Editing Exponentiation (section)

==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 {{mvar|n}}th roots, that is, of exponents <math>1/n,</math> where {{mvar|n}} is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to {{mvar|n}}th roots, this case deserves to be considered first, since it does not need to use [[complex logarithm]]s, and is therefore easier to understand.

==={{mvar|n}}th roots of a complex number===
Every nonzero complex number {{mvar|z}} 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 {{mvar|z}}, and <math>\theta</math> is its [[argument (complex analysis)|argument]]. The argument is defined [[up to]] an integer multiple of {{math|2{{pi}}}}; 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 {{mvar|n}}th root of a complex number can be obtained by taking the {{mvar|n}}th root of the absolute value and dividing its argument by {{mvar|n}}:
: <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 {{mvar|n}}th root, and provides a new {{mvar|n}}th root. This can be done {{mvar|n}} times (<math>k=0,1,...,n-1</math>), and provides the {{mvar|n}} {{mvar|n}}th 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 {{mvar|n}} {{mvar|n}}th root as the [[principal root]]. The common choice is to choose the {{mvar|n}}th root for which <math>-\pi<\theta\le \pi,</math> that is, the {{mvar|n}}th root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal {{mvar|n}}th root a [[continuous function]] in the whole complex plane, except for negative real values of the [[radicand]]. This function equals the usual {{mvar|n}}th root for positive real radicands. For negative real radicands, and odd exponents, the principal {{mvar|n}}th root is not real, although the usual {{mvar|n}}th root is real. [[Analytic continuation]] shows that the principal {{mvar|n}}th root is the unique [[complex differentiable]] function that extends the usual {{mvar|n}}th 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 {{mvar|n}}th roots are [[circular permutation|permuted circularly]] (they are multiplied by <math DISPLAY=textstyle>e^{2i\pi/n}</math>). This shows that it is not possible to define a {{mvar|n}}th root function that is continuous in the whole complex plane.

====Roots of unity====
{{Main|Root of unity}}

[[File:One3Root.svg|thumb|right|The three third roots of {{math|1}}]]
The {{mvar|n}}th roots of unity are the {{mvar|n}} complex numbers such that {{math|1=''w''<sup>''n''</sup> = 1}}, where {{mvar|n}} 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 {{mvar|n}} {{mvar|n}}th roots of unity are the {{mvar|n}} 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 {{mvar|n}}th roots of unity that have this generating property are called ''primitive {{mvar|n}}th roots of unity''; they have the form <math>\omega^k=e^\frac{2k\pi i}{n},</math> with {{mvar|k}} [[coprime integers|coprime]] with {{mvar|n}}. 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 {{mvar|n}}th roots of unity allow expressing all {{mvar|n}}th roots of a complex number {{mvar|z}} as the {{mvar|n}} products of a given {{mvar|n}}th roots of {{mvar|z}} with a {{mvar|n}}th root of unity.

Geometrically, the {{mvar|n}}th roots of unity lie on the [[unit circle]] of the [[complex plane]] at the vertices of a [[regular polygon|regular {{mvar|n}}-gon]] with one vertex on the real number 1.

As the number <math>e^\frac{2k\pi i}{n}</math> is the primitive {{mvar|n}}th root of unity with the smallest positive [[argument (complex analysis)|argument]], it is called the ''principal primitive {{mvar|n}}th root of unity'', sometimes shortened as ''principal {{mvar|n}}th root of unity'', although this terminology can be confused with the [[principal value]] of <math>1^{1/n}</math>, which is 1.<ref>{{cite book |title=Introduction to Algorithms |edition=second |author-last1=Cormen |author-first1=Thomas H. |author-last2=Leiserson |author-first2=Charles E. |author-last3=Rivest |author-first3=Ronald L. |author-last4=Stein |author-first4=Clifford |publisher=[[MIT Press]] |date=2001 |isbn=978-0-262-03293-3}} [http://highered.mcgraw-hill.com/sites/0070131511/student_view0/chapter30/glossary.html Online resource] {{webarchive|url=https://web.archive.org/web/20070930201902/http://highered.mcgraw-hill.com/sites/0070131511/student_view0/chapter30/glossary.html |date=2007-09-30}}.</ref><ref>{{cite book |title=Difference Equations: From Rabbits to Chaos |title-link= Difference Equations: From Rabbits to Chaos |edition=[[Undergraduate Texts in Mathematics]] |author-last1=Cull |author-first1=Paul |author-last2=Flahive |author-first2=Mary |author-link2=Mary Flahive |author-last3=Robson |author-first3=Robby |date=2005 |publisher=Springer |isbn=978-0-387-23234-8}} Defined on p. 351.</ref><ref>{{MathWorld |title=Principal root of unity |id=PrincipalRootofUnity}}</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 {{mvar|z}} 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 {{mvar|z}} in its [[domain of a function|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 {{mvar|z}},
: <math>e^{\log z}=z,</math>
and the [[argument (complex analysis)|argument]] of {{mvar|z}} 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 [[continuous function|discontinuous]] at negative real values of {{mvar|z}}, and it is [[holomorphic]] (that is, complex differentiable) elsewhere. If {{mvar|z}} 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 {{mvar|z}} is real and nonpositive.

If {{mvar|z}} 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 {{mvar|n}} 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 {{mvar|z}}. In this case, it is useful to consider these functions as [[multivalued function]]s.

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 {{mvar|k}} 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 {{mvar|k}} is any integer.

Different values of {{mvar|k}} give different values of <math>z^w</math> unless {{mvar|w}} is a [[rational number]], that is, there is an integer {{mvar|d}} such that {{mvar|dw}} is an integer. This results from the [[periodic function|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 {{mvar|m}} and {{mvar|n}} [[coprime integers]] with <math>n>0,</math> then <math>z^w</math> has exactly {{mvar|n}} values. In the case <math>m=1,</math> these values are the same as those described in [[#nth roots of a complex number|§ {{mvar|n}}th roots of a complex number]]. If {{mvar|w}} is an integer, there is only one value that agrees with that of {{slink||Integer exponents}}.

The multivalued exponentiation is holomorphic for <math>z\ne 0,</math> in the sense that its [[graph of a function|graph]] consists of several sheets that define each a holomorphic function in the neighborhood of every point. If {{mvar|z}} varies continuously along a circle around {{math|0}}, 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 {{mvar|z}} and {{mvar|w}}. Although this can be described by a single formula, it is clearer to split the computation in several steps.
* ''[[Polar form]] of {{mvar|z}}''. If <math>z=a+ib</math> is the canonical form of {{mvar|z}} ({{mvar|a}} and {{mvar|b}} 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 {{tmath|\operatorname{atan2} }} is the [[atan2|two-argument arctangent]] function.
* ''[[complex logarithm|Logarithm]] of {{mvar|z}}''. 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 {{mvar|k}}.
* ''Canonical form of <math>w\log z.</math>'' If <math>w=c+di</math> with {{mvar|c}} and {{mvar|d}} 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> <br>The polar form of {{mvar|i}} 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><br>Similarly, the polar form of {{math|−2}} 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 {{mvar|w}} 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:

{{bulleted list
| The identity {{math|1=log(''b''<sup>''x''</sup>) = ''x'' ⋅ log ''b''}} holds whenever {{mvar|b}} is a positive real number and {{mvar|x}} is a real number. But for the [[principal branch]] of the complex logarithm one has
<math display="block"> \log((-i)^2) = \log(-1) = i\pi \neq 2\log(-i) = 2\log(e^{-i\pi/2})=2\,\frac{-i\pi}{2} = -i\pi</math>

Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:
<math display="block">\log w^z \equiv z \log w \pmod{2 \pi i}</math>

This identity does not hold even when considering log as a multivalued function. The possible values of {{math|log(''w''<sup>''z''</sup>)}} contain those of {{math|''z'' ⋅ log ''w''}} as a [[proper subset]]. Using {{math|Log(''w'')}} for the principal value of {{math|log(''w'')}} and {{mvar|m}}, {{mvar|n}} as any integers the possible values of both sides are:
<math display="block">\begin{align}
      \left\{\log w^z \right\} &= \left\{ z \cdot \operatorname{Log} w + z \cdot 2 \pi i n + 2 \pi i m \mid m,n\in\Z\right\} \\
      \left\{z \log w \right\} &= \left\{ z \operatorname{Log} w + z \cdot 2 \pi i n \mid n\in \Z\right\}
\end{align}</math>
| The identities {{math|1=(''bc'')<sup>''x''</sup> = ''b''<sup>''x''</sup>''c''<sup>''x''</sup>}} and {{math|1=(''b''/''c'')<sup>''x''</sup> = ''b''<sup>''x''</sup>/''c''<sup>''x''</sup>}} are valid when {{mvar|b}} and {{mvar|c}} are positive real numbers and {{mvar|x}} is a real number. But, for the principal values, one has
<math display="block">(-1 \cdot -1)^\frac{1}{2} =1 \neq (-1)^\frac{1}{2} (-1)^\frac{1}{2} =i \cdot i=i^2 =-1</math>
and
<math display="block">\left(\frac{1}{-1}\right)^\frac{1}{2} = (-1)^\frac{1}{2} = i \neq \frac{1^\frac{1}{2}}{(-1)^\frac{1}{2}} = \frac{1}{i} = -i</math>

On the other hand, when {{mvar|x}} 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 {{math|(−1 ⋅ −1)<sup>1/2</sup>}} are {{math|{{mset|1, −1}}}}. The identity holds, but saying {{math|1={1} = {{mset|(−1 ⋅ −1)<sup>1/2</sup>}}}} is incorrect.
| The identity {{math|1=(''e''<sup>''x''</sup>)<sup>''y''</sup> = ''e''<sup>''xy''</sup>}} holds for real numbers {{mvar|x}} and {{mvar|y}}, but assuming its truth for complex numbers leads to the following [[mathematical fallacy|paradox]], discovered in 1827 by [[Thomas Clausen (mathematician)|Clausen]]:<ref name="Clausen1827">{{cite journal |author-last1=Steiner |author-first1=J. |author-last2=Clausen |author-first2=T. |author-last3=Abel |author-first3=Niels Henrik |author-link3=Niels Henrik Abel |title=Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen |trans-title=Problems and propositions, the former to solve, the later to prove |journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]] |volume=2 |date=1827 |pages=286–287 |url=https://www.digizeitschriften.de/dms/img/?PID=PPN243919689_0002%7Clog33&physid=phys301#navi}}</ref>

For any integer {{mvar|n}}, we have:
# <math>e^{1 + 2 \pi i n} = e^1 e^{2 \pi i n} = e \cdot 1 = e</math>
# <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)
# <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)
# <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>)
# <math>e^{-4 \pi^2 n^2} = 1\qquad</math> (dividing by {{mvar|e}})

but this is false when the integer {{mvar|n}} 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 {{math|1=''x'' = ''e''}}. 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 {{mvar|z}}, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity {{math|1=(''e''<sup>''x''</sup>)<sup>''y''</sup> = ''e''<sup>''xy''</sup>}} 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.
}}