Exponentiation

File:Expo02.svg

Graphs of Template:Math for various bases Template:Math: Template:Nobr Template:Nobr Template:Nobr Template:Nobr Each curve passes through the point Template:Math because any nonzero number raised to the power of Template:Math is Template:Math. At Template:Math, the value of Template:Math equals the base because any number raised to the power of Template:Math is the number itself.

In mathematics, exponentiation, denoted Template:Math, is an operation involving two numbers: the base, Template:Mvar, and the exponent or power, Template:Mvar.<ref name=":1">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> When Template:Mvar is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, Template:Math is the product of multiplying Template:Mvar bases:<ref name=":1"/> <math display="block">b^n = \underbrace{b \times b \times \dots \times b \times b}_{n \text{ times}}.</math>In particular, <math>b^1=b</math>.

The exponent is usually shown as a superscript to the right of the base as Template:Math or in computer code as b^n. This binary operation is often read as "Template:Mvar to the power Template:Mvar"; it may also be referred to as "Template:Mvar raised to the Template:Mvarth power", "the Template:Mvarth power of Template:Mvar",<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Power%7CPower.html}} |title = Power |author = Weisstein, Eric W. |website = MathWorld |access-date = 2020-08-27 |ref = Template:SfnRef }}</ref> or, most briefly, "Template:Mvar to the Template:Mvar".

The above definition of <math>b^n</math> immediately implies several properties, in particular the multiplication rule:<ref group="nb">There are three common notations for multiplication: <math>x\times y</math> is most commonly used for explicit numbers and at a very elementary level; <math>xy</math> is most common when variables are used; <math>x\cdot y</math> is used for emphasizing that one talks of multiplication or when omitting the multiplication sign would be confusing.</ref>

<math display="block">\begin{align} b^n \times b^m & = \underbrace{b \times \dots \times b}_{n \text{ times}} \times \underbrace{b \times \dots \times b}_{m \text{ times}} \\[1ex] & = \underbrace{b \times \dots \times b}_{n+m \text{ times}} \ =\ b^{n+m} . \end{align}</math>

That is, when multiplying a base raised to one power times the same base raised to another power, the powers add. Extending this rule to the power zero gives <math>b^0 \times b^n = b^{0+n} = b^n</math>, and, where Template:Mvar is non-zero, dividing both sides by <math>b^n</math> gives <math>b^0 = b^n / b^n = 1</math>. That is the multiplication rule implies the definition <math display="block"> b^0=1. </math>A similar argument implies the definition for negative integer powers: <math display="block">b^{-n} = 1/b^n.</math>That is, extending the multiplication rule gives <math>b^{-n} \times b^n = b^{-n+n} = b^0 = 1 </math>. Dividing both sides by <math>b^n</math> gives <math>b^{-n} = 1 / b^n</math>. This also implies the definition for fractional powers: <math display="block">b^{n/m} = \sqrt[m]{b^n}.</math>For example, <math> b^{1/2} \times b^{1/2} = b^{1/2 \,+\, 1/2} = b^1 = b </math>, meaning <math> (b^{1/2})^2 = b </math>, which is the definition of square root: <math>b^{1/2} = \sqrt{b} </math>.

The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define <math>b^x</math> for any positive real base <math>b</math> and any real number exponent <math>x</math>. More involved definitions allow complex base and exponent, as well as certain types of matrices as base or exponent.

Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.

Etymology

The term exponent originates from the Latin exponentem, the present participle of exponere, meaning "to put forth".<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The term power (Template:Langx) is a mistranslation<ref name="Rotman">Template:Cite book</ref><ref>Template:Cite book</ref> of the ancient Greek δύναμις (dúnamis, here: "amplification"<ref name="Rotman"/>) used by the Greek mathematician Euclid for the square of a line,<ref name="MacTutor"/> following Hippocrates of Chios.<ref>Template:Cite book</ref>

The word exponent was coined in 1544 by Michael Stifel.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite book</ref> In the 16th century, Robert Recorde used the terms "square", "cube", "zenzizenzic" (fourth power), "sursolid" (fifth), "zenzicube" (sixth), "second sursolid" (seventh), and "zenzizenzizenzic" (eighth).<ref name="worldwidewords">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> "Biquadrate" has been used to refer to the fourth power as well.

History

In The Sand Reckoner, Archimedes proved the law of exponents, Template:Math, necessary to manipulate powers of Template:Math.<ref> Archimedes. (2009). THE SAND-RECKONER. In T. Heath (Ed.), The Works of Archimedes: Edited in Modern Notation with Introductory Chapters (Cambridge Library Collection - Mathematics, pp. 229-232). Cambridge: Cambridge University Press. {{#invoke:doi|main}}.</ref> He then used powers of Template:Math to estimate the number of grains of sand that can be contained in the universe.

In the 9th century, the Persian mathematician Al-Khwarizmi used the terms مَال (māl, "possessions", "property") for a square—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"<ref name="worldwidewords"/>—and كَعْبَة (Kaʿbah, "cube") for a cube, which later Islamic mathematicians represented in mathematical notation as the letters mīm (m) and kāf (k), respectively, by the 15th century, as seen in the work of Abu'l-Hasan ibn Ali al-Qalasadi.<ref>Template:MacTutor</ref> Nicolas Chuquet used a form of exponential notation in the 15th century, for example Template:Math to represent Template:Math.<ref>Template:Cite book</ref> This was later used by Henricus Grammateus and Michael Stifel in the 16th century. In the late 16th century, Jost Bürgi would use Roman numerals for exponents in a way similar to that of Chuquet, for example Template:Overset for Template:Math.<ref name="cajori">Template:Cite book</ref>

In 1636, James Hume used in essence modern notation, when in L'algèbre de Viète he wrote Template:Math for Template:Math.<ref>Template:Cite book</ref> Early in the 17th century, the first form of our modern exponential notation was introduced by René Descartes in his text titled La Géométrie; there, the notation is introduced in Book I.<ref>Template:Cite book (And Template:Math, or Template:Math, in order to multiply Template:Math by itself; and Template:Math, in order to multiply it once more by Template:Math, and thus to infinity).</ref>

I designate ... Template:Math, or Template:Math in multiplying Template:Math by itself; and Template:Math in multiplying it once more again by Template:Math, and thus to infinity.{{#if:René DescartesLa Géométrie|{{#if:|}}

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Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as Template:Math.

Samuel Jeake introduced the term indices in 1696.<ref name="MacTutor">Template:MacTutor</ref> The term involution was used synonymously with the term indices, but had declined in usage<ref>The most recent usage in this sense cited by the OED is from 1806 (Template:Cite OED).</ref> and should not be confused with its more common meaning.

In 1748, Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing:

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20th century

As calculation was mechanized, notation was adapted to numerical capacity by conventions in exponential notation. For example Konrad Zuse introduced floating-point arithmetic in his 1938 computer Z1. One register contained representation of leading digits, and a second contained representation of the exponent of 10. Earlier Leonardo Torres Quevedo contributed Essays on Automation (1914) which had suggested the floating-point representation of numbers. The more flexible decimal floating-point representation was introduced in 1946 with a Bell Laboratories computer. Eventually educators and engineers adopted scientific notation of numbers, consistent with common reference to order of magnitude in a ratio scale.<ref>Janet Shiver & Terri Wiilard "Scientific notation: working with orders of magnitude from Visionlearning</ref>

For instance, in 1961 the School Mathematics Study Group developed the notation in connection with units used in the metric system.<ref>School Mathematics Study Group (1961) Mathematics for Junior High School, volume 2, part 1, Yale University Press</ref><ref>Cecelia Callanan (1967) "Scientific Notation", The Mathematics Teacher 60: 252–6 JSTOR</ref>

Exponents also came to be used to describe units of measurement and quantity dimensions. For instance, since force is mass times acceleration, it is measured in kg m/sec². Using M for mass, L for length, and T for time, the expression M L T^–2 is used in dimensional analysis to describe force.<ref>Edwin Bidwell Wilson (1920) Theory of Dimensions, chapter 11 in Aeronautics: A Class Text, via Internet Archive</ref><ref>Template:Cite book</ref>

Terminology

The expression Template:Math is called "the square of Template:Math" or "Template:Math squared", because the area of a square with side-length Template:Math is Template:Math. (It is true that it could also be called "Template:Math to the second power", but "the square of Template:Math" and "Template:Math squared" are more traditional)

Similarly, the expression Template:Math is called "the cube of Template:Math" or "Template:Math cubed", because the volume of a cube with side-length Template:Math is Template:Math.

When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example, Template:Math. The base Template:Math appears Template:Math times in the multiplication, because the exponent is Template:Math. Here, Template:Math is the 5th power of 3, or 3 raised to the 5th power.

The word "raised" is usually omitted, and sometimes "power" as well, so Template:Math can be simply read "3 to the 5th", or "3 to the 5".

Integer exponents

The exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations.

Positive exponents

The definition of the exponentiation as an iterated multiplication can be formalized by using induction,<ref>Template:Cite book</ref> and this definition can be used as soon as one has an associative multiplication:

The base case is

<math>b^1 = b</math>

and the recurrence is

<math>b^{n+1} = b^n \cdot b.</math>

The associativity of multiplication implies that for any positive integers Template:Mvar and Template:Mvar,

<math>b^{m+n} = b^m \cdot b^n,</math>

and

<math>(b^m)^n=b^{mn}.</math>

Zero exponent

As mentioned earlier, a (nonzero) number raised to the Template:Math power is Template:Math:<ref>Template:Cite book</ref><ref name=":1"/>

<math>b^0=1.</math>

This value is also obtained by the empty product convention, which may be used in every algebraic structure with a multiplication that has an identity. This way the formula

<math>b^{m+n}=b^m\cdot b^n</math>

also holds for <math>n=0</math>.

The case of Template:Math is controversial. In contexts where only integer powers are considered, the value Template:Math is generally assigned to Template:Math but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context. Template:Crossreference

Negative exponents

Exponentiation with negative exponents is defined by the following identity, which holds for any integer Template:Mvar and nonzero Template:Mvar:

<math>b^{-n} = \frac{1}{b^n}</math>.<ref name=":1"/>

Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (<math>\infty</math>).<ref>Template:Cite book</ref>

This definition of exponentiation with negative exponents is the only one that allows extending the identity <math>b^{m+n}=b^m\cdot b^n</math> to negative exponents (consider the case <math>m=-n</math>).

The same definition applies to invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted Template:Math (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element Template:Mvar is standardly denoted <math>x^{-1}.</math>

Identities and properties

Template:Redirect The following identities, often called Template:Vanchor, hold for all integer exponents, provided that the base is non-zero:<ref name=":1"/>

<math>\begin{align}

          b^m \cdot b^n &= b^{m + n} \\
 \left(b^m\right)^n &= b^{m \cdot n} \\
      b^n \cdot c^n &= (b \cdot c)^n

\end{align}</math>

Unlike addition and multiplication, exponentiation is not commutative: for example, <math>2^3 = 8</math>, but reversing the operands gives the different value <math>3^2=9</math>. Also unlike addition and multiplication, exponentiation is not associative: for example, Template:Math, whereas Template:Math. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up<ref name="Bronstein_1987"/><ref name="NIST_2010"/><ref name="Zeidler_2013"/> (or left-associative). That is,

<math>b^{p^q} = b^{\left(p^q\right)},</math>

which, in general, is different from

<math>\left(b^p\right)^q = b^{p q} .</math>

Powers of a sum

The powers of a sum can normally be computed from the powers of the summands by the binomial formula

<math>(a+b)^n=\sum_{i=0}^n \binom{n}{i}a^ib^{n-i}=\sum_{i=0}^n \frac{n!}{i!(n-i)!}a^ib^{n-i}.</math>

However, this formula is true only if the summands commute (i.e. that Template:Math), which is implied if they belong to a structure that is commutative. Otherwise, if Template:Mvar and Template:Mvar are, say, square matrices of the same size, this formula cannot be used. It follows that in computer algebra, many algorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose computer algebra systems use a different notation (sometimes Template:Math instead of Template:Math) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.

Combinatorial interpretation

For nonnegative integers Template:Mvar and Template:Mvar, the value of Template:Math is the number of functions from a set of Template:Mvar elements to a set of Template:Mvar elements (see cardinal exponentiation). Such functions can be represented as Template:Mvar-tuples from an Template:Mvar-element set (or as Template:Mvar-letter words from an Template:Mvar-letter alphabet). Some examples for particular values of Template:Mvar and Template:Mvar are given in the following table:

Template:Math	The Template:Math possible Template:Mvar-tuples of elements from the set Template:Math
05 = 0	Template:CNone
14 = 1	(1, 1, 1, 1)
23 = 8	(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)
32 = 9	(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)
41 = 4	(1), (2), (3), (4)
50 = 1	()

Particular bases

Powers of ten

Template:See also {{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In the base ten (decimal) number system, integer powers of Template:Math are written as the digit Template:Math followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, Template:Math and Template:Math.

Exponentiation with base Template:Math is used in scientific notation to denote large or small numbers. For instance, Template:Val (the speed of light in vacuum, in metres per second) can be written as Template:Val and then approximated as Template:Val.

SI prefixes based on powers of Template:Math are also used to describe small or large quantities. For example, the prefix kilo means Template:Math, so a kilometre is Template:Val.

Template:AnchorPowers of two

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The first negative powers of Template:Math have special names: <math>2^{-1}</math>is a half; <math>2^{-2}</math> is a quarter.

Powers of Template:Math appear in set theory, since a set with Template:Math members has a power set, the set of all of its subsets, which has Template:Math members.

Integer powers of Template:Math are important in computer science. The positive integer powers Template:Math give the number of possible values for an Template:Math-bit integer binary number; for example, a byte may take Template:Math different values. The binary number system expresses any number as a sum of powers of Template:Math, and denotes it as a sequence of Template:Math and Template:Math, separated by a binary point, where Template:Math indicates a power of Template:Math that appears in the sum; the exponent is determined by the place of this Template:Math: the nonnegative exponents are the rank of the Template:Math on the left of the point (starting from Template:Math), and the negative exponents are determined by the rank on the right of the point.

Powers of one

Every power of one equals: Template:Math.

Powers of zero

For a positive exponent Template:Math, the Template:Mvarth power of zero is zero: Template:Math. For a negative exponent, <math>0^{-n}=1/0^n=1/0</math> is undefined.

In some contexts (e.g., combinatorics), the expression [[zero to the power of zero|Template:Math]] is defined to be equal to <math>1</math>; in others (e.g., analysis), it is often undefined.

Powers of negative one

Since a negative number times another negative is positive, we have:

<math>(-1)^n = \left\{\begin{array}{rl}

1 & \text{for even } n, \\ -1 & \text{for odd } n. \\

\end{array}\right.</math>

Because of this, powers of Template:Math are useful for expressing alternating sequences. For a similar discussion of powers of the complex number Template:Math, see Template:Slink.

Large exponents

The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound:

Template:Math as Template:Math when Template:Math

This can be read as "b to the power of n tends to +∞ as n tends to infinity when b is greater than one".

Powers of a number with absolute value less than one tend to zero:

Template:Math as Template:Math when Template:Math

Any power of one is always one:

Template:Math for all Template:Math for Template:Math

Powers of a negative number <math>b\leq -1</math> alternate between positive and negative as Template:Math alternates between even and odd, and thus do not tend to any limit as Template:Math grows.

If the exponentiated number varies while tending to Template:Math as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is

Template:Math as Template:Math

See Template:Slink below.

Other limits, in particular those of expressions that take on an indeterminate form, are described in Template:Slink below.

Power functions

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

File:Potenssi 1 3 5.svg

Power functions for Template:Math

File:Potenssi 2 4 6.svg

Power functions for Template:Math

Real functions of the form <math>f(x) = cx^n</math>, where <math>c \ne 0</math>, are sometimes called power functions.<ref>Template:Cite book</ref> When <math>n</math> is an integer and <math>n \ge 1</math>, two primary families exist: for <math>n</math> even, and for <math>n</math> odd. In general for <math>c > 0</math>, when <math>n</math> is even <math>f(x) = cx^n</math> will tend towards positive infinity with increasing <math>x</math>, and also towards positive infinity with decreasing <math>x</math>. All graphs from the family of even power functions have the general shape of <math>y=cx^2</math>, flattening more in the middle as <math>n</math> increases.<ref name="Calculus: Early Transcendentals">Template:Cite book</ref> Functions with this kind of symmetry Template:Nobr are called even functions.

When <math>n</math> is odd, <math>f(x)</math>'s asymptotic behavior reverses from positive <math>x</math> to negative <math>x</math>. For <math>c > 0</math>, <math>f(x) = cx^n</math> will also tend towards positive infinity with increasing <math>x</math>, but towards negative infinity with decreasing <math>x</math>. All graphs from the family of odd power functions have the general shape of <math>y=cx^3</math>, flattening more in the middle as <math>n</math> increases and losing all flatness there in the straight line for <math>n=1</math>. Functions with this kind of symmetry Template:Nobr are called odd functions.

For <math>c < 0</math>, the opposite asymptotic behavior is true in each case.<ref name="Calculus: Early Transcendentals"/>

Table of powers of decimal digits

n	n2	n3	n4	n5	n6	n7	n8	n9	n10
1	1	1	1	1	1	1	1	1	1
2	4	8	16	32	64	128	256	512	1024
3	9	27	81	243	729	Template:Val	Template:Val	Template:Val	Template:Val
4	16	64	256	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
5	25	125	625	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
6	36	216	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
7	49	343	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
8	64	512	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
9	81	729	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
10	100	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val

Rational exponents

File:Mplwp roots 01.svg

From top to bottom: Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math.

If Template:Mvar is a nonnegative real number, and Template:Mvar is a positive integer, <math>x^{1/n}</math> or <math>\sqrt[n]x</math> denotes the unique nonnegative real [[nth root|Template:Mvarth root]] of Template:Mvar, that is, the unique nonnegative real number Template:Mvar such that <math>y^n=x.</math>

If Template:Mvar is a positive real number, and <math>\frac pq</math> is a rational number, with Template:Mvar and Template:Mvar integers, then <math display="inline">x^{p/q}</math> is defined as

<math>x^\frac pq= \left(x^p\right)^\frac 1q=(x^\frac 1q)^p.</math>

The equality on the right may be derived by setting <math>y=x^\frac 1q,</math> and writing <math>(x^\frac 1q)^p=y^p=\left((y^p)^q\right)^\frac 1q=\left((y^q)^p\right)^\frac 1q=(x^p)^\frac 1q.</math>

If Template:Mvar is a positive rational number, Template:Math, by definition.

All these definitions are required for extending the identity <math>(x^r)^s = x^{rs}</math> to rational exponents.

On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real Template:Mvarth root, which is negative, if Template:Mvar is odd, and no real root if Template:Mvar is even. In the latter case, whichever complex Template:Mvarth root one chooses for <math>x^\frac 1n,</math> the identity <math>(x^a)^b=x^{ab}</math> cannot be satisfied. For example,

<math>\left((-1)^2\right)^\frac 12 = 1^\frac 12= 1\neq (-1)^{2\cdot\frac 12} =(-1)^1=-1.</math>

See Template:Slink and Template:Slink for details on the way these problems may be handled.

Real exponents

For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (Template:Slink, below), or in terms of the logarithm of the base and the exponential function (Template:Slink, below). The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to complex exponents.

On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values. One may choose one of these values, called the principal value, but there is no choice of the principal value for which the identity

<math>\left(b^r\right)^s = b^{r s}</math>

is true; see Template:Slink. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.

Limits of rational exponents

File:Continuity of the Exponential at 0.svg

The limit of Template:Math is Template:Math when Template:Mvar tends to the infinity.

Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number Template:Mvar with an arbitrary real exponent Template:Mvar can be defined by continuity with the rule<ref name="Denlinger">Template:Cite book</ref>

<math> b^x = \lim_{r (\in \mathbb{Q}) \to x} b^r \quad (b \in \mathbb{R}^+,\, x \in \mathbb{R}),</math>

where the limit is taken over rational values of Template:Mvar only. This limit exists for every positive Template:Mvar and every real Template:Mvar.

For example, if Template:Math, the non-terminating decimal representation Template:Math and the monotonicity of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain <math>b^\pi:</math>

<math>\left[b^3, b^4\right], \left[b^{3.1}, b^{3.2}\right], \left[b^{3.14}, b^{3.15}\right], \left[b^{3.141}, b^{3.142}\right], \left[b^{3.1415}, b^{3.1416}\right], \left[b^{3.14159}, b^{3.14160}\right], \ldots</math>

So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted <math>b^\pi.</math>

This defines <math>b^x</math> for every positive Template:Mvar and real Template:Mvar as a continuous function of Template:Mvar and Template:Mvar. See also Well-defined expression.<ref>Template:Cite book</ref>

Exponential function

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The exponential function may be defined as <math>x\mapsto e^x,</math> where <math>e\approx 2.718</math> is Euler's number, but to avoid circular reasoning, this definition cannot be used here. Rather, we give an independent definition of the exponential function <math>\exp(x),</math> and of <math>e=\exp(1)</math>, relying only on positive integer powers (repeated multiplication). Then we sketch the proof that this agrees with the previous definition: <math>\exp(x)=e^x.</math>

There are many equivalent ways to define the exponential function, one of them being

<math>\exp(x) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n.</math>

One has <math>\exp(0)=1,</math> and the exponential identity (or multiplication rule) <math>\exp(x)\exp(y)=\exp(x+y)</math> holds as well, since

<math>\exp(x)\exp(y) = \lim_{n\rightarrow\infty} \left(1 + \frac{x}{n}\right)^n\left(1 + \frac{y}{n}\right)^n = \lim_{n\rightarrow\infty} \left(1 + \frac{x+y}{n} + \frac{xy}{n^2}\right)^n,</math>

and the second-order term <math>\frac{xy}{n^2}</math> does not affect the limit, yielding <math>\exp(x)\exp(y) = \exp(x+y)</math>.

Euler's number can be defined as <math>e=\exp(1)</math>. It follows from the preceding equations that <math>\exp(x)=e^x</math> when Template:Mvar is an integer (this results from the repeated-multiplication definition of the exponentiation). If Template:Mvar is real, <math>\exp(x)=e^x</math> results from the definitions given in preceding sections, by using the exponential identity if Template:Mvar is rational, and the continuity of the exponential function otherwise.

The limit that defines the exponential function converges for every complex value of Template:Mvar, and therefore it can be used to extend the definition of <math>\exp(z)</math>, and thus <math>e^z,</math> from the real numbers to any complex argument Template:Mvar. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.

Powers via logarithms

The definition of Template:Math as the exponential function allows defining Template:Math for every positive real numbers Template:Mvar, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm Template:Math is the inverse of the exponential function Template:Math means that one has

<math>b = \exp(\ln b)=e^{\ln b}</math>

for every Template:Math. For preserving the identity <math>(e^x)^y=e^{xy},</math> one must have

<math>b^x=\left(e^{\ln b} \right)^x = e^{x \ln b}</math>

So, <math>e^{x \ln b}</math> can be used as an alternative definition of Template:Math for any positive real Template:Mvar. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.

Complex exponents with a positive real base

If Template:Mvar is a positive real number, exponentiation with base Template:Mvar and complex exponent Template:Mvar is defined by means of the exponential function with complex argument (see the end of Template:Slink, above) as

<math>b^z = e^{(z\ln b)},</math>

where <math>\ln b</math> denotes the natural logarithm of Template:Mvar.

This satisfies the identity

<math>b^{z+t} = b^z b^t,</math>

In general, <math DISPLAY=inline>\left(b^z\right)^t</math> is not defined, since Template:Math is not a real number. If a meaning is given to the exponentiation of a complex number (see Template:Slink, below), one has, in general,

<math>\left(b^z\right)^t \ne b^{zt},</math>

unless Template:Mvar is real or Template:Mvar is an integer.

Euler's formula,

<math>e^{iy} = \cos y + i \sin y,</math>

allows expressing the polar form of <math>b^z</math> in terms of the real and imaginary parts of Template:Mvar, namely

<math>b^{x+iy}= b^x(\cos(y\ln b)+i\sin(y\ln b)),</math>

where the absolute value of the trigonometric factor is one. This results from

<math>b^{x+iy}=b^x b^{iy}=b^x e^{iy\ln b} =b^x(\cos(y\ln b)+i\sin(y\ln b)).</math>

Non-integer exponents with a complex base

In the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only. For other bases, difficulties appear already with the apparently simple case of Template:Mvarth roots, that is, of exponents <math>1/n,</math> where Template:Mvar is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to Template:Mvarth roots, this case deserves to be considered first, since it does not need to use complex logarithms, and is therefore easier to understand.

Template:Mvarth roots of a complex number

Every nonzero complex number Template:Mvar may be written in polar form as

<math>z=\rho e^{i\theta}=\rho(\cos \theta +i \sin \theta),</math>

where <math>\rho</math> is the absolute value of Template:Mvar, and <math>\theta</math> is its argument. The argument is defined up to an integer multiple of Template:Math; this means that, if <math>\theta</math> is the argument of a complex number, then <math>\theta +2k\pi</math> is also an argument of the same complex number for every integer <math>k</math>.

The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an Template:Mvarth root of a complex number can be obtained by taking the Template:Mvarth root of the absolute value and dividing its argument by Template:Mvar:

<math>\left(\rho e^{i\theta}\right)^\frac 1n=\sqrt[n]\rho \,e^\frac{i\theta}n.</math>

If <math>2\pi</math> is added to <math>\theta</math>, the complex number is not changed, but this adds <math>2i\pi/n</math> to the argument of the Template:Mvarth root, and provides a new Template:Mvarth root. This can be done Template:Mvar times (<math>k=0,1,...,n-1</math>), and provides the Template:Mvar Template:Mvarth roots of the complex number:

<math>\left(\rho e^{i(\theta+2k\pi)}\right)^\frac 1n=\sqrt[n]\rho \,e^\frac{i(\theta+2k\pi)}n.</math>

It is usual to choose one of the Template:Mvar Template:Mvarth root as the principal root. The common choice is to choose the Template:Mvarth root for which <math>-\pi<\theta\le \pi,</math> that is, the Template:Mvarth root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal Template:Mvarth root a continuous function in the whole complex plane, except for negative real values of the radicand. This function equals the usual Template:Mvarth root for positive real radicands. For negative real radicands, and odd exponents, the principal Template:Mvarth root is not real, although the usual Template:Mvarth root is real. Analytic continuation shows that the principal Template:Mvarth root is the unique complex differentiable function that extends the usual Template:Mvarth root to the complex plane without the nonpositive real numbers.

If the complex number is moved around zero by increasing its argument, after an increment of <math>2\pi,</math> the complex number comes back to its initial position, and its Template:Mvarth roots are permuted circularly (they are multiplied by <math DISPLAY=textstyle>e^{2i\pi/n}</math>). This shows that it is not possible to define a Template:Mvarth root function that is continuous in the whole complex plane.

Roots of unity

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

File:One3Root.svg

The three third roots of Template:Math

The Template:Mvarth roots of unity are the Template:Mvar complex numbers such that Template:Math, where Template:Mvar is a positive integer. They arise in various areas of mathematics, such as in discrete Fourier transform or algebraic solutions of algebraic equations (Lagrange resolvent).

The Template:Mvar Template:Mvarth roots of unity are the Template:Mvar first powers of <math>\omega =e^\frac{2\pi i}{n}</math>, that is <math>1=\omega^0=\omega^n, \omega=\omega^1, \omega^2,..., \omega^{n-1}.</math> The Template:Mvarth roots of unity that have this generating property are called primitive Template:Mvarth roots of unity; they have the form <math>\omega^k=e^\frac{2k\pi i}{n},</math> with Template:Mvar coprime with Template:Mvar. The unique primitive square root of unity is <math>-1;</math> the primitive fourth roots of unity are <math>i</math> and <math>-i.</math>

The Template:Mvarth roots of unity allow expressing all Template:Mvarth roots of a complex number Template:Mvar as the Template:Mvar products of a given Template:Mvarth roots of Template:Mvar with a Template:Mvarth root of unity.

Geometrically, the Template:Mvarth roots of unity lie on the unit circle of the complex plane at the vertices of a [[regular polygon|regular Template:Mvar-gon]] with one vertex on the real number 1.

As the number <math>e^\frac{2k\pi i}{n}</math> is the primitive Template:Mvarth root of unity with the smallest positive argument, it is called the principal primitive Template:Mvarth root of unity, sometimes shortened as principal Template:Mvarth root of unity, although this terminology can be confused with the principal value of <math>1^{1/n}</math>, which is 1.<ref>Template:Cite book Online resource Template:Webarchive.</ref><ref>Template:Cite book Defined on p. 351.</ref><ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:PrincipalRootofUnity%7CPrincipalRootofUnity.html}} |title = Principal root of unity |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref>

Complex exponentiation

Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for <math DISPLAY=textstyle>z^w</math>. So, either a principal value is defined, which is not continuous for the values of Template:Mvar that are real and nonpositive, or <math DISPLAY=textstyle>z^w</math> is defined as a multivalued function.

In all cases, the complex logarithm is used to define complex exponentiation as

<math>z^w=e^{w\log z},</math>

where <math>\log z</math> is the variant of the complex logarithm that is used, which is a function or a multivalued function such that

<math>e^{\log z}=z</math>

for every Template:Mvar in its domain of definition.

Principal value

The principal value of the complex logarithm is the unique continuous function, commonly denoted <math>\log,</math> such that, for every nonzero complex number Template:Mvar,

<math>e^{\log z}=z,</math>

and the argument of Template:Mvar satisfies

<math>-\pi <\operatorname{Arg}z \le \pi.</math>

The principal value of the complex logarithm is not defined for <math>z=0,</math> it is discontinuous at negative real values of Template:Mvar, and it is holomorphic (that is, complex differentiable) elsewhere. If Template:Mvar is real and positive, the principal value of the complex logarithm is the natural logarithm: <math>\log z=\ln z.</math>

The principal value of <math>z^w</math> is defined as <math>z^w=e^{w\log z},</math> where <math>\log z</math> is the principal value of the logarithm.

The function <math>(z,w)\to z^w</math> is holomorphic except in the neighbourhood of the points where Template:Mvar is real and nonpositive.

If Template:Mvar is real and positive, the principal value of <math>z^w</math> equals its usual value defined above. If <math>w=1/n,</math> where Template:Mvar is an integer, this principal value is the same as the one defined above.

Multivalued function

In some contexts, there is a problem with the discontinuity of the principal values of <math>\log z</math> and <math>z^w</math> at the negative real values of Template:Mvar. In this case, it is useful to consider these functions as multivalued functions.

If <math>\log z</math> denotes one of the values of the multivalued logarithm (typically its principal value), the other values are <math>2ik\pi +\log z,</math> where Template:Mvar is any integer. Similarly, if <math>z^w</math> is one value of the exponentiation, then the other values are given by

<math>e^{w(2ik\pi +\log z)} = z^we^{2ik\pi w},</math>

where Template:Mvar is any integer.

Different values of Template:Mvar give different values of <math>z^w</math> unless Template:Mvar is a rational number, that is, there is an integer Template:Mvar such that Template:Mvar is an integer. This results from the periodicity of the exponential function, more specifically, that <math>e^a=e^b</math> if and only if <math>a-b</math> is an integer multiple of <math>2\pi i.</math>

If <math>w=\frac mn</math> is a rational number with Template:Mvar and Template:Mvar coprime integers with <math>n>0,</math> then <math>z^w</math> has exactly Template:Mvar values. In the case <math>m=1,</math> these values are the same as those described in [[#nth roots of a complex number|§ Template:Mvarth roots of a complex number]]. If Template:Mvar is an integer, there is only one value that agrees with that of Template:Slink.

The multivalued exponentiation is holomorphic for <math>z\ne 0,</math> in the sense that its graph consists of several sheets that define each a holomorphic function in the neighborhood of every point. If Template:Mvar varies continuously along a circle around Template:Math, then, after a turn, the value of <math>z^w</math> has changed of sheet.

Computation

The canonical form <math>x+iy</math> of <math>z^w</math> can be computed from the canonical form of Template:Mvar and Template:Mvar. Although this can be described by a single formula, it is clearer to split the computation in several steps.

• Polar form of Template:Mvar. If <math>z=a+ib</math> is the canonical form of Template:Mvar (Template:Mvar and Template:Mvar being real), then its polar form is <math display=block>z=\rho e^{i\theta}= \rho (\cos\theta + i \sin\theta),</math> with <math display=inline>\rho=\sqrt{a^2+b^2}</math> and <math>\theta=\operatorname{atan2}(b,a)</math>, where Template:Tmath is the two-argument arctangent function.
• Logarithm of Template:Mvar. The principal value of this logarithm is <math>\log z=\ln \rho+i\theta,</math> where <math>\ln</math> denotes the natural logarithm. The other values of the logarithm are obtained by adding <math>2ik\pi</math> for any integer Template:Mvar.
• Canonical form of <math>w\log z.</math> If <math>w=c+di</math> with Template:Mvar and Template:Mvar real, the values of <math>w\log z</math> are <math display=block>w\log z = (c\ln \rho - d\theta-2dk\pi) +i (d\ln \rho + c\theta+2ck\pi),</math> the principal value corresponding to <math>k=0.</math>
• Final result. Using the identities <math>e^{x+y}=e^xe^y</math> and <math>e^{y\ln x} = x^y,</math> one gets <math DISPLAY=block>z^w=\rho^c e^{-d(\theta+2k\pi)} \left(\cos (d\ln \rho + c\theta+2ck\pi) +i\sin(d\ln \rho + c\theta+2ck\pi)\right),</math> with <math>k=0</math> for the principal value.

Examples

• <math>i^i</math>
  The polar form of Template:Mvar is <math>i=e^{i\pi/2},</math> and the values of <math>\log i</math> are thus <math DISPLAY=block>\log i=i\left(\frac \pi 2 +2k\pi\right).</math> It follows that <math DISPLAY=block>i^i=e^{i\log i}=e^{-\frac \pi 2} e^{-2k\pi}.</math>So, all values of <math>i^i</math> are real, the principal one being <math DISPLAY=block> e^{-\frac \pi 2} \approx 0.2079.</math>
• <math>(-2)^{3+4i}</math>
  Similarly, the polar form of Template:Math is <math>-2 = 2e^{i \pi}.</math> So, the above described method gives the values <math DISPLAY=block>\begin{align}

(-2)^{3 + 4i} &= 2^3 e^{-4(\pi+2k\pi)} (\cos(4\ln 2 + 3(\pi +2k\pi)) +i\sin(4\ln 2 + 3(\pi+2k\pi)))\\ &=-2^3 e^{-4(\pi+2k\pi)}(\cos(4\ln 2) +i\sin(4\ln 2)). \end{align}</math>In this case, all the values have the same argument <math>4\ln 2,</math> and different absolute values.

In both examples, all values of <math>z^w</math> have the same argument. More generally, this is true if and only if the real part of Template:Mvar is an integer.

Failure of power and logarithm identities

Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions. For example:

Template:Bulleted list{(-1)^\frac{1}{2}} = \frac{1}{i} = -i</math>

On the other hand, when Template:Mvar is an integer, the identities are valid for all nonzero complex numbers.

If exponentiation is considered as a multivalued function then the possible values of Template:Math are Template:Math. The identity holds, but saying Template:Math is incorrect. | The identity Template:Math holds for real numbers Template:Mvar and Template:Mvar, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:<ref name="Clausen1827">Template:Cite journal</ref>

For any integer Template:Mvar, we have:

1. <math>e^{1 + 2 \pi i n} = e^1 e^{2 \pi i n} = e \cdot 1 = e</math>
2. <math>\left(e^{1 + 2\pi i n}\right)^{1 + 2 \pi i n} = e\qquad</math> (taking the <math>(1 + 2 \pi i n)</math>-th power of both sides)
3. <math>e^{1 + 4 \pi i n - 4 \pi^2 n^2} = e\qquad</math> (using <math>\left(e^x\right)^y = e^{xy}</math> and expanding the exponent)
4. <math>e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e\qquad</math> (using <math>e^{x+y} = e^x e^y</math>)
5. <math>e^{-4 \pi^2 n^2} = 1\qquad</math> (dividing by Template:Mvar)

but this is false when the integer Template:Mvar is nonzero.

The error is the following: by definition, <math>e^y</math> is a notation for <math>\exp(y),</math> a true function, and <math>x^y</math> is a notation for <math>\exp(y\log x),</math> which is a multi-valued function. Thus the notation is ambiguous when Template:Math. Here, before expanding the exponent, the second line should be <math display="block">\exp\left((1 + 2\pi i n)\log \exp(1 + 2\pi i n)\right) = \exp(1 + 2\pi i n).</math>

Therefore, when expanding the exponent, one has implicitly supposed that <math>\log \exp z =z</math> for complex values of Template:Mvar, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity Template:Math must be replaced by the identity <math display="block">\left(e^x\right)^y = e^{y\log e^x},</math> which is a true identity between multivalued functions. }}

Irrationality and transcendence

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} If Template:Mvar is a positive real algebraic number, and Template:Mvar is a rational number, then Template:Math is an algebraic number. This results from the theory of algebraic extensions. This remains true if Template:Mvar is any algebraic number, in which case, all values of Template:Math (as a multivalued function) are algebraic. If Template:Mvar is irrational (that is, not rational), and both Template:Mvar and Template:Mvar are algebraic, Gelfond–Schneider theorem asserts that all values of Template:Math are transcendental (that is, not algebraic), except if Template:Mvar equals Template:Math or Template:Math.

In other words, if Template:Mvar is irrational and <math>b\not\in \{0,1\},</math> then at least one of Template:Mvar, Template:Mvar and Template:Math is transcendental.

Integer powers in algebra

The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication.<ref group="nb">More generally, power associativity is sufficient for the definition.</ref> The definition of Template:Math requires further the existence of a multiplicative identity.<ref>Template:Cite book</ref>

An algebraic structure consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by Template:Math is a monoid. In such a monoid, exponentiation of an element Template:Mvar is defined inductively by

• <math>x^0 = 1,</math>
• <math>x^{n+1} = x x^n</math> for every nonnegative integer Template:Mvar.

If Template:Mvar is a negative integer, <math>x^n</math> is defined only if Template:Mvar has a multiplicative inverse.<ref>Template:Cite book</ref> In this case, the inverse of Template:Mvar is denoted Template:Math, and Template:Math is defined as <math>\left(x^{-1}\right)^{-n}.</math>

Exponentiation with integer exponents obeys the following laws, for Template:Mvar and Template:Mvar in the algebraic structure, and Template:Mvar and Template:Mvar integers:

<math>\begin{align}

x^0&=1\\ x^{m+n}&=x^m x^n\\ (x^m)^n&=x^{mn}\\ (xy)^n&=x^n y^n \quad \text{if } xy=yx, \text{and, in particular, if the multiplication is commutative.} \end{align}</math>

These definitions are widely used in many areas of mathematics, notably for groups, rings, fields, square matrices (which form a ring). They apply also to functions from a set to itself, which form a monoid under function composition. This includes, as specific instances, geometric transformations, and endomorphisms of any mathematical structure.

When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if Template:Mvar is a real function whose valued can be multiplied, <math>f^n</math> denotes the exponentiation with respect of multiplication, and <math>f^{\circ n}</math> may denote exponentiation with respect of function composition. That is,

<math>(f^n)(x)=(f(x))^n=f(x) \,f(x) \cdots f(x),</math>

and

<math>(f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots)).</math>

Commonly, <math>(f^n)(x)</math> is denoted <math>f(x)^n,</math> while <math>(f^{\circ n})(x)</math> is denoted <math>f^n(x).</math>

In a group

A multiplicative group is a set with as associative operation denoted as multiplication, that has an identity element, and such that every element has an inverse.

So, if Template:Mvar is a group, <math>x^n</math> is defined for every <math>x\in G</math> and every integer Template:Mvar.

The set of all powers of an element of a group form a subgroup. A group (or subgroup) that consists of all powers of a specific element Template:Mvar is the cyclic group generated by Template:Mvar. If all the powers of Template:Mvar are distinct, the group is isomorphic to the additive group <math>\Z</math> of the integers. Otherwise, the cyclic group is finite (it has a finite number of elements), and its number of elements is the order of Template:Mvar. If the order of Template:Mvar is Template:Mvar, then <math>x^n=x^0=1,</math> and the cyclic group generated by Template:Mvar consists of the Template:Mvar first powers of Template:Mvar (starting indifferently from the exponent Template:Math or Template:Math).

Order of elements play a fundamental role in group theory. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the order of the group). The possible orders of group elements are important in the study of the structure of a group (see Sylow theorems), and in the classification of finite simple groups.

Superscript notation is also used for conjugation; that is, Template:Math, where Template:Math and Template:Math are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely <math>(g^h)^k=g^{hk}</math> and <math>(gh)^k=g^kh^k.</math>

In a ring

In a ring, it may occur that some nonzero elements satisfy <math>x^n=0</math> for some integer Template:Mvar. Such an element is said to be nilpotent. In a commutative ring, the nilpotent elements form an ideal, called the nilradical of the ring.

If the nilradical is reduced to the zero ideal (that is, if <math>x\neq 0</math> implies <math>x^n\neq 0</math> for every positive integer Template:Mvar), the commutative ring is said to be reduced. Reduced rings are important in algebraic geometry, since the coordinate ring of an affine algebraic set is always a reduced ring.

More generally, given an ideal Template:Mvar in a commutative ring Template:Mvar, the set of the elements of Template:Mvar that have a power in Template:Mvar is an ideal, called the radical of Template:Mvar. The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring <math>k[x_1, \ldots, x_n]</math> over a field Template:Mvar, an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of Hilbert's Nullstellensatz).

Matrices and linear operators

If Template:Math is a square matrix, then the product of Template:Math with itself Template:Math times is called the matrix power. Also <math>A^0</math> is defined to be the identity matrix,<ref>Chapter 1, Elementary Linear Algebra, 8E, Howard Anton.</ref> and if Template:Math is invertible, then <math>A^{-n} = \left(A^{-1}\right)^n</math>.

Matrix powers appear often in the context of discrete dynamical systems, where the matrix Template:Math expresses a transition from a state vector Template:Math of some system to the next state Template:Math of the system.<ref>Template:Citation</ref> This is the standard interpretation of a Markov chain, for example. Then <math>A^2x</math> is the state of the system after two time steps, and so forth: <math>A^nx</math> is the state of the system after Template:Math time steps. The matrix power <math>A^n</math> is the transition matrix between the state now and the state at a time Template:Math steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors.

Apart from matrices, more general linear operators can also be exponentiated. An example is the derivative operator of calculus, <math>d/dx</math>, which is a linear operator acting on functions <math>f(x)</math> to give a new function <math>(d/dx)f(x) = f'(x)</math>. The Template:Mathth power of the differentiation operator is the Template:Mathth derivative:

<math>\left(\frac{d}{dx}\right)^nf(x) = \frac{d^n}{dx^n}f(x) = f^{(n)}(x).</math>

These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups.<ref>E. Hille, R. S. Phillips: Functional Analysis and Semi-Groups. American Mathematical Society, 1975.</ref> Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the heat equation, Schrödinger equation, wave equation, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.

Finite fields

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A field is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is associative and every nonzero element has a multiplicative inverse. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of Template:Math. Common examples are the field of complex numbers, the real numbers and the rational numbers, considered earlier in this article, which are all infinite.

A finite field is a field with a finite number of elements. This number of elements is either a prime number or a prime power; that is, it has the form <math>q=p^k,</math> where Template:Mvar is a prime number, and Template:Mvar is a positive integer. For every such Template:Mvar, there are fields with Template:Mvar elements. The fields with Template:Mvar elements are all isomorphic, which allows, in general, working as if there were only one field with Template:Mvar elements, denoted <math>\mathbb F_q.</math>

One has

<math>x^q=x</math>

for every <math>x\in \mathbb F_q.</math>

A primitive element in <math>\mathbb F_q</math> is an element Template:Mvar such that the set of the Template:Math first powers of Template:Mvar (that is, <math>\{g^1=g, g^2, \ldots, g^{p-1}=g^0=1\}</math>) equals the set of the nonzero elements of <math>\mathbb F_q.</math> There are <math>\varphi (p-1)</math> primitive elements in <math>\mathbb F_q,</math> where <math>\varphi</math> is Euler's totient function.

In <math>\mathbb F_q,</math> the freshman's dream identity

<math>(x+y)^p = x^p+y^p</math>

is true for the exponent Template:Mvar. As <math>x^p=x</math> in <math>\mathbb F_q,</math> It follows that the map

<math>\begin{align}

F\colon{} & \mathbb F_q \to \mathbb F_q\\ & x\mapsto x^p \end{align}</math> is linear over <math>\mathbb F_q,</math> and is a field automorphism, called the Frobenius automorphism. If <math>q=p^k,</math> the field <math>\mathbb F_q</math> has Template:Mvar automorphisms, which are the Template:Mvar first powers (under composition) of Template:Mvar. In other words, the Galois group of <math>\mathbb F_q</math> is cyclic of order Template:Mvar, generated by the Frobenius automorphism.

The Diffie–Hellman key exchange is an application of exponentiation in finite fields that is widely used for secure communications. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the discrete logarithm, is computationally expensive. More precisely, if Template:Mvar is a primitive element in <math>\mathbb F_q,</math> then <math>g^e</math> can be efficiently computed with exponentiation by squaring for any Template:Mvar, even if Template:Mvar is large, while there is no known computationally practical algorithm that allows retrieving Template:Mvar from <math>g^e</math> if Template:Mvar is sufficiently large.

Powers of sets Template:Anchor

The Cartesian product of two sets Template:Mvar and Template:Mvar is the set of the ordered pairs <math>(x,y)</math> such that <math>x\in S</math> and <math>y\in T.</math> This operation is not properly commutative nor associative, but has these properties up to canonical isomorphisms, that allow identifying, for example, <math>(x,(y,z)),</math> <math>((x,y),z),</math> and <math>(x,y,z).</math>

This allows defining the Template:Mvarth power <math>S^n</math> of a set Template:Mvar as the set of all Template:Mvar-tuples <math>(x_1, \ldots, x_n)</math> of elements of Template:Mvar.

When Template:Mvar is endowed with some structure, it is frequent that <math>S^n</math> is naturally endowed with a similar structure. In this case, the term "direct product" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example <math>\R^n</math> (where <math>\R</math> denotes the real numbers) denotes the Cartesian product of Template:Mvar copies of <math>\R,</math> as well as their direct product as vector space, topological spaces, rings, etc.

Sets as exponents

Template:See also A Template:Mvar-tuple <math>(x_1, \ldots, x_n)</math> of elements of Template:Mvar can be considered as a function from <math>\{1,\ldots, n\}.</math> This generalizes to the following notation.

Given two sets Template:Mvar and Template:Mvar, the set of all functions from Template:Mvar to Template:Mvar is denoted <math>S^T</math>. This exponential notation is justified by the following canonical isomorphisms (for the first one, see Currying):

<math>(S^T)^U\cong S^{T\times U},</math>

<math>S^{T\sqcup U}\cong S^T\times S^U,</math>

where <math>\times</math> denotes the Cartesian product, and <math>\sqcup</math> the disjoint union.

One can use sets as exponents for other operations on sets, typically for direct sums of abelian groups, vector spaces, or modules. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, <math>\R^\N</math> denotes the vector space of the infinite sequences of real numbers, and <math>\R^{(\N)}</math> the vector space of those sequences that have a finite number of nonzero elements. The latter has a basis consisting of the sequences with exactly one nonzero element that equals Template:Math, while the Hamel bases of the former cannot be explicitly described (because their existence involves Zorn's lemma).

In this context, Template:Math can represents the set <math>\{0,1\}.</math> So, <math>2^S</math> denotes the power set of Template:Mvar, that is the set of the functions from Template:Mvar to <math>\{0,1\},</math> which can be identified with the set of the subsets of Template:Mvar, by mapping each function to the inverse image of Template:Math.

This fits in with the exponentiation of cardinal numbers, in the sense that Template:Math, where Template:Math is the cardinality of Template:Math.

In category theory

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In the category of sets, the morphisms between sets Template:Mvar and Template:Mvar are the functions from Template:Mvar to Template:Mvar. It results that the set of the functions from Template:Mvar to Template:Mvar that is denoted <math>Y^X</math> in the preceding section can also be denoted <math>\hom(X,Y).</math> The isomorphism <math>(S^T)^U\cong S^{T\times U}</math> can be rewritten

<math>\hom(U,S^T)\cong \hom(T\times U,S).</math>

This means the functor "exponentiation to the power Template:Mvar" is a right adjoint to the functor "direct product with Template:Mvar".

This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor <math>X\to X^T</math> is, if it exists, a right adjoint to the functor <math>Y\to T\times Y.</math> A category is called a Cartesian closed category, if direct products exist, and the functor <math>Y\to X\times Y</math> has a right adjoint for every Template:Mvar.

Repeated exponentiation

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation. This sequence of operations is expressed by the Ackermann function and Knuth's up-arrow notation. Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at Template:Math, the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and Template:Val (Template:Math) respectively.

Limits of powers

Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 0⁰. The limits in these examples exist, but have different values, showing that the two-variable function Template:Math has no limit at the point Template:Math. One may consider at what points this function does have a limit.

More precisely, consider the function <math>f(x,y) = x^y</math> defined on <math> D = \{(x, y) \in \mathbf{R}^2 : x > 0 \}</math>. Then Template:Math can be viewed as a subset of Template:Math (that is, the set of all pairs Template:Math with Template:Math, Template:Math belonging to the extended real number line Template:Math, endowed with the product topology), which will contain the points at which the function Template:Math has a limit.

In fact, Template:Math has a limit at all accumulation points of Template:Math, except for Template:Math, Template:Math, Template:Math and Template:Math.<ref>Nicolas Bourbaki, Topologie générale, V.4.2.</ref> Accordingly, this allows one to define the powers Template:Math by continuity whenever Template:Math, Template:Math, except for Template:Math, Template:Math, Template:Math and Template:Math, which remain indeterminate forms.

Under this definition by continuity, we obtain:

• Template:Math and Template:Math, when Template:Math.
• Template:Math and Template:Math, when Template:Math.
• Template:Math and Template:Math, when Template:Math.
• Template:Math and Template:Math, when Template:Math.

These powers are obtained by taking limits of Template:Math for positive values of Template:Math. This method does not permit a definition of Template:Math when Template:Math, since pairs Template:Math with Template:Math are not accumulation points of Template:Math.

On the other hand, when Template:Math is an integer, the power Template:Math is already meaningful for all values of Template:Math, including negative ones. This may make the definition Template:Math obtained above for negative Template:Math problematic when Template:Math is odd, since in this case Template:Math as Template:Math tends to Template:Math through positive values, but not negative ones.

Efficient computation with integer exponents

Computing Template:Math using iterated multiplication requires Template:Math multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute Template:Math, apply Horner's rule to the exponent 100 written in binary:

<math>100 = 2^2 +2^5 + 2^6 = 2^2(1+2^3(1+2))</math>.

Then compute the following terms in order, reading Horner's rule from right to left. Template:Static row numbers

22 = 4

2 (22) = 23 = 8

(23)2 = 26 = 64

(26)2 = 212 = Template:Val

(212)2 = 224 = Template:Val

2 (224) = 225 = Template:Val

(225)2 = 250 = Template:Val

(250)2 = 2100 = Template:Val

This series of steps only requires 8 multiplications instead of 99.

In general, the number of multiplication operations required to compute Template:Math can be reduced to <math>\sharp n +\lfloor \log_{2} n\rfloor -1,</math> by using exponentiation by squaring, where <math>\sharp n</math> denotes the number of Template:Maths in the binary representation of Template:Mvar. For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal addition-chain exponentiation. Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for Template:Math is a difficult problem, for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available.<ref>Template:Cite journal</ref> However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.

Iterated functions

Template:See also Function composition is a binary operation that is defined on functions such that the codomain of the function written on the right is included in the domain of the function written on the left. It is denoted <math>g\circ f,</math> and defined as

<math>(g\circ f)(x)=g(f(x))</math>

for every Template:Mvar in the domain of Template:Mvar.

If the domain of a function Template:Mvar equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the Template:Mvarth power of the function under composition, commonly called the Template:Mvarth iterate of the function. Thus <math>f^n</math> denotes generally the Template:Mvarth iterate of Template:Mvar; for example, <math>f^3(x)</math> means <math>f(f(f(x))).</math><ref name="Peano_1903"/>

When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the pointwise multiplication, which induces another exponentiation. When using functional notation, the two kinds of exponentiation are generally distinguished by placing the exponent of the functional iteration before the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication after the parentheses. Thus <math>f^2(x)= f(f(x)),</math> and <math>f(x)^2= f(x)\cdot f(x).</math> When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example <math>f^{\circ 3}=f\circ f \circ f,</math> and <math>f^3=f\cdot f\cdot f.</math> For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the trigonometric functions. So, <math>\sin^2 x</math> and <math>\sin^2(x)</math> both mean <math>\sin(x)\cdot\sin(x)</math> and not <math>\sin(\sin(x)),</math> which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.<ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Cajori_1929"/>

In this context, the exponent <math>-1</math> denotes always the inverse function, if it exists. So <math>\sin^{-1}x=\sin^{-1}(x) = \arcsin x.</math> For the multiplicative inverse fractions are generally used as in <math>1/\sin(x)=\frac 1{\sin x}.</math>

In programming languages

Programming languages generally express exponentiation either as an infix operator or as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is the caret (^). The original version of ASCII included an uparrow symbol (↑), intended for exponentiation, but this was replaced by the caret in 1967, so the caret became usual in programming languages.<ref>Template:Cite book</ref> The notations include:

• x ^ y: AWK, BASIC, J, MATLAB, Wolfram Language (Mathematica), R, Microsoft Excel, Analytica, TeX (and its derivatives), TI-BASIC, bc (for integer exponents), Haskell (for nonnegative integer exponents), Lua, and most computer algebra systems.
• x ** y. The Fortran character set did not include lowercase characters or punctuation symbols other than +-*/()&=.,' and so used ** for exponentiation<ref name="Sayre_1956"/><ref>Template:Citation</ref> (the initial version used a xx b instead.<ref name="Backus_1954"/>). Many other languages followed suit: Ada, Z shell, KornShell, Bash, COBOL, CoffeeScript, Fortran, FoxPro, Gnuplot, Groovy, JavaScript, OCaml, ooRexx, F#, Perl, PHP, PL/I, Python, Rexx, Ruby, SAS, Seed7, Tcl, ABAP, Mercury, Haskell (for floating-point exponents), Turing, and VHDL.
• x ↑ y: Algol Reference language, Commodore BASIC, TRS-80 Level II/III BASIC.<ref name="InfoWorld_1982">Template:Cite news</ref><ref name="80Micro_1983">Template:Cite journal</ref>
• x ^^ y: Haskell (for fractional base, integer exponents), D.
• x⋆y: APL.

In most programming languages with an infix exponentiation operator, it is right-associative, that is, a^b^c is interpreted as a^(b^c).<ref>Template:Cite book</ref> This is because (a^b)^c is equal to a^(b*c) and thus not as useful. In some languages, it is left-associative, notably in Algol, MATLAB, and the Microsoft Excel formula language.

Other programming languages use functional notation:

• (expt x y): Common Lisp.
• pown x y: F# (for integer base, integer exponent).

Still others only provide exponentiation as part of standard libraries:

• pow(x, y): C, C++ (in math library).
• Math.Pow(x, y): C#.
• math:pow(X, Y): Erlang.
• Math.pow(x, y): Java.
• [Math]::Pow(x, y): PowerShell.

In some statically typed languages that prioritize type safety such as Rust, exponentiation is performed via a multitude of methods:

• x.pow(y) for x and y as integers
• x.powf(y) for x and y as floating-point numbers
• x.powi(y) for x as a float and y as an integer

See also

• Double exponential function
• Exponential decay
• Exponential field
• Exponential growth
• Pentation
• List of exponential topics
• Modular exponentiation
• Unicode subscripts and superscripts
• x^y = y^x