Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

Several elementary functions which are defined via power series may be defined in any unital Banach algebra ; examples include the exponential function and the trigonometric functions, and more generally any entire function.

* Conjugate prior, in Bayesian statistics, a family of probability distributions that contains a prior and the posterior distributions for a particular likelihood function ( particularly for one-parameter exponential families )

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function.

This complex exponential function is sometimes denoted The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.

It was Euler ( presumably around 1740 ) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now named after him.

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 ).

Euler's formula provides a powerful connection between analysis and trigonometry, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function:

The exponential function e < sup > x </ sup > for real values of x may be defined in a few different equivalent ways ( see Characterizations of the exponential function ).

Note that this derivation does assume that ƒ ( x ) is sufficiently differentiable and well-behaved ; specifically, that ƒ may be approximated by polynomials ; equivalently, that ƒ is a real analytic function of exponential type.

This is the essentially the reason for the restriction to exponential type of less than 2π: the function sin ( 2πnz ) grows faster than e < sup > 2π | z |</ sup > along the imaginary axis!

The natural exponential function

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.

The exponential time hypothesis is that no algorithm can solve 3-Sat in time.

In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 10 < sup > 78 </ sup > in volume, driven by a negative-pressure vacuum energy density.

In category theory, currying can be found in the universal property of an exponential object, which gives rise to the following adjunction in cartesian closed categories: There is a natural isomorphism between the morphisms from a binary product and the morphisms to an exponential object.

The result is in fact an exponential growth, thus giving rise to explosive increases in reaction rates, and indeed to chemical explosions themselves.

If is a constant, the solution is particularly simple, and describes, e. g., if, the exponential decay of radioactive material at the macroscopic level.

While the technique extends to higher dimension ( as proved by Edelsbrunner and Shah ), the runtime can be exponential in the dimension even if the final Delaunay triangulation is small.

The most widely used method of discounting is exponential discounting, which values future cash flows as " how much money would have to be invested currently, at a given rate of return, to yield the cash flow in future.

For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x < sub > 0 </ sub >,

If a Diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential Diophantine equation.

The reason for this is that the complex exponential is the eigenfunction of differentiation.

( In particular, the exponential map can be used to define abstract index groups.

The exponential function is widely used in physics, chemistry and mathematics.

The latter action is often called the conjugation action, and an exponential notation is commonly used for the right-action variant: ; it satisfies.

It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment.

For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems.

) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently.

It has been demonstrated that changes in biodiversity through the Phanerozoic correlate much better with the hyperbolic model ( widely used in demography and macrosociology ) than with exponential and logistic models ( traditionally used in population biology and extensively applied to fossil biodiversity as well ).

The transmission coefficient for FTIR is highly sensitive to the spacing between the high index media ( the function is approximately exponential until the gap is almost closed ), so this effect has often been used to modulate optical transmission and reflection with a large dynamic range.

But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p. m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives.

In hydrology, the exponential distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.

Other distributions which can be used to characterise particle size are: the Rosin-Rammler distribution, applied to coarsely dispersed dusts and sprays ; the Nukiyama-Tanasawa distribution, for sprays having extremely broad size ranges ; the power function distribution, which has been applied to atmospheric aerosols ; the exponential distribution, applied to powdered materials and for cloud droplets the Khrgian-Mazin distribution.

** Exponential family, sometimes used in place of " exponential family "

If it is desired to sample more of the exponential electron portion of the characteristic, an asymmetric double probe may be used, with one electrode larger than the other.

* used the Weil conjectures to prove estimates for exponential sums.

It is also known as the log-Weibull distribution and the double exponential distribution ( which is sometimes used to refer to the Laplace distribution ).

Both the terms " Markov property " and " strong Markov property " have been used in connection with a particular " memoryless " property of the exponential distribution.

This is the form of the equation that is most commonly used to describe exponential decay.

These rules apply to exponential growth and are therefore used for compound interest as opposed to simple interest calculations.

The term exponential class is sometimes used in place of " exponential family ".