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| [[Image:Conformal map.svg|right|thumb|A rectangular grid (top) and its image under a conformal map ''f'' (bottom).]]
| | Hello, I'm Galen, a 26 year old from Bermagui South, Australia.<br>My hobbies include (but are not limited to) Book collecting, Billiards and watching Two and a Half Men.<br><br>My weblog Friv Games Online ([http://www.scribd.com/doc/235967505 read the full info here]) |
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| In [[mathematics]], '''holomorphic functions''' are the central objects of study in [[complex analysis]]. A holomorphic function is a [[complex number|complex]]-valued [[function (mathematics)|function]] of one or more complex variables that is complex differentiable in a [[neighborhood (mathematics)|neighborhood]] of every point in its [[Domain (mathematical analysis)|domain]]. The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually [[smooth function|infinitely differentiable]] and equal to its own [[Taylor series]].
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| The term '''[[analytic function]]''' is often used interchangeably with “holomorphic function”, although the word “analytic” is also used in a broader sense to describe any function (real, complex, or of more general type) that can be written as a convergent power series in a neighborhood of each point in its [[domain of a function|domain]]. The fact that the class of ''complex analytic functions'' coincides with the class of ''holomorphic functions'' is a [[Holomorphic functions are analytic|major theorem in complex analysis]].
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| Holomorphic functions are also sometimes referred to as '''[[regular functions]]'''<ref>[http://eom.springer.de/a/a012240.htm Springer Online Reference Books], [http://mathworld.wolfram.com/RegularFunction.html Wolfram MathWorld]</ref> or as '''[[conformal map]]s'''. A holomorphic function whose domain is the whole complex plane is called an [[entire function]]. The phrase "holomorphic at a point ''z''<sub>0</sub>" means not just differentiable at ''z''<sub>0</sub>, but differentiable everywhere within some neighborhood of ''z''<sub>0</sub> in the complex plane.
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| == Definition ==
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| [[File:Non-holomorphic complex conjugate.svg|thumb|The function <math>f(z) = \bar{z}</math> is not complex-differentiable at zero, because as shown above, the value of (f(z)-f(0))/(z-0) varies depending on the direction from which zero is approached. Along the real axis, f equals the function g(z) = z and the limit is 1, while along the imaginary axis, f equals h(z) = −z and the limit is −1. Other directions yield yet other limits.]]
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| Given a complex-valued function ''f'' of a single complex variable, the '''derivative''' of ''f'' at a point ''z''<sub>0</sub> in its domain is defined by the [[limit of a function|limit]]
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| :<math>f'(z_0) = \lim_{z \to z_0} {f(z) - f(z_0) \over z - z_0 }. </math>
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| This is the same as the [[Derivative#Definition via difference quotients|definition of the derivative]] for real functions, except that all of the quantities are complex. In particular, the limit is taken as the complex number ''z'' approaches ''z''<sub>0</sub>, and must have the same value for any sequence of complex values for ''z'' that approach ''z''<sub>0</sub> on the complex plane. If the limit exists, we say that ''f'' is '''complex-differentiable''' at the point ''z''<sub>0</sub>. This concept of complex differentiability shares several properties with [[derivative|real differentiability]]: it is [[linear transformation|linear]] and obeys the [[product rule]], [[quotient rule]], and [[chain rule]].
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| If ''f'' is ''complex differentiable'' at ''every'' point ''z''<sub>0</sub> in an open set ''U'', we say that ''f'' is '''holomorphic on U'''. We say that ''f'' is '''holomorphic at the point ''z''<sub>0</sub>''' if it is holomorphic on some neighborhood of ''z''<sub>0</sub>. We say that ''f'' is holomorphic on some non-open set ''A'' if it is holomorphic in an open set containing ''A''.
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| The relationship between real differentiability and complex differentiability is the following. If a complex function {{nowrap|''f''(''x'' + i ''y'')}} = {{nowrap|''u''(''x'', ''y'') + i ''v''(''x'', ''y'')}} is holomorphic, then ''u'' and ''v'' have first partial derivatives with respect to ''x'' and ''y'', and satisfy the [[Cauchy–Riemann equations]]:
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| :<math>\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad \mbox{and} \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\,</math>
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| or, equivalently, the [[Wirtinger derivative]] of ''f'' with respect to the complex conjugate of ''z'' is zero:
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| :<math>\frac{\partial f}{\partial\overline{z}} = 0,</math>
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| which is to say that, roughly, ''f'' is functionally independent from the complex conjugate of ''z''.
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| If continuity is not a given, the converse is not necessarily true. A simple converse is that if ''u'' and ''v'' have ''continuous'' first partial derivatives and satisfy the Cauchy–Riemann equations, then ''f'' is holomorphic. A more satisfying converse, which is much harder to prove, is the [[Looman–Menchoff theorem]]: if ''f'' is continuous, ''u'' and ''v'' have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then ''f'' is holomorphic.
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| == Terminology ==
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| The word "holomorphic" was introduced by two of [[Cauchy]]'s students, [[Charles Auguste Briot|Briot]] (1817–1882) and Bouquet (1819–1895), and derives from the Greek [[wikt:ὅλος|ὅλος]] (''holos'') meaning "entire", and [[wikt:μορφή|μορφή]] (''morphē'') meaning "form" or "appearance".<ref>{{cite book |last=Markushevich |first=A. I. |editor-last=Silverman |editor-first=Richard A. |title=Theory of functions of a Complex Variable |publisher=[[American Mathematical Society]] |location=New York |origyear=1977 |year=2005 |edition=2nd |isbn=0-8218-3780-X |url=http://books.google.com/books?id=H8xfPRhTOcEC&dq |page=112}}</ref>
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| Today, the term "holomorphic function" is sometimes preferred to "analytic function", as the latter is a more general concept. This is also because an important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow directly from the definitions. The term "analytic" is however also in wide use.
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| == Properties ==
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| Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.
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| If one identifies '''C''' with '''R'''<sup>'''2'''</sup>, then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the [[Cauchy-Riemann equations]], a set of two [[partial differential equation]]s.
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| Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of [[Laplace's equation]] on '''R<sup>2</sup>'''. In other words, if we express a holomorphic function ''f''(''z'') as ''u''(''x'', ''y'') + ''i v''(''x'', ''y'') both ''u'' and ''v'' are [[harmonic function]]s, where v is the [[harmonic conjugate]] of u and vice-versa.
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| [[Cauchy's integral theorem]] implies that the [[line integral]] of every holomorphic function along a [[loop (topology)|loop]] vanishes:
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| :<math>\oint_\gamma f(z)\,dz = 0. </math>
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| Here ''γ'' is a [[rectifiable path]] in a [[simply connected]] [[open subset]] ''U'' of the [[complex plane]] '''C''' whose start point is equal to its end point, and {{nowrap|''f'' : ''U'' → '''C'''}} is a holomorphic function.
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| [[Cauchy's integral formula]] states that every function holomorphic inside a [[Disk (mathematics)|disk]] is completely determined by its values on the disk's boundary. Furthermore: Suppose ''U'' is an open subset of '''C''', {{nowrap|''f'' : ''U'' → '''C'''}} is a holomorphic function and the closed disk {{nowrap begin}}''D'' = {''z'' : |''z'' − ''z''<sub>0</sub>| ≤ ''r''}{{nowrap end}} is completely contained in ''U''. Let γ be the circle forming the [[boundary (topology)|boundary]] of ''D''. Then for every ''a'' in the [[interior (topology)|interior]] of ''D'':
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| :<math>f(a) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\, dz </math>
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| where the [[contour integral]] is taken [[Curve orientation|counter-clockwise]].
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| The derivative ''f''′(''a'') can be written as a contour integral using '''[[Cauchy's differentiation formula]]''':
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| :<math>f'(a) = {1 \over 2\pi i} \oint_\gamma {f(z) \over (z-a)^{2}}\, dz,</math>
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| for any simple loop positively winding once around ''a'', and
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| :<math>f'(a) = \lim\limits_{\gamma\to a}\frac i{2\mathcal{A}(\gamma)}\oint_{\gamma}f(z) d\bar z,</math>
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| for infinitesimal positive loops γ around ''a''.
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| In regions where the first derivative is not zero, holomorphic functions are [[conformal map|conformal]] in the sense that they preserve angles and the shape (but not size) of small figures.
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| Every [[holomorphic functions are analytic|holomorphic function is analytic]]. That is, a holomorphic function ''f'' has derivatives of every order at each point ''a'' in its domain, and it coincides with its own [[Taylor series]] at ''a'' in a neighborhood of ''a''. In fact, ''f'' coincides with its Taylor series at ''a'' in any disk centered at that point and lying within the domain of the function.
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| From an algebraic point of view, the set of holomorphic functions on an open set is a [[commutative ring]] and a [[complex vector space]]. In fact, it is a [[locally convex topological vector space]], with the [[norm (mathematics)|seminorms]] being the [[suprema]] on [[compact set|compact subsets]].
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| From a geometric perspective, a function ''f'' is holomorphic at ''z''<sub>0</sub> if and only if its [[exterior derivative]] ''df'' in a neighborhood ''U'' of ''z''<sub>0</sub> is equal to ''f''′(''z'') ''dz'' for some continuous function ''f''′. It follows from
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| :<math>\textstyle 0 = d^2 f = d(f^\prime dz) = df^\prime \wedge dz</math>
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| that ''df''′ is also proportional to ''dz'', implying that the derivative ''f''′ is itself holomorphic and thus that ''f'' is infinitely differentiable. Similarly, the fact that ''d''(''f dz'') = ''f''′ ''dz'' ∧ ''dz'' = 0 implies that any function ''f'' that is holomorphic on the simply connected region ''U'' is also integrable on ''U''. (For a path γ from ''z''<sub>0</sub> to ''z'' lying entirely in ''U'', define
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| :<math>\textstyle F_\gamma(z) = F_0 + \int_\gamma f dz</math>;
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| in light of the [[Jordan curve theorem]] and the [[Stokes' theorem|generalized Stokes' theorem]], ''F''<sub>γ</sub>(''z'') is independent of the particular choice of path γ, and thus ''F''(''z'') is a well-defined function on ''U'' having ''F''(''z''<sub>0</sub>) = ''F''<sub>0</sub> and ''dF'' = ''f dz''.)
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| == Examples ==
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| All [[polynomial]] functions in ''z'' with complex [[coefficient]]s are holomorphic on '''C''', and so are [[sine]], [[cosine]] and the [[exponential function]]. (The trigonometric functions are in fact closely related to and can be defined via the exponential function using [[Eulers formula in complex analysis|Euler's formula]]). The principal branch of the [[complex logarithm]] function is holomorphic on the [[Set (mathematics)|set]] '''C''' \ {''z'' ∈ '''R''' : z ≤ 0}. The [[square root]] function can be defined as
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| :<math>\sqrt{z} = e^{\frac{1}{2}\log z}</math>
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| and is therefore holomorphic wherever the logarithm log(''z'') is. The function 1/''z'' is holomorphic on {''z'' : ''z'' ≠ 0}.
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| As a consequence of the [[Cauchy–Riemann equations]], a real-valued holomorphic function must be constant. Therefore, the absolute value of ''z'', the argument of ''z'', the [[real part]] of ''z'' and the [[imaginary part]] of ''z'' are not holomorphic. Another typical example of a continuous function which is not holomorphic is the complex conjugate {{overline|''z''}} formed by [[complex conjugation]].
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| == Several variables ==
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| A complex analytic function of [[several complex variables]] is defined to be analytic and holomorphic at a point if it is locally expandable (within a [[polydisk]], a [[Cartesian product]] of [[disk (mathematics)|disk]]s, centered at that point) as a convergent power series in the variables. This condition is stronger than the [[Cauchy–Riemann equations]]; in fact it can be stated
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| as follows:
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| A function of several complex variables is holomorphic [[if and only if]] it satisfies the Cauchy–Riemann equations{{Clarify|date=June 2013}} and is locally [[square-integrable]]. {{Citation needed|date=June 2013}}
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| == Extension to functional analysis ==
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| {{Main|infinite-dimensional holomorphy}}
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| The concept of a holomorphic function can be extended to the infinite-dimensional spaces of [[functional analysis]]. For instance, the [[Fréchet derivative|Fréchet]] or [[Gâteaux derivative]] can be used to define a notion of a holomorphic function on a [[Banach space]] over the field of complex numbers.
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| == See also ==
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| * [[Antiderivative (complex analysis)]]
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| * [[Antiholomorphic function]]
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| * [[Biholomorphy]]
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| * [[Meromorphic function]]
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| * [[Quadrature domains]]
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| == References ==
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| <references/>
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| ==External links==
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| * {{springer|title=Analytic function|id=p/a012240}}
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| {{Link FA|lmo}}
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| [[Category:Analytic functions]]
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