Exp

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

The function (e^z) (read as the exponential function) is defined for all z. It presents the zth power of the Euler number e ≈2.71828.

Specific values

Specialized values

Values at fixed points

exp(((n i π)/m)) can be expressed using only square roots if n∈ℤ and m is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.

Values at infinities

General characteristics

Domain and analyticity

(e^z) is an entire analytical function of z which is defined over the whole complex z-plane.

Symmetries and periodicities

Mirror symmetry

Periodicity

(e^z) is a periodic function with period 2π  i.

Homogeneity

Poles and essential singularities

The function (e^z) has only one singular point at z = (OverTilde[∞]).  It is an essential singular point.

Branch points

The function (e^z) does not have branch points.

Branch cuts

The function (e^z) does not have branch cuts.

Series representations

Generalized power series

Expansions at z==(Subscript[z, 0])

Expansions at z==0

Other series representations

H. J. Brothers

H. J. Brothers

H. J. Brothers

Asymptotic series expansions

Residue representations

Padé approximants

Other series representations

Dual Taylor series representations

Integral representations

Contour integral representations

Product representations

Limit representations

Direct Limits

Iterative Limits

Continued fraction representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Equations involving this function

Algebraic equations

Differential equations

First-order equations

Schrödinger equations

The spectrum is purely continuous, the wavefunctions (Subscript[ψ, k])(x)== ((Sqrt[sinh(π k)])/π)(Subscript[K, 𝕚  k])((e^x))/;k>0 are normalized to δ(k-k').

Transformations

Transformations and argument simplifications

Argument involving complex components

Argument involving basic arithmetic operations

Argument involving inverse trigonometric and hyperbolic functions

Addition formulas

Half-angle formulas

Multiple arguments

Argument involving numeric multiples of variable

Argument involving symbolic multiples of variable

Some functions of arguments

Products, sums, and powers of the direct function

Products of the direct function

Powers of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Symbolic differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving functions of the direct function

Involving functions of the direct function and a power function

Definite integration

For the direct function itself

Involving the direct function

Multiple integrals

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Inverse Laplace transforms

Mellin transforms

Hankel transforms

Summation

Infinite summation

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQ

Involving Hypergeometric1F1

Through Meijer G

Classical cases for the direct function itself

Classical cases for the direct function itself minus parts of its series expansion

Classical cases involving Sin

Classical cases involving Cos

Classical cases involving Sinh

Classical cases involving Cosh

Classical cases involving Erf

Classical cases involving Erfc

Classical cases involving Erfi

Classical cases involving ExpIntegralEi

Classical cases involving exponential integral ExpIntegralE

Classical cases involving incomplete gamma functions ||| Classical cases involving incomplete gamma functions

Classical cases involving regularized incomplete gamma functions GammaRegularized

Classical cases involving AiryAi

Classical cases involving AiryAiPrime

Classical cases involving AiryBi

Classical cases involving AiryBiPrime

Classical cases involving Bessel BesselI

Classical cases involving Bessel BesselK

Classical cases involving ParabolicCylinderD

Classical cases involving Hermite HermiteH

Classical cases involving Cosh and Hermite HermiteH

Classical cases involving Sinh and Hermite HermiteH

Classical cases involving products of Hermite HermiteH

Classical cases involving ParabolicCylinderD and Hermite HermiteH

Classical cases involving Laguerre LaguerreL

Classical cases involving  Cosh and Laguerre LaguerreL

Classical cases involving  Sinh and Laguerre LaguerreL

Classical cases involving  products of LaguerreL LaguerreL

Classical cases involving Hypergeometric0F1

Classical cases involving Hypergeometric0F1Regularized

Classical cases involving Hypergeometric1F1

Classical cases involving  Cosh and Hypergeometric1F1

Classical cases involving  Sinh and Hypergeometric1F1

Classical cases involving  products of Hypergeometric1F1

Classical cases involving Hypergeometric1F1Regularized

Classical cases involving  Hypergeometric1F1 and Hypergeometric1F1Regularized

Classical cases involving  Hypergeometric1F1 and hypergeometric HypergeometricU

Classical cases involving Hypergeometric1F1Regularized

Classical cases involving  Cosh and Hypergeometric1F1Regularized

Classical cases involving  Sinh and Hypergeometric1F1Regularized

Classical cases involving  products of Hypergeometric1F1Regularized

Classical cases involving hypergeometric HypergeometricU

Classical cases involving  Cosh and hypergeometric HypergeometricU

Classical cases involving  Sinh and hypergeometric HypergeometricU

Classical cases involving products of hypergeometric HypergeometricU

Classical cases involving  Hypergeometric1F1Regularized and hypergeometric HypergeometricU

Classical cases involving  Hypergeometric1F1 and Laguerre LaguerreL

Classical cases involving  Hypergeometric1F1Regularized and Laguerre LaguerreL

Classical cases involving hypergeometric HypergeometricU and Laguerre LaguerreL

Classical cases involving WhittakerM WhittakerM

Classical cases involving WhittakerW WhittakerW

Generalized cases for the direct function itself

Generalized cases involving Cos

Generalized cases involving Sin

Generalized cases involving Erf

Generalized cases involving Erfc

Generalized cases involving Erfi

Generalized cases involving incomplete gamma functions ||| Generalized cases involving incomplete gamma functions

Generalized cases involving regularized incomplete gamma functions GammaRegularized

Generalized cases involving AiryAi

Generalized cases involving AiryAiPrime

Generalized cases involving AiryBi

Generalized cases involving AiryBiPrime

Generalized cases involving ParabolicCylinderD

Generalized cases involving Hermite HermiteH

Generalized cases involving Cosh and Hermite HermiteH

Generalized cases involving Sinh and Hermite HermiteH

Generalized cases involving ParabolicCylinderD and Hermite HermiteH

Generalized cases involving  products of Hetmite HermiteH

Generalized cases involving  products of LaguerreL LaguerreL

Generalized cases involving  products of Hypergeometric1F1

Generalized cases involving  Hypergeometric1F1 and Hypergeometric1F1Regularized

Generalized cases involving  Hypergeometric1F1 and hypergeometric HypergeometricU

Generalized cases involving  products of Hypergeometric1F1Regularized

Generalized cases involving  Hypergeometric1F1 and Laguerre LaguerreL

Generalized cases involving  Hypergeometric1F1Regularized and Laguerre LaguerreL

Generalized cases involving hypergeometric HypergeometricU and Laguerre LaguerreL

Generalized cases involving products of hypergeometric HypergeometricU

Generalized cases involving  Hypergeometric1F1Regularized and hypergeometric HypergeometricU

Generalized cases involving WhittakerM WhittakerM

Through other functions

Involving Bessel functions

Involving Jacobi functions

Involving Mathieu functions

Involving some elliptic-type functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With inverse function Exp

The left side of above formula corresponds to composition f((f^((-1)))(z))/; f(z)==(e^z), which generically equal to z.

The left side of above formula corresponds to composition (f^((-1)))(f(z))/; f(z)==(e^z), which generically does not equal to z.

The left side of above formula corresponds to composition (f^((-1)))(f(z))/; f(z)==(e^z), which equal to z under restriction -π<Im(z)≤π.

With related functions

Involving power function

Involving Tan

Involving Cot

Involving Tanh

Involving Coth

Involving trigonometric functions

Involving hyperbolic functions

Involving other related functions

Inequalities

Theorems

Fourier transformation and Parseval relation

(OverHat[f])(y)==(1/(Sqrt[2π]))(∫_(-∞)^∞)f(x)(e^(i y x))dx⇔ f(x)==(1/(Sqrt[2π]))(∫_(-∞)^∞)(OverHat[f])(y)(e^(-i y x))dy;(∫_(-∞)^∞)(Subscript[f, 1])(t)(Subscript[f, 2])(x-t)dt==(∫_(-∞)^∞)(OverHat[(Subscript[f, 1])])(y)(OverHat[(Subscript[f, 2])])(y)(e^(-i y x))dy.

Laplace transformation and Parseval relation

(OverHat[f])(y)==(∫_0^∞)f(x)(e^(-p x))dx⇔ f(x)==(1/(2π i ))(∫_(γ-i ∞)^(γ+i ∞))(OverHat[f])(p)(e^(p x))dp;(∫_(-∞)^∞)(Subscript[f, 1])(t)(Subscript[f, 2])(x-t)dt==(1/(2π i ))(∫_(γ-i ∞)^(γ+i ∞))(OverHat[(Subscript[f, 1])])(p)(OverHat[(Subscript[f, 2])])(p)(e^(p x))dp.

Fourier series (exponential form)

f(x)∼(∑_(k=-∞)^∞)(Subscript[A, k]) exp(((i π k x)/r))/;(Subscript[A, k])==(1/(2r))(∫_(-r)^r)f(t) exp(((i π k t)/r))dt.

The Lindemann-Weierstrass theorem

If the algebraic numbers (Subscript[z, 1]) , (Subscript[z, 2]), ... (Subscript[z, n]) are linearly independent over ℚ, then the numbers (e^(Subscript[z, 1])),  (e^(Subscript[z, 2])), … (e^(Subscript[z, n])) are algebraically independent over ℚ.

Schanuel's conjecture

If (Subscript[z, 1]),(Subscript[z, 2]),…,(Subscript[z, n])∈ℂ are linearly independent over ℚ, then the transcendence degree of the set of numbers (Subscript[z, 1]),(Subscript[z, 2]),…,(Subscript[z, n]),(e^(Subscript[z, 1])),(e^(Subscript[z, 2])),…,(e^(Subscript[z, n])) is at least n.

Wigner surmise

The probability density p(s) for a level spacing s of a complicated nucleus is given by p(s)==((π s)/2)exp(-(π/4)(s^2)).

The number of surviving atoms in radioactive decay

The number of surviving atoms N(t) at time t in radioactive decay obeys the law N(t)==N(0)exp(-(t/τ)).

Other information

Sometimes in the literature (e^z) is defined as the inverse of the logarithmic function; that is w = (e^z) if and only if log(w) =  z.

Sometimes in the literature (e^z) is defined as the only function f(z) that remains unchanged by differentiation, so that  ((d f(z))/(d z))==f(z), and also satisfies the property f(0)==1.

History

– R. Cotes (1714) found the formula (e^(i ϕ))==cos(ϕ)+i sin(ϕ) 
– L. Euler (1740–1748) found a series expansion for (e^z)
– Cayley (around 1880) used the notation "exp"

The function exp is encountered often in mathematics and the natural sciences.

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