Editing Complex number (section)

==Complex analysis==

{{main|Complex analysis}}
<!--
[[File:Color complex plot.jpg|upright=0.8|right|thumb|[[Domain coloring]] plot of the function
<br /><math>f(x) = \tfrac{(x^2 - 1)(x - 2 - i)^2}{x^2 + 2 + 2 i}</math><br />
The hue represents the function argument, while the saturation and [[Lightness (color)|value]] represent the magnitude.]]

The absolute value has three important properties:

<math display=block> | z | \geq 0, \,</math> where <math> | z | = 0 \,</math> [[if and only if]] <math> z = 0</math>

<math display=block> | z + w | \leq | z | + | w | \,</math> ([[triangle inequality]])

<math display=block> | z \cdot w | = | z | \cdot | w | </math>

for all complex numbers {{mvar|z}} and {{mvar|w}}. These imply that {{math|1={{!}}1{{!}} = 1}} and {{math|1={{!}}''z''/''w''{{!}} = {{!}}''z''{{!}}/{{!}}''w''{{!}}}}. By defining the '''distance''' function {{math|1=''d''(''z'', ''w'') = {{!}}''z'' − ''w''{{!}}}}, we turn the set of complex numbers into a [[metric space]] and we can therefore talk about [[limit (mathematics)|limits]] and [[continuous function|continuity]].

In general, distances between complex numbers are given by the distance function {{math|1=''d''(''z'', ''w'') = {{!}}''z'' − ''w''{{!}}}}, which turns the complex numbers into a [[metric space]] and introduces the ideas of [[limit (mathematics)|limits]] and [[continuous function|continuity]]. All of the standard properties of two dimensional space therefore hold for the complex numbers, including important properties of the modulus such as non-negativity, and the [[triangle inequality]] (<math>| z + w | \leq | z | + | w |</math> for all {{mvar|z}} and {{mvar|w}}).

-->

The study of functions of a complex variable is known as ''[[complex analysis]]'' and has enormous practical use in [[applied mathematics]] as well as in other branches of mathematics. Often, the most natural proofs for statements in [[real analysis]] or even [[number theory]] employ techniques from complex analysis (see [[prime number theorem]] for an example). 

[[File:Complex-plot.png|right|thumb|A [[domain coloring]] graph of the function
{{math|{{sfrac|(''z''<sup>2</sup> − 1)(''z'' − 2 − ''i'')<sup>2</sup>|''z''<sup>2</sup> + 2 + 2''i''}}}}. Darker spots mark moduli near zero, brighter spots are farther away from the origin. The color encodes the argument. The function has zeros for {{math|±1, (2 + ''i'')}} and [[pole (complex analysis)|poles]] at <math>\pm \sqrt{{-2-2i}}.</math>]]
Unlike real functions, which are commonly represented as two-dimensional graphs, [[complex function]]s have four-dimensional graphs and may usefully be illustrated by color-coding a [[graph of a function of two variables|three-dimensional graph]] to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

===Convergence===
[[File:ComplexPowers.svg|right|thumb|Illustration of the behavior of the sequence <math>z^n</math> for three different values of ''z'' (all having the same argument): for <math>|z|<1</math> the sequence converges to 0 (inner spiral), while it diverges for <math>|z|>1</math> (outer spiral).]]
The notions of [[convergent series]] and [[continuous function]]s in (real) analysis have natural analogs in complex analysis. A sequence <!--(''a''<sub>''n''</sub>)<sub>''n'' ≥ 0</sub>--> of complex numbers is said to [[convergent sequence|converge]] if and only if its real and imaginary parts do. This is equivalent to the [[(ε, δ)-definition of limit]]s, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, <math>\mathbb{C}</math>, endowed with the [[metric (mathematics)|metric]]
<math display=block>\operatorname{d}(z_1, z_2) = |z_1 - z_2|</math>
is a complete [[metric space]], which notably includes the [[triangle inequality]]
<math display=block>|z_1 + z_2| \le |z_1| + |z_2|</math>
for any two complex numbers {{math|''z''<sub>1</sub>}} and {{math|''z''<sub>2</sub>}}.

===Complex exponential===
[[File:ComplexExpMapping.svg|thumb|right|Illustration of the complex exponential function mapping the complex plane, ''w'' = exp ⁡(''z''). The left plane shows a square mesh with mesh size 1, with the three complex numbers 0, 1, and ''i'' highlighted. The two rectangles (in magenta and green) are mapped to circular segments, while the lines parallel to the ''x''-axis are mapped to rays emanating from, but not containing the origin. Lines parallel to the ''y''-axis are mapped to circles.]]
Like in real analysis, this notion of convergence is used to construct a number of [[elementary function]]s: the ''[[exponential function]]'' {{math|exp ''z''}}, also written {{math|''e''<sup>''z''</sup>}}, is defined as the [[infinite series]], which can be shown to [[radius of convergence|converge]] for any ''z'':
<math display=block>\exp z:= 1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!}. </math>
For example, <math>\exp (1)</math> is [[E (mathematical constant)|Euler's number]] <math>e \approx 2.718</math>. ''[[Euler's formula]]'' states:
<math display=block>\exp(i\varphi) = \cos \varphi + i\sin \varphi </math>
for any real number {{mvar|φ}}. This formula is a quick consequence of general basic facts about convergent power series and the definitions of the involved functions as power series. As a special case, this includes [[Euler's identity]]
<math display=block>\exp(i \pi) = -1. </math>

===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.

===Complex sine and cosine===
The series defining the real trigonometric functions [[sine|{{math|sin}}]] and [[cosine|{{math|cos}}]], as well as the [[hyperbolic functions]] {{math|sinh}} and {{math|cosh}}, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as [[tangent (function)|{{math|tan}}]], things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of [[analytic continuation]].

The value of a trigonometric or hyperbolic function of a complex number can be expressed in terms of those functions evaluated on real numbers, via angle-addition formulas.  For {{math|1=''z'' = ''x'' + ''iy''}},

<math display=block>\sin{z} = \sin{x} \cosh{y} + i \cos{x} \sinh{y}</math>

<math display=block>\cos{z} = \cos{x} \cosh{y} - i \sin{x} \sinh{y}</math>

<math display=block>\tan{z} = \frac{\tan{x} + i \tanh{y}}{1 - i \tan{x} \tanh{y}}</math>

<math display=block>\cot{z} = -\frac{1 + i \cot{x} \coth{y}}{\cot{x} -i \coth{y}}</math>

<math display=block>\sinh{z} = \sinh{x} \cos{y} + i \cosh{x} \sin{y}</math>

<math display=block>\cosh{z} = \cosh{x} \cos{y} + i \sinh{x} \sin{y}</math>

<math display=block>\tanh{z} = \frac{\tanh{x} + i \tan{y}}{1 + i \tanh{x} \tan{y}}</math>

<math display=block>\coth{z} = \frac{1 - i \coth{x} \cot{y}}{\coth{x} - i \cot{y}}</math>

Where these expressions are not well defined, because a trigonometric or hyperbolic function evaluates to infinity or there is division by zero, they are nonetheless correct as [[Limit (mathematics)|limit]]s.

===Holomorphic functions===
[[File:Sin1z-cplot.svg|thumb|Color wheel graph of the function {{math|sin(1/''z'')}} that is holomorphic except at ''z'' = 0, which is an essential singularity of this function. White parts inside refer to numbers having large absolute values.]]
A function <math>f: \mathbb{C}</math> → <math>\mathbb{C}</math> is called [[Holomorphic function|holomorphic]] or ''complex differentiable'' at a point <math>z_0</math> if the limit
:<math>\lim_{z \to z_0} {f(z) - f(z_0) \over z - z_0 }</math>
exists (in which case it is denoted by <math>f'(z_0)</math>). This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching <math>z_0</math> in different directions imposes a much stronger condition than being (real) differentiable. For example, the function
:<math>f(z) = \overline z</math>
is differentiable as a function <math>\R^2 \to \R^2</math>, but is ''not'' complex differentiable.
A real differentiable function is complex differentiable [[if and only if]] it satisfies the [[Cauchy–Riemann equations]], which are sometimes abbreviated as
:<math>\frac{\partial f}{\partial \overline z} = 0.</math>

Complex analysis shows some features not apparent in real analysis. For example, the [[identity theorem]] asserts that two holomorphic functions {{mvar|f}} and {{mvar|g}} agree if they agree on an arbitrarily small [[open subset]] of <math>\mathbb{C}</math>. [[Meromorphic function]]s, functions that can locally be written as {{math|''f''(''z'')/(''z'' − ''z''<sub>0</sub>)<sup>''n''</sup>}} with a holomorphic function {{mvar|f}}, still share some of the features of holomorphic functions. Other functions have [[essential singularity|essential singularities]], such as {{math|sin(1/''z'')}} at {{math|1=''z'' = 0}}.