The original proof is based on the Taylor series expansions of the exponential function e < sup > z </ sup > ( where z is a complex number ) and of sin x and cos x for real numbers x ( see below ).

In mathematics, the exponential function is the function e < sup > x </ sup >, where e is the number ( approximately 2. 718281828 ) such that the function e < sup > x </ sup > is its own derivative.

According to Technorati, the number of blogs doubles about every 6 months with a total of 35. 3 million blogs. This is an example of the early stages of logistic growth, where growth is approximately exponential, since blogs are a recent innovation.

where now A ( k < sub > 1 </ sub >) ( the integral is the inverse fourier transform of A ( k1 )) is a function exhibiting a sharp peak in a region of wave vectors Δk surrounding the point k < sub > 1 </ sub > = k. In exponential form:

The exponential function inside the integral for ψ oscillates rapidly with its argument, say φ ( k < sub > 1 </ sub >), and where it varies rapidly, the exponentials cancel each other out, interfere destructively, contributing little to ψ.

However, an exception occurs at the location where the argument φ of the exponential varies slowly.

where D denotes the continuum derivative operator, mapping f to its derivative f. The expansion is valid when both sides act on analytic functions, for sufficiently small h. Thus, T < sub > h </ sub >= e < sup > D </ sup >, and formally inverting the exponential yields

Futurists such as Ray Kurzweil, Bruce Sterling, and Vernor Vinge believe that the exponential improvement described by Moore's law will ultimately lead to a technological singularity: a period where progress in technology occurs almost instantly.

That is, where the number of elements which are exposed to mutation is large there is often an exponential increase in search space size.

where the last right-hand term of the previous equation is easily obtained by remembering that the Fourier transform of a Dirac pulse is a complex exponential function, an eigenfunction of every linear system.

has an exponential distribution, where F < sup > − 1 </ sup > is the quantile function, defined by

where the expectation is taken with respect to the exponential distribution with rate parameter, and is the digamma function.

On the other hand, if T is written as a unary number ( a string of n ones, where n = T ), then it only takes time n. By writing T in unary rather than binary, we have reduced the obvious sequential algorithm from exponential time to linear time.

where is the starting frequency ( at ), and is the rate of exponential increase in frequency.

namely the branches of the inverse relation of the function f ( w ) = we < sup > w </ sup > where e < sup > w </ sup > is the exponential function and w is any complex number.

While the decay of a single atom occurs spontaneously, the decay of an initial population of identical atoms over time, t, follows a decaying exponential distribution, e < sup >− λt </ sup >, where λ is called a decay constant.

where is the entire exponential integral.

where the exponential is defined by,

As the total biomass in a specific area becomes depleted to the point where the entire sustainability of the biomass is threatened, then the exponential part of the curve comes into play.

For the element A ( t ) from the algebra ( set g with the non-commutative product *), where t is the " time parameter ", the ordered exponential

The radius of the largest ball about the origin in T < sub > p </ sub > M that can be mapped diffeomorphically via exp < sub > p </ sub > is called the injectivity radius of M at p. The cut locus of the exponential map is, roughly speaking, the set of all points where the exponential map fails to have a unique minimum.

Extreme examples of the behaviour is the Ornstein – Uhlenbeck covariance function and the squared exponential where the former is never differentiable and the latter infinitely differentiable.

By using the de Sitter universe instead, where the expansion is truly exponential, there are many simplifications.

There was only one place where Jake Carwood's description had gone badly awry: the peace and quiet.

There was only one place where the mountain might receive her -- that unnamed, unnameable pool harbored in its secret bosom.

They were west of the Sabine, but only God knew where.

The answers derived by these means may determine not only the temporal organization of the dance but also its spatial design, special slips designating the location on the stage where the movement is to be performed.

it was mud in wet weather and dust, ankle-deep, in dry, and could be crossed only at the corner where there were stepping stones.

In a society where everything is for sale, Marlowe is the only man who cannot be bought.

Ideally speaking, it should be allowed to operate only where the public has a great stake in the continuity of supply or services, and where the actions of a single proprietor are secondary to the needs of society.

The countrywide success of `` Lazybones '' and `` And The Angels Sing '' could only lead to Hollywood, where, besides Harold Arlen, Mercer collaborated with Harry Warren, Jimmy Van Heusen, Richard Whiting, Walter Donaldson, Jerome Kern, and Arthur Schwartz.

the Low Countries, where the Middle Ages were to last for another two centuries and die out only when Charles the Bold of Burgundy met his first defeat in the fields and forests below the walls of Grandson.

He concluded that selective service would not only prevent the disorganization of essential war industries but would avoid the undesirable moral effects of the British reliance on enlistment only -- `` where the feeling of the people was whipped into a frenzy by girls pinning white feathers on reluctant young men, orators preaching hate of the Germans, and newspapers exaggerating enemy outrages to make men enlist out of motives of revenge and retaliation ''.

Lewis gave him a guidebook tour of London and, motoring and walking, took him to Stratford, but the London stay was for only ten days, and on the twentieth they took the train for Southampton, where they spent the night for an early morning Channel crossing.

Such kids only wind up among the unemployed on relief or in jail where they become a much bigger burden.

The United States was engaged in a military attack on a peaceful, orderly people governed by a regime that had proved itself the most pro-Western and anti-Communist within any of the new nations -- the only place in Africa, moreover, where a productive relationship between whites and blacks had apparently been achieved.

Off-Broadway, where production is still comparatively cheap, is proving itself only slightly more original.

That exchange was not only possible but commonplace last week in Manhattan, as more and more New Yorkers were discovering 29th Street and Eighth Avenue, where half a dozen small nightclubs with names like Arabian Nights, Grecian Palace and Egyptian Gardens are the American inpost of belly dancing.

How foolish it was to try to fathom the truth in an area where only faith would suffice.

But during the last several years boats were launched in areas where, a short time ago, the only water to be found was in wells and watering troughs for livestock.

When the automobile was in its embryonic stage, such roads as existed were pretty much open roads with the tacit understanding that horses should not be unduly terrified being about the only rule governing where, when and how fast a car could go.

In other instances where there is no automatic block signaling, the distant has only green and yellow aspects.

The times I can recall when I was publicly humiliated by him -- lovely dinner parties in our Trianon Suite where the collation was postponed and postponed and postponed, only to be served dry and overcooked at a table where the host's chair was vacant ; ;

`` It's only that I like to know where you go ''.

However, the possible absence of a center of symmetry not only moves the hydrogen atom off Af, but also allows the oxygen atoms to become nonequivalent, with Af at Af and Af at Af ( space group Af ), where Af represents the oxygens on one side of the Af layers and Af those on the other side.