Power

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed z

For fixed a

Values at fixed points

Values at infinities

General characteristics

Domain and analyticity

(z^a) is an analytical function of z and a which is defined over (ℂ^2). For fixed z, it is an entire function of a.  For positive integer a, (z^a) degenerates to a polynomial in z.

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

(z^a) is a periodic function with respect to a with period ((2 i π)/(log(z))).

Homogeneity

Scale symmetry

Poles and essential singularities

With respect to a

For fixed z, the function (z^a) has only one singular point at a=(OverTilde[∞]).  It is an essential singular point.

With respect to z

Branch points

With respect to a

For fixed z, the function (z^a) does not have branch points.

With respect to z

For fixed noninteger a, the function  (z^a) has two singular branch points: z==0, z==(OverTilde[∞]).

For integer a, the function (z^a) does not have branch points.

Branch cuts

With respect to a

For fixed z, the function (z^a)does not have branch cuts.

With respect to z

For fixed noninteger a, the function  (z^a) is a single-valued function on the z-plane cut along the interval (-∞, 0), where it is continuous from above.

For integer a, the function (z^a) does not have branch cuts.

Series representations

Generalized power series

Expansions at generic point a==(Subscript[a, 0])

Expansions at a==0

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

Expansions on branch cuts

Expansions at z==1

Expansions of (((1+z))^a) at z==0

Expansions of (((1+z))^a) at z==∞

Asymptotic series expansions

Residue representations

Representations of (z^a)

Representations of (((1+z))^a)

Integral representations

Contour integral representations

Continued fraction representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Ordinary nonlinear differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Addition formulas

With respect to z

With respect to a

Half-angle formulas

With respect to z

With respect to a

Multiple arguments

With respect to z

With respect to a

Power of arguments

With respect to z

Exponent of arguments

With respect to z

Some functions of arguments

With respect to z

Products, sums, and powers of the direct function

Products of the direct function

Quotients of the direct function

Power of the direct function

Sum of the direct function

Power of the product of the direct function

Related transformations

Identities

Functional identities

Univariate functional identities

Biivariate functional identities

Other identities

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

With respect to z

With respect to a

Symbolic differentiation

With respect to z

With respect to a

Fractional integro-differentiation

With respect to z

With respect to a

Integration

Indefinite integration

For the direct function with respect to z

For the direct function with respect to a

Definite integration

For the direct function itself

Involving the direct function

Involving related functions

Multiple integration

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Hankel transforms

Summation

Finite summation

Various sums

Summed truncated generalized hypergeometric series

Infinite summation

Multidimensional summation

Products

Finite products

Infinite products

Regularized infinite products

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Involving Hypergeometric2F1

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 unit step UnitStep

Classical cases involving Sign

Classical cases involving Sqrt in the arguments

Classical cases involving Sqrt in the arguments and unit step UnitStep

Generalized cases involving Sqrt in the arguments

Generalized cases involving Sqrt in the arguments and unit step UnitStep

Through other functions

Representations through equivalent functions

With inverse function

The left side of the above formula corresponds to the composition f((f^((-1)))(z))/; f(z)==(z^a), which generically does not equal z. This relation also reflects the invertibility of the power function (z^a) with respect to the variable z because its inverse (f^((-1))) can coincide with the original function f , but for another value of the parameter a.

The left side of above formula corresponds to composition f((f^((-1)))(z))/; f(z)==(z^a), which equal to z under restriction -π<Im(((log(z))/a))≤π. This relation also reflects invertibility of power function (z^a) with respect to variable z because its inversion (f^((-1))) can coincides with original f but for other values of parameter a.

The left side of above formula corresponds to composition (f^((-1)))(f(z))/; f(z)==(z^a), which generically does not equal to z. This relation also reflects invertibility of power function (z^a) with respect to variable z because its inversion (f^((-1))) can coincides with original f but for other values of parameter a.

The left side of above formula corresponds to composition (f^((-1)))(f(z))/; f(z)==(z^a), which equal to z under restriction -π<Im(a log(z))≤π. This relation also reflects invertibility of power function (z^a) with respect to variable z because its inversion (f^((-1))) can coincides with original f but for other values of parameter a.

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

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

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

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

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

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

With related functions

With orthogonal polynomials

Inequalities

Zeros

Theorems

Gelfond-Schneider theorem

The number (α^β), where α, β are algebraic numbers, α≠0,1, β ∉ ℚ, is transcendental.

Mellin transformation and Parseval relation

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

Hilbert transformation

(OverHat[f])(y)==(1/π)(∫_(-∞)^∞)((f(x))/(x-y))dx⇔f(x)==-(1/π)(∫_(-∞)^∞)(((OverHat[f])(y))/(y-x))dy.

The eigenfunctions of the Hilbert transformation are given by the following:

(1/π)𝒫(∫_(-∞)^∞)((f(x,n))/(x-y))dx==sgn(n) f(y,n)/;f(x,n)==((((1+i x))^n)/(((1-i x))^(n+1)))∧n∈ℤ

Zipf's law

The frequency f versus the rank r in a text has the form f=(r^(-β)), where β≈0.6 for natural languages.

The quantum mechanical density of states of a  d-dimensional lattice

The quantum mechanical density of states 𝒟(ε) of a  d-dimensional lattice exhibits van Hove singularities of the form 𝒟(ε)∝(ε^((d-1)/2)), where ε is the energy.

Infinite tetration

The sequence

(x^x),(x^(x^x)),(x^(x^(x^x))),…

converges for real x only if (e^(-e))≤x≤(e^(1/e)).

Differential-Algebraic Constants

To ensure the correctness of many formulas given in this collection over the whole complex plane, it is often necessary to work with expressions of the form ((((z^2)))^a)==(((i z))^a)(((-i z))^a) , ((((z^a)))^(1/a)), ((((z^a)))^(1/a))/z, z ((((z^(-a))))^(1/a)), (((-(z^a)))^(1/a)), (((-(z^2)))^a)==(((z^a)(-z))^a),etc. (where a is a generic complex number). While in a textbook-mathematics setting these expressions are often simplified to z, ±1, 1,  (((-1))^(1/a))z, etc, this cannot be done inside Mathematica. From a complex function point of view the Riemann surface of such functions are made from disconnected sheets. Inside Mathematica all branch cuts of all functions (that have branch cuts) follow uniquely from the branch cut of the single power function (the logarithm function respectively). As a result the branch cuts related to more complicated functions such as ((((z^2)))^a), ((((z^a)))^(1/a)), ((((z^a)))^(1/a))/z, z ((((z^(-a))))^(1/a)), (((-(z^a)))^(1/a)), (((-(z^2)))^a) exist, although they do not start and end at branch points, but mostly extend from the origin to (OverTilde[∞]) along opposite rays.
For instance we have:

(Subscript[ℬ𝒞, z])(((((z^2)))^a))==(Subscript[ℬ𝒞, z])((((i z))^a)(((-i z))^a))=={{(-i ∞,0),1},{(0,i ∞),-1}}

(lim_(ϵ->+0)) ((((((x+ϵ))^2)))^a)==(e^(-2i π a)) ((((x^2)))^a)/;i x>0

(lim_(ϵ->+0)) ((((((x-ϵ))^2)))^a)==(e^(-2i π a)) ((((x^2)))^a)/;i x<0

An expression of the form  ((((z^a)))^(1/a))/z , z ((((z^(-a))))^(1/a))are called differential-algebraic constants because their derivative vanishes generically everywhere as a complex function (but not as a generalized function).

History

– Ancient Egyptian, Babylonian, and Greek mathematicians
– Euclid knew (a^m) (a^n)== (a^(m+n))  for integers m and  n
– N. Oresme (14th century) knew this rule even for fractional positive  m and n, and also knew (((a b))^(1/n))==(a^(1/n))(b^(1/n))  and ((((a^m)))^(p/q))==((((a^(m p))))^(1/q))
– F. Viete (1593) derived the famous geometric series for 1/(1+x)
– Joh. Bernoulli (1697) defined arbitrary powers as  (z^a)==((((e^(log(z)))))^a)==(e^(a log(z)))
– L. Euler (1728) proved that (i^i)==(e^(-π/2)) and in 1748 used a series expansion for (((1+x))^a) with rational a
–  Martin Ohm (1823) was the first who fully develop the theory of the exponential (z^a), when both z and a are complex numbers
–  N. H. Abel (1826) first proved correctness of the series expansion of (((1+x))^a) in the unit circle for arbitrary a

The function (z^a) is encountered often in mathematics and the natural sciences.

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