Editing Holomorphic function (section)

== Definition ==
[[File:Non-holomorphic complex conjugate.svg|thumb|The function {{tmath|1=f(z) = \bar{z} }} is not complex differentiable at zero, because as shown above, the value of {{tmath|\frac{f(z) - f(0)}{z - 0} }} varies depending on the direction from which zero is approached. On the real axis only, {{tmath|f}} equals the function {{tmath|1=g(z) = z}} and the limit is {{tmath|1}}, while along the imaginary axis only, {{tmath|f}} equals the different function {{tmath|1=h(z) = -z}} and the limit is {{tmath|-1}}. Other directions yield yet other limits.]]
Given a complex-valued function {{tmath|f}} of a single complex variable, the '''derivative''' of {{tmath|f}} at a point {{tmath|z_0}} in its domain is defined as the [[limit of a function|limit]]<ref>[[Lars Ahlfors|Ahlfors, L.]], ''Complex Analysis, 3 ed.'' (McGraw-Hill, 1979).</ref>
:<math>f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{ z - z_0 }.</math>

This is the same definition as for the [[derivative]] of a [[real function]], except that all quantities are complex. In particular, the limit is taken as the complex number {{tmath|z}} tends to {{tmath|z_0}}, and this means that the same value is obtained for any sequence of complex values for {{tmath|z}} that tends to {{tmath|z_0}}. If the limit exists, {{tmath|f}} is said to be '''complex differentiable''' at {{tmath|z_0}}. 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]].<ref>{{cite book |author-link=Peter Henrici (mathematician) |last=Henrici |first=P. |title=Applied and Computational Complex Analysis |publisher=Wiley |year=1986 |orig-year=1974, 1977 }} Three volumes, publ.: 1974, 1977, 1986.</ref>

A function is '''holomorphic''' on an [[open set]] {{tmath|U}} if it is ''complex differentiable'' at ''every'' point of {{tmath|U}}. A function {{tmath|f}} is ''holomorphic'' at a point {{tmath|z_0}} if it is holomorphic on some [[neighbourhood (mathematics)|neighbourhood]] of {{tmath|z_0}}.<ref>
{{cite book
 |first1=Peter     |last1=Ebenfelt      |first2=Norbert  |last2=Hungerbühler
 |first3=Joseph J. |last3=Kohn          |first4=Ngaiming |last4=Mok
 |first5=Emil J.   |last5=Straube
 |year=2011
 |url=https://books.google.com/books?id=3GeUgafFRgMC&q=holomorphic |via=Google
 |title=Complex Analysis
 |publisher=Springer
 |series=Science & Business Media
|isbn=978-3-0346-0009-5 }}
</ref>
A function is ''holomorphic'' on some non-open set {{tmath|A}} if it is holomorphic at every point of {{tmath|A}}. 

A function may be complex differentiable at a point but not holomorphic at this point. For example, the function <math>\textstyle f(z) = |z|\vphantom{l}^2 = z\bar{z}</math> ''is'' complex differentiable at {{tmath|0}}, but ''is not'' complex differentiable anywhere else, esp. including in no place close to {{tmath|0}} (see the Cauchy–Riemann equations, below). So, it is ''not'' holomorphic at {{tmath|0}}. 

The relationship between real differentiability and complex differentiability is the following: If a complex function {{tmath|1= f(x+ iy) = u(x,y) + i\,v(x, y)}} is holomorphic, then {{tmath|u}} and {{tmath|v}} have first partial derivatives with respect to {{tmath|x}} and {{tmath|y}}, and satisfy the [[Cauchy–Riemann equations]]:<ref name=Mark>
{{cite book
 |last=Markushevich |first=A.I.
 |year=1965
 |title=Theory of Functions of a Complex Variable
 |publisher=Prentice-Hall
}} [In three volumes.]
</ref>
:<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>
or, equivalently, the [[Wirtinger derivative]] of {{tmath|f}} with respect to {{tmath|\bar z}}, the [[complex conjugate]] of {{tmath|z}}, is zero:<ref name=Gunning>
{{cite book
 |last1 = Gunning |first1 = Robert C. |author1-link = Robert Gunning (mathematician)
 |last2 = Rossi   |first2 = Hugo
 |year = 1965
 |title = Analytic Functions of Several Complex Variables
 |series = Modern Analysis
 |place = Englewood Cliffs, NJ
 |publisher = [[Prentice-Hall]]
 |mr = 0180696 |zbl = 0141.08601 |isbn = 9780821869536
 |url = https://books.google.com/books?id=L0zJmamx5AAC  |via=Google
}}
</ref>
:<math>\frac{\partial f}{\partial\bar{z}} = 0,</math>
which is to say that, roughly, {{tmath|f}} is functionally independent from {{tmath|\bar z}}, the complex conjugate of {{tmath|z}}.

If continuity is not given, the converse is not necessarily true. A simple converse is that if {{tmath|u}} and {{tmath|v}} have ''continuous'' first partial derivatives and satisfy the Cauchy–Riemann equations, then {{tmath|f}} is holomorphic. A more satisfying converse, which is much harder to prove, is the [[Looman–Menchoff theorem]]: if {{tmath|f}} is continuous, {{tmath|u}} and {{tmath|v}} have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then {{tmath|f}} is holomorphic.<ref>
{{cite journal
 |first1=J.D. |last1=Gray
 |first2=S.A. |last2=Morris
 |date=April 1978
 |title=When is a function that satisfies the Cauchy-Riemann equations analytic?
 |journal=[[The American Mathematical Monthly]]
 |volume=85 |issue=4 |pages=246–256
 |jstor=2321164  |doi=10.2307/2321164
}}
</ref>