabha which calms down lion-man Narasimha avatar of Vishnu, and Shiva then gives Vishnu a chakra (not to be confused with Sudarshan Chakra) as gift. A similar story is told in the late medieval era Sharabha Upanishad. However, Vaishnava Dvaita school refutes this Shaivite view of Narasimha. According to the Shiva Purana, Shiva has 19 avatars. According to the Kurma Purana, he has 28.

The vanara god Hanuman who helped Rama (the Vishnu avatar) is considered by some to be the eleventh avatar of Rudra (Shiva). Some regional deities like Khandoba are also believed by some to be avatars of Shiva. Ashwatthama, the son of Drona is also considered to be an avatar of Shiva.

Shesha and his avatars (Balarama and Lakshmana) are occasionally linked to Shiva. Adi Shankara, the formulator of Advaita Vedanta, is also occasionally regarded as an avatar of Shiva.

In Dasam Granth, Guru Gobind Singh mentioned two avatars of Rudra: Dattatreya Avatar and Parasnath Avatar.

Avatars of Devis
Avatars are also observed in Shaktism, the sect dedicated to the worship of the Goddess (Devi), but they do not have universal acceptance in the sect. The Devi Bhagavata Purana describes the descent of Devi avatars to punish the wicked and defend the righteous as – much as the Bhagavata Purana does with the avatars of Vishnu.

Nilakantha, an 18th-century commentator on the Devi Bhagavata Purana – which includes the Devi Gita – says that various avatars of the Goddess includes Shakambhari and even the masculine Krishna and Rama – generally thought to be Vishnu's avatars. Parvati, Lakshmi and Saraswati are main goddesses worshipped as Devi avatars.

Avatars of Parvati
Devi is popular in her form as Parvati. In Devi Mahatmya she is seen as the Goddess Mahakali, and in Uma Samhita, she is seen as Devi herself. Regarding her incarnations, it varies per sect in Hinduism. She could be all Goddesses as said in Shaivism and some main Shatism interpretations like the Sri kula and Kali Kula families, or just a form of Devi in some other Shaktism interpretations and many Vaishnava interpretations. With this in mind, Parvati's forms include:
 * Mahakali
 * Sati
 * Mahavidyas- Kali, Tara, Tripurasundari, Bhuvaneshwari, Bhairavi, Chinnamasta, Dhumavati, Bagalamukhi
 * The Shakti Peethas
 * Navadurgas - Shailaputri, Brahmacharini, Chandraghanta, Kushmanda, Skandamata, Katyayani, Kaalratri, Mahagauri, Siddhidhatri
 * Annapurna
 * Kaushiki
 * Durga
 * Shitala
 * Maheshwari
 * Ardhanarishvara
 * Vindhyavasini
 * Bhramari
 * Shakambhari
 * Raktadantika
 * Twarita
 * In North India- Khodiyar, Bahuchara Mata, Naina Devi, Jwala etc.
 * In Western India- Bhavani, Saptashrungi, Ekvira, Amba etc.
 * In East India-Dakshina Kali, Kamakhya, Jagadhatri etc.
 * In South India- Meenakshi, Vishalakshi, Kamakshi, Abhirami, Mookambika, Akilandeswari, Mariamman, Karumariamman etc.

All of these incarnations helped provide security to the world and even brought Shiva into the participation of worldly affairs.

Avatars of Lakshmi
Like Vishnu, his consort Lakshmi incarnates as in many forms to help provide order and to enlighten the world with her consort. She has many forms, and just like Parvati, some of her forms are not consistent throughout all sects and interpretations of Hinduism. In Vaishnavism and some interpretations of Shaktism, Lakshmi is seen as Devi herself. She could be every Goddess as said in Vaishnavism and some interpretations of Shaktism, or just another form of Devi as seen in other interpretations of Shaktism and in Shaivism. With this in mind, Lakshmi's forms include:
 * Mahalakshmi
 * At 108 Divya Desam
 * Ashtalakshmi - Adi Lakshmi, Dhanya Lakshmi, Dhana Lakshmi, Gaja Lakshmi, Santana Lakshmi, Vidya Lakshmi, Veera Lakshmi, Vijaya Lakshmi
 * With avatars of Vishnu - Sita, Radha, Gopi, Rukmini, Ashtabharya, Junior wives of Krishna, Padmavathi, Lakshmi Narayana, Andal, Dharini
 * Singular forms - Vedavati, Chottanikkara Bhagavathy, Mookambika devi, Vaishno Devi, As Kolhapur ambabai
 * Matrikas - Vaishnavi, Varahi, Narasimhi
 * Wives of Vishnu - Sri, Bhumi, Nila
 * Mahavidya Kamalatmika

Avatars of Brahma
In Dasam Granth, second scriptures of Sikhs written by Guru Gobind Singh, mentioned seven Brahma Avatars. Khat avatar in this list refers to six different scholars who are considered to be founders of six schools of Indian philosophy. According to the Skanda Purana, Brahma incarnated himself as Yajnavalkya in response to a curse from Shiva.
 * 1) Valmiki
 * 2) Kashyapa
 * 3) Shukra
 * 4) Baches
 * 5) Vyasa
 * 6) Khat
 * 7) Kalidasa

Avatars of Ganesha
The Linga Purana declares that Ganesha incarnates to destroy demons and to help the gods and pious people. The two Upapuranas – Ganesha Purana and Mudgala Purana – detail the avatars of Ganesha. Both these upapuranas are core scriptures of the Ganapatya sect – exclusively dedicated to Ganesha worship.

Four avatars of Ganesha are listed in the Ganesha Purana: Mohotkata, Mayūreśvara, Gajanana and Dhumraketu. Each avatar corresponds to a different yuga, has a different mount and different skin complexion, but all the avatars have a common purpose – to slay demons.

The Mudgala Puranam describes eight avatars of Ganesha:
 * 1) Vakratunda ("twisting trunk"), his mount is a lion.
 * 2) Ekadanta ("single tusk"), his mount is a mouse.
 * 3) Mahodara ("big belly"), his mount is a mouse.
 * 4) Gajavaktra (or Gajānana) ("elephant face"), his mount is a mouse.
 * 5) Lambodara ("pendulous belly"), his mount is a mouse.
 * 6) Vikata ("unusual form", "misshapen"), his mount is a peacock.
 * 7) Vighnaraja ("king of obstacles"), his mount is the celestial serpent.
 * 8) Dhumravarna ("grey color") corresponds to Śiva, his mount is a horse.

Avatars of Varuna
Jhulelal, the Iṣṭa-devatā (most-revered deity) of Sindhi Hindus, is considered the incarnation of Varuna.

General bibliography

 * Short introduction and bibliography of sources about Avatāra.

title: Avatar


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Balance sheet

In financial accounting, a balance sheet (also known as statement of financial position or statement of financial condition) is a summary of the financial balances of an individual or organization, whether it be a sole proprietorship, a business partnership, a corporation, private limited company or other organization such as government or not-for-profit entity. Assets, liabilities and ownership equity are listed as of a specific date, such as the end of its financial year. A balance sheet is often described as a "snapshot of a company's financial condition". It is the summary of each and every financial statement of an organization.

Of the four basic financial statements, the balance sheet is the only statement which applies to a single point in time of a business's calendar year.

A standard company balance sheet has two sides: assets on the left, and financing on the right–which itself has two parts; liabilities and ownership equity. The main categories of assets are usually listed first, and typically in order of liquidity. Assets are followed by the liabilities. The difference between the assets and the liabilities is known as equity or the net assets or the net worth or capital of the company and according to the accounting equation, net worth must equal assets minus liabilities.

Another way to look at the balance sheet equation is that total assets equals liabilities plus owner's equity. Looking at the equation in this way shows how assets were financed: either by borrowing money (liability) or by using the owner's money (owner's or shareholders' equity). Balance sheets are usually presented with assets in one section and liabilities and net worth in the other section with the two sections "balancing".

A business operating entirely in cash can measure its profits by withdrawing the entire bank balance at the end of the period, plus any cash in hand. However, many businesses are not paid immediately; they build up inventories of goods and acquire buildings and equipment. In other words: businesses have assets and so they cannot, even if they want to, immediately turn these into cash at the end of each period. Often, these businesses owe money to suppliers and to tax authorities, and the proprietors do not withdraw all their original capital and profits at the end of each period. In other words, businesses also have liabilities.

Types
A balance sheet summarizes an organization's or individual's assets, equity and liabilities at a specific point in time. Two forms of balance sheet exist. They are the report form and account form. Individuals and small businesses tend to have simple balance sheets. Larger businesses tend to have more complex balance sheets, and these are presented in the organization's annual report. Large businesses also may prepare balance sheets for segments of their businesses. A balance sheet is often presented alongside one for a different point in time (typically the previous year) for comparison.

Personal
A personal balance sheet lists current assets such as cash in checking accounts and savings accounts, long-term assets such as common stock and real estate, current liabilities such as loan debt and mortgage debt due, or overdue, long-term liabilities such as mortgage and other loan debt. Securities and real estate values are listed at market value rather than at historical cost or cost basis. Personal net worth is the difference between an individual's total assets and total liabilities.

US small business
A small business balance sheet lists current assets such as cash, accounts receivable, and inventory, fixed assets such as land, buildings, and equipment, intangible assets such as patents, and liabilities such as accounts payable, accrued expenses, and long-term debt. Contingent liabilities such as warranties are noted in the footnotes to the balance sheet. The small business's equity is the difference between total assets and total liabilities.

Charities
In England and Wales, smaller charities which are not also companies are permitted to file a statement of assets and liabilities instead of a balance sheet. This statement lists the charity's main assets and liabilities as at the end of its financial year.

Public business entities structure
Guidelines for balance sheets of public business entities are given by the International Accounting Standards Board and numerous country-specific organizations/companies. The standard used by companies in the US adheres to U.S. Generally Accepted Accounting Principles (GAAP). The Federal Accounting Standards Advisory Board (FASAB) is a United States federal advisory committee whose mission is to develop generally accepted accounting principles (GAAP) for federal financial reporting entities.

Balance sheet account names and usage depend on the organization's country and the type of organization. Government organizations do not generally follow standards established for individuals or businesses.

If applicable to the business, summary values for the following items should be included in the balance sheet: Assets are all the things the business owns. This will include property, tools, vehicles, furniture, machinery, and so on.

Assets
Current assets
 * 1) Accounts receivable
 * 2) Cash and cash equivalents
 * 3) Inventories
 * 4) Cash at bank, Petty Cash, Cash On Hand
 * 5) Prepaid expenses for future services that will be used within a year
 * 6) Revenue Earned In Arrears (Accrued Revenue) for services done but not yet received for the year
 * 7) Loan To (Less than one financial period)

Non-current assets (Fixed assets)
 * 1) Property, plant and equipment
 * 2) Investment property, such as real estate held for investment purposes
 * 3) Intangible assets, such as patents, copyrights and goodwill
 * 4) Financial assets (excluding investments accounted for using the equity method, accounts receivables, and cash and cash equivalents), such as notes receivables
 * 5) Investments accounted for using the equity method
 * 6) Biological assets, which are living plants or animals. Bearer biological assets are plants or animals which bear agricultural produce for harvest, such as apple trees grown to produce apples and sheep raised to produce wool.
 * 7) Loan To (More than one financial period)

Liabilities

 * 1) Accounts payable
 * 2) Provisions for warranties or court decisions (contingent liabilities that are both probable and measurable)
 * 3) Financial liabilities (excluding provisions and accounts payables), such as promissory notes and corporate bonds
 * 4) Liabilities and assets for current tax
 * 5) Deferred tax liabilities and deferred tax assets
 * 6) Unearned revenue for services paid for by customers but not yet provided
 * 7) Interests on loan stock
 * 8) Creditors' equity

Net current assets
Net current assets means current assets minus current liabilities.

Equity / capital
The net assets shown by the balance sheet equals the third part of the balance sheet, which is known as the shareholders' equity. It comprises:
 * 1) Issued capital and reserves attributable to equity holders of the parent company (controlling interest)
 * 2) Non-controlling interest in equity

Formally, shareholders' equity is part of the company's liabilities: they are funds "owing" to shareholders (after payment of all other liabilities); usually, however, "liabilities" are used in the more restrictive sense of liabilities excluding shareholders' equity. The balance of assets and liabilities (including shareholders' equity) is not a coincidence. Records of the values of each account in the balance sheet are maintained using a system of accounting known as double-entry bookkeeping. In this sense, shareholders' equity by construction must equal assets minus liabilities, and thus the shareholders' equity is considered to be a residual. Regarding the items in the equity section, the following disclosures are required:
 * 1) Numbers of shares authorized, issued and fully-paid, and issued but not fully paid
 * 2) Par value of shares
 * 3) Reconciliation of shares outstanding at the beginning and the end of the period
 * 4) Description of rights, preferences, and restrictions of shares
 * 5) Treasury shares, including shares held by subsidiaries and associates
 * 6) Shares reserved for issuance under options and contracts
 * 7) A description of the nature and purpose of each reserve within owners' equity

Substantiation
Balance sheet substantiation is the accounting process conducted by businesses on a regular basis to confirm that the balances held in the primary accounting system of record (e.g. SAP, Oracle, other ERP system's General Ledger) are reconciled (in balance with) with the balance and transaction records held in the same or supporting sub-systems.

Balance sheet substantiation includes multiple processes including reconciliation (at a transactional or at a balance level) of the account, a process of review of the reconciliation and any pertinent supporting documentation and a formal certification (sign-off) of the account in a predetermined form driven by corporate policy.

Balance sheet substantiation is an important process that is typically carried out on a monthly, quarterly and year-end basis. The results help to drive the regulatory balance sheet reporting obligations of the organization.

Historically, balance sheet substantiation has been a wholly manual process, driven by spreadsheets, email and manual monitoring and reporting. In recent years software solutions have been developed to bring a level of process automation, standardization and enhanced control to the balance sheet substantiation or account certification process. These solutions are suitable for organizations with a high volume of accounts and/or personnel involved in the Balance Sheet Substantiation process and can be used to drive efficiencies, improve transparency and help to reduce risk.

Balance sheet substantiation is a key control process in the SOX 404 top-down risk assessment.

Sample
The following balance sheet is a very brief example prepared in accordance with IFRS. It does not show all possible kinds of assets, liabilities and equity, but it shows the most usual ones. Because it shows goodwill, it could be a consolidated balance sheet. Monetary values are not shown, summary (subtotal) rows are missing as well.

Under IFRS items are always shown based on liquidity from the least liquid assets at the top, usually land and buildings to the most liquid, i.e. cash. Then liabilities and equity continue from the most immediate liability to be paid (usual account payable) to the least i.e. long-term debt such as mortgages and owner's equity at the very bottom.

Consolidated Statement of Finance Position of XYZ, Ltd.  As of 31 December 2025

ASSETS Non-Current Assets (Fixed Assets) Property, Plant and Equipment (PPE) Less : Accumulated Depreciation Goodwill Intangible Assets (Patent, Copyright, Trademark, etc.) Less : Accumulated Amortization Investments in Financial assets due after one year Investments in Associates and Joint Ventures Other Non-Current Assets, e.g. Deferred Tax Assets, Lease Receivable and Receivables due after one year Current Assets Inventories Prepaid Expenses Investments in Financial assets due within one year Non-Current and Current Assets Held for sale Accounts Receivable (Debtors) due within one year Less : Allowances for Doubtful debts Cash and Cash Equivalents

TOTAL ASSETS (this will match/balance the total for Liabilities and Equity below)

LIABILITIES and EQUITY Current Liabilities (Creditors: amounts falling due within one year) Accounts Payable Current Income Tax Payable Current portion of Loans Payable Short-term Provisions Other Current Liabilities, e.g. Deferred income, Security deposits Non-Current Liabilities (Creditors: amounts falling due after more than one year) Loans Payable Issued Debt Securities, e.g. Notes/Bonds Payable Deferred Tax Liabilities Provisions, e.g. Pension Obligations Other Non-Current Liabilities, e.g. Lease Obligations EQUITY Paid-in Capital Share Capital (Ordinary Shares, Preference Shares) Share Premium Less: Treasury Shares Retained Earnings '''Revaluation Reserve Other Accumulated Reserves Accumulated Other Comprehensive Income Non-Controlling Interest

TOTAL LIABILITIES and EQUITY (this will match/balance the total for Assets above)

title: Balance sheet


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Apalala

Apalāla is a water-dwelling Nāga in Buddhist mythology. It is said that Apalāla lived near the Swat River, this area is currently located in Peshawar, Khyber-Pakhtunkhwa Province, Pakistan. He is known to be a Naga King.

Apalāla was converted to Buddhism by the Buddha; this is one of the most popular legends in Buddhist lore and art. The tale is often told to children of Buddhist parents for them to learn their happiness lies in the Buddhist faith.

In Buddhist texts
The story of Apalāla's conversion (Pali: Apalāladamana) does not seem to be found in the Pali Canon, although his name does appear with other beings that honor the Buddha.

The Samantapāsādikā mentions that this story was among those not included in the Three Councils. It is evidenced that it was known in Sri Lanka as it is mentioned among the scenes depicted in the relic-chamber of the Mahāthūpa. The Divyāvadāna also mentions that Apalāla's conversion took place shortly before the Buddha's death.

title: Apalala


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Apam Napat

Apam Napat is a deity in the Indo-Iranian pantheon associated with water. His names in the Vedas, Apām Napāt, and in Zoroastrianism, Apąm Napāt, mean "child of the waters" in Sanskrit and Avestan respectively. Napāt ("grandson", "progeny") is cognate with Latin nepos and English nephew. In the Rig Veda, he is described as the creator of all things. It is considered to originate from the Proto-Indo-European Hepom Nepōts.

In the Vedas it is often apparent that Apām Napāt is being used as a title, not a proper name. This is most commonly applied to Agni, god of fire, and occasionally to Savitr, god of the sun. A correspondence has also been posited by Boyce between both the Vedic and Avestic traditions of Apam Napat, and Varuna, who is also addressed as "Child of the Waters", and is considered a god of the sea. In the Iranian tradition, he is also called Burz ("high one," ) and is a yazad.

Role
As a member of the Iranian ahuric triad, along with Ahura Mazda and Mithra, Apąm Napāt – also named Ahura Berezaiti – is an exalted figure. In Yasht 19 of the Zoroastrian Avesta Apąm Napāt appears as the creator of mankind. However, since in Zoroastrianism Ahura Mazdā is venerated as supreme creator, this function of Apąm Napāt has become reduced. This is one reason Apąm Napāt is no longer widely worshipped, though he is still honoured daily through the Zoroastrian liturgies. The creator-god status is also seen in a hymn in honour of the Vedic Apām Napāt.

Alongside Mithra, Apąm Napāt maintains order in society, as well as Khvarenah, by which legitimate rule is maintained among the Iranian peoples. It is his duty to distribute water from the sea to all regions.

Fire and water
In one Vedic hymn Apām Napāt is described as emerging from the water, golden, and "clothed in lightning", which has been conjectured to be a reference to fire. His regular identification with Agni, who is described a number of times as hiding or residing in water,   and comparison with other Indo-European texts, has led some to speculate about the existence of a Proto-Indo-European myth featuring a fire deity born from water.

Other such mentions include the ninth-century Skaldic poem Ynglingatal, which uses the kenning sævar niðr 'kinsman of the sea' to refer to fire, and an old Armenian poem in which a reed in the middle of the sea spontaneously catches fire, from which springs the hero Vahagn, with fiery hair and eyes that blaze like sun.

Conjectured original fireless myth
Whether fire was an original part of Apam Napat's nature remains a matter of debate, especially since this connection is absent from the Iranian version. Hermann Oldenberg believed Apam Napat was originally an independent water deity who later came to be associated with Agni, in part because of an ancient Indian belief that water contained fire within itself, fire appearing to "enter into" water when quenched by it.

Associations with Savitr could be understood as similarly deriving from an image of the setting sun sinking into the ocean. Another theory explains the connection between fire and water through lightning, "the flash of fire born from the rainbearing clouds".

'Swamp gas' conjecture
Based on the idea that this fire-from-water image was inspired by flaming seepage natural gas, attempts have been made to connect the name "Apam Napat" to the word "naphtha", which passed into Greek – and thence English – from an Iranian language.

However, there is only a modest amount of evidence for a link between the sacred fires of Iranian religion and petroleum or natural gas – although the account of the blowing of the 3 sacred fires out to sea from the back of the ox Srishok where, unquenched, they continue to burn on the water is suggestive – particularly in relation to hydrocarbon deposits in the Southwestern part of the Caspian Sea, exploited currently by the Absheron gas field near Baku in Azerbaijan.

The etymology of the word "naphtha" has been claimed likely to relate to the Akkadian napṭu, "petroleum".

title: Apam Napat