Editing Complex number (section)

===Complex logarithm===
{{main|Complex logarithm}}
[[File:ComplexExpStrips.svg|right|thumb|The exponential function maps complex numbers ''z'' differing by a multiple of <math>2\pi i</math> to the same complex number ''w''.]]
For any positive real number ''t'', there is a unique real number ''x'' such that <math>\exp(x) = t</math>. This leads to the definition of the [[natural logarithm]] as the [[inverse function|inverse]] 
<math>\ln \colon \R^+ \to \R ; x \mapsto \ln x </math> of the exponential function. The situation is different for complex numbers, since
:<math>\exp(z+2\pi i) = \exp z \exp (2 \pi i) = \exp z</math>
by the functional equation and Euler's identity. 
For example, {{math|1=''e''{{sup|''iπ''}} = ''e''{{sup|3''iπ''}} = −1}} , so both {{mvar|iπ}} and {{math|3''iπ''}} are possible values for the complex logarithm of {{math|−1}}. 

In general, given any non-zero complex number ''w'', any number ''z'' solving the equation
:<math>\exp z = w</math>
is called a [[complex logarithm]] of {{mvar|w}}, denoted <math>\log w</math>. It can be shown that these numbers satisfy
<math display=block>z = \log w = \ln|w| + i\arg w, </math>
where <math>\arg</math> is the [[arg (mathematics)|argument]] defined [[#Polar form|above]], and <math>\ln</math> the (real) [[natural logarithm]]. As arg is a [[multivalued function]], unique only up to a multiple of {{math|2''π''}}, log is also multivalued. The [[principal value]] of log is often taken by restricting the imaginary part to the [[interval (mathematics)|interval]] {{open-closed|−''π'', ''π''}}. This leads to the complex logarithm being a [[bijective]] function taking values in the strip <math>\R^+ + \; i \, \left(-\pi, \pi\right]</math> (that is denoted <math>S_0</math> in the above illustration)
<math display=block>\ln \colon \; \Complex^\times \; \to \; \; \; \R^+ + \; i \, \left(-\pi, \pi\right] .</math>

If <math>z \in \Complex \setminus \left( -\R_{\ge 0} \right)</math> is not a non-positive real number (a positive or a non-real number), the resulting [[principal value]] of the complex logarithm is obtained with {{math|−''π'' < ''φ'' < ''π''}}. It is an [[analytic function]] outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number <math>z \in -\R^+ </math>, where the principal value is {{math|1=ln ''z'' = ln(−''z'') + ''iπ''}}.{{efn|However for another inverse function of the complex exponential function (and not the above defined principal value), the branch cut could be taken at any other [[Line (geometry)#Ray|ray]] thru the origin.}}

Complex [[exponentiation]] {{math|''z''<sup>''ω''</sup>}} is defined as
<math display=block>z^\omega = \exp(\omega \ln z), </math>
and is multi-valued, except when {{mvar|ω}} is an integer. For {{math|1=''ω'' = 1 / ''n''}}, for some natural number {{mvar|n}}, this recovers the non-uniqueness of {{mvar|n}}th roots mentioned above. If {{math|''z'' > 0}} is real (and {{mvar|ω}} an arbitrary complex number), one has a preferred choice of <math>\ln x</math>, the real logarithm, which can be used to define a preferred exponential function.

Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see [[Exponentiation#Failure of power and logarithm identities|failure of power and logarithm identities]]. For example, they do not satisfy
<math display=block>a^{bc} = \left(a^b\right)^c.</math>
Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.