Editing Holomorphic function (section)

== Properties ==
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.<ref>
{{cite book
 | last = Henrici  | first = Peter  | author-link = Peter Henrici (mathematician)
 | year = 1993  | orig-year = 1986
 | title = Applied and Computational Complex Analysis
 | volume = 3
 | place = New York - Chichester - Brisbane - Toronto - Singapore
 | publisher = [[John Wiley & Sons]]
 | series = Wiley Classics Library
 | edition = Reprint
 | mr = 0822470 | zbl = 1107.30300 | isbn = 0-471-58986-1
 | url = https://books.google.com/books?id=vKZPsjaXuF4C  |via=Google
}}
</ref> That is, if functions {{tmath|f}} and {{tmath|g}} are holomorphic in a domain {{tmath|U}}, then so are {{tmath|f+g}}, {{tmath|f-g}}, {{tmath| fg}}, and {{tmath|f \circ g}}. Furthermore, {{tmath|f/g }} is holomorphic if {{tmath|g}} has no zeros in {{tmath|U}};  otherwise it is [[meromorphic]].

If one identifies {{tmath|\C}} with the real [[plane (geometry)|plane]] {{tmath|\textstyle \R^2}}, 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.<ref name=Mark/>

Every holomorphic function can be separated into its real and imaginary parts {{tmath|1=f(x + iy) = u(x, y) + i\,v(x,y)}}, and each of these is a [[harmonic function]] on {{tmath|\textstyle \R^2}} (each satisfies [[Laplace's equation]] {{tmath|1=\textstyle \nabla^2 u = \nabla^2 v = 0}}), with {{tmath|v}} the [[harmonic conjugate]] of {{tmath|u}}.<ref>
{{cite book
 |first=L.C. |last=Evans |author-link=Lawrence C. Evans
 |year=1998
 |title=Partial Differential Equations
 |publisher=American Mathematical Society
}}
</ref>
Conversely, every harmonic function {{tmath|u(x, y)}} on a [[Simply connected space|simply connected]] domain {{tmath|\textstyle \Omega \subset \R^2}} is the real part of a holomorphic function: If {{tmath|v}} is the harmonic conjugate of {{tmath|u}}, unique up to a constant, then {{tmath|1=f(x + iy) = u(x, y) + i\,v(x, y)}} is holomorphic.

[[Cauchy's integral theorem]] implies that the [[contour integral]] of every holomorphic function along a [[loop (topology)|loop]] vanishes:<ref name=Lang>
{{cite book
 |first = Serge  |last = Lang  | author-link = Serge Lang
 | year = 2003
 | title = Complex Analysis
 | series = Springer Verlag GTM
 | publisher = [[Springer Verlag]]
}}
</ref>

:<math>\oint_\gamma f(z)\,\mathrm{d}z = 0.</math>

Here {{tmath|\gamma}} is a [[rectifiable path]] in a simply connected [[domain (mathematical analysis)|complex domain]] {{tmath|U \subset \C}} whose start point is equal to its end point, and {{tmath|f \colon U \to \C}} is a holomorphic function.

[[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.<ref name=Lang/> Furthermore: Suppose {{tmath|U \subset \C}} is a complex domain, {{tmath|f\colon U \to \C}} is a holomorphic function and the closed disk <math> D \equiv \{ z : | z - z_0 | \leq r \} </math> is [[neighbourhood (mathematics)#Neighbourhood of a set|completely contained]] in {{tmath|U}}. Let {{tmath|\gamma}} be the circle forming the [[boundary (topology)|boundary]] of {{tmath|D}}. Then for every {{tmath|a}} in the [[interior (topology)|interior]] of {{tmath|D}}:

:<math>f(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{z-a}\,\mathrm{d}z</math>

where the contour integral is taken [[curve orientation|counter-clockwise]].

The derivative {{tmath|{f'}(a)}} can be written as a contour integral<ref name=Lang /> using [[Cauchy's differentiation formula]]:

:<math> f'\!(a) = \frac{ 1 }{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^2}\,\mathrm{d}z,</math>

for any simple loop positively winding once around {{tmath|a}}, and

:<math> f'\!(a) = \lim\limits_{\gamma\to a} \frac{ i }{2\mathcal{A}(\gamma)} \oint_{\gamma}f(z)\,\mathrm{d}\bar{z},</math>

for [[infinitesimal]] positive loops {{tmath|\gamma}} around {{tmath|a}}.

In regions where the first derivative is not zero, holomorphic functions are [[conformal map|conformal]]: they preserve angles and the shape (but not size) of small figures.<ref>
{{cite book
 | last =Rudin | first =Walter | author-link = Walter Rudin
 | year=1987
 | title=Real and Complex Analysis
 | publisher=McGraw–Hill Book Co.
 | location=New York
 | edition=3rd
 | isbn=978-0-07-054234-1 | mr=924157
}}
</ref>

Every [[holomorphic functions are analytic|holomorphic function is analytic]]. That is, a holomorphic function {{tmath|f}} has derivatives of every order at each point {{tmath|a}} in its domain, and it coincides with its own [[Taylor series]] at {{tmath|a}} in a neighbourhood of {{tmath|a}}. In fact, {{tmath|f}} coincides with its Taylor series at {{tmath|a}} in any disk centred at that point and lying within the domain of the function.

From an algebraic point of view, the set of holomorphic functions on an open set is a [[commutative ring]] and a [[complex vector space]]. Additionally, the set of holomorphic functions in an open set {{tmath|U}} is an [[integral domain]] [[if and only if]] the open set {{tmath|U}} is connected. <ref name=Gunning/> In fact, it is a [[locally convex topological vector space]], with the [[norm (mathematics)|seminorms]] being the [[suprema]] on [[compact subset]]s.

From a geometric perspective, a function {{tmath|f}} is holomorphic at {{tmath|z_0}} if and only if its [[exterior derivative]] {{tmath|\mathrm{d}f}} in a neighbourhood {{tmath|U}} of {{tmath|z_0}} is equal to {{tmath| f'(z)\,\mathrm{d}z}} for some continuous function {{tmath|f'}}. It follows from

:<math>0 = \mathrm{d}^2 f = \mathrm{d}(f'\,\mathrm{d}z) = \mathrm{d}f' \wedge \mathrm{d}z</math>

that {{tmath|\mathrm{d}f'}} is also proportional to {{tmath|\mathrm{d}z}}, implying that the derivative {{tmath|\mathrm{d}f'}} is itself holomorphic and thus that {{tmath|f}} is infinitely differentiable. Similarly, {{tmath|1= \mathrm{d}(f\,\mathrm{d}z ) = f'\,\mathrm{d}z \wedge \mathrm{d}z = 0}} implies that any function {{tmath|f}} that is holomorphic on the simply connected region {{tmath|U}} is also integrable on {{tmath|U}}.

(For a path {{tmath|\gamma}} from {{tmath|z_0}} to {{tmath|z}} lying entirely in {{tmath|U}}, define {{tmath|1= F_\gamma(z) = F(0) + \int_\gamma f\,\mathrm{d}z }}; in light of the [[Jordan curve theorem]] and the [[Stokes' theorem|generalized Stokes' theorem]], {{tmath|F_\gamma(z)}} is independent of the particular choice of path {{tmath|\gamma}}, and thus {{tmath|F(z)}} is a well-defined function on {{tmath|U}} having {{tmath|1= \mathrm{d}F = f\,\mathrm{d}z}} or {{tmath|1= f = \frac{\mathrm{d}F}{\mathrm{d}z} }}.)