Holomorphic function

## Holomorphic Function

[edit | edit source]

Holomorphic (of gr. ὅλος holos, 'whole' and μορφή morphe, 'form') is a property of certain complex valued functions which are analyzed in the Complex Analysis as a function  f : U -> C {\textstyle f\colon U\rightarrow \mathbb {C} } with an open set U ⊆ C {\textstyle U\subseteq \mathbb {C} } is called holomorphic if  f {\displaystyle f} is differentiable at each point of  U {\textstyle U} . In this topic also equivalent criteria for a function being holomorphic are discussed.

## Subtopics

[edit | edit source]

* Criteria \- Equivalent criteria for a function being holomorphic

## Real and complex differentiation

[edit | edit source]

Even if the definition is analogous to real differentiation, the function theory shows that the holomorphy is a very strong property. It produces a variety of phenomena that do not have a counterpart in the real. For example, each holomorphic function can be differentiated as often as desired (continuous) and can be developed locally at each point into a power series.

### Definition: Complex differentiation

[edit | edit source]

It is  U ⊆ C {\textstyle U\subseteq \mathbb {C} } an open subset of the complex plane and  z 0 ∈ U {\textstyle z_{0}\in U} a point of this subset. A function  f : U -> C {\textstyle f\colon U\to \mathbb {C} } is called complex differentiable in point  z 0 {\textstyle z_{0}} , if the limit

lim h -> 0 f ( z 0 + h ) − f ( z 0 ) h {\displaystyle \lim _{h\to 0}{\frac {f(z_{0}+h)-f(z_{0})}{h}}}

with  h ∈ C ∖ { 0 } {\textstyle h\in \mathbb {C} \setminus \\{0\\}} . If the limit exists, then the limit is denoted with  f ′ ( z 0 ) {\textstyle f'(z_{0})} .

### Definition: Holomorphic in one point

[edit | edit source]

Let  U ⊂ C {\textstyle U\subset \mathbb {C} } be an open set and  f : U -> C {\textstyle f:U\to \mathbb {C} } a function.  f {\textstyle f} is called holomorphic in point  z 0 ∈ U {\textstyle z_{0}\in U} , if a neighbourhood of  z 0 {\textstyle z_{0}} exists, in which  f {\textstyle f} is complex differentiable.

### Definition: Full function

[edit | edit source]

If  f : C -> C {\textstyle f:\mathbb {C} \to \mathbb {C} } is complexly differentiable to the whole  C {\textstyle \mathbb {C} } , then  f {\textstyle f} is called a entire function.

## Explanatory notes

[edit | edit source]

### Link between complex and real differentiation

[edit | edit source]

C {\textstyle \mathbb {C} } can be interpreted as a two-dimensional real vector space with the canonical base  { 1 , i } {\textstyle \\{1,i\\}} and so one can examine a function  f : U -> C {\textstyle f\colon U\to \mathbb {C} } on an open set  U ⊆ C {\textstyle U\subseteq \mathbb {C} } . In multivariable calculus it is known that  f {\textstyle f} total differentiable in  z 0 {\textstyle z_{0}} if there exists a  R {\textstyle \mathbb {R} } -linear mapping  L {\textstyle L} , so that

f ( z 0 + h ) = f ( z 0 ) + L ( h ) + r ( h ) {\displaystyle f(z_{0}+h)=f(z_{0})+L(h)+r(h)}

where  r {\textstyle r} is a function with the property

lim h -> 0 r ( h ) | h | = 0. {\displaystyle \lim _{h\to 0}{\frac {r(h)}{|h|}}=0.}

It can now be seen that the function  f {\textstyle f} is complex differentiable in  z 0 {\textstyle z_{0}} , if  f {\displaystyle f} is total differentiable in  z 0 {\textstyle z_{0}} and  L {\textstyle L} is even  C {\textstyle \mathbb {C} } -linear. The latter is a sufficient condition. It means that the transformation matrix L {\textstyle L} with respect to the canonical base  { 1 , i } {\textstyle \\{1,i\\}} has the form

L ( z 1 + i z 2 ) = C ( ( a − b b a ) ⋅ ( z 1 z 2 ) ) {\displaystyle L(z_{1}+iz_{2})=C\left({\begin{pmatrix}a&-b\\\b&a\end{pmatrix}}\cdot {\begin{pmatrix}z_{1}\\\z_{2}\end{pmatrix}}\right)}

with  C ( y 1 y 2 ) := y 1 + i y 2 {\textstyle C{\begin{pmatrix}y_{1}\\\y_{2}\end{pmatrix}}:=y_{1}+iy_{2}} .

### Jacobi Matrix

[edit | edit source]

Main article: Cauchy-Riemann equations

If a function  f ( x + i y ) = u ( x , y ) + i ⋅ v ( x , y ) {\textstyle f\left(x+iy\right)=u\left(x,y\right)+i\cdot v\left(x,y\right)} is decompose into functions of its real and imaginary parts with real-valued functions  u , v {\textstyle u,v} , the total derivative  L {\textstyle L} with tranformation matrix has the Jacobian matrix

( ∂ u ∂ x ∂ u ∂ y ∂ v ∂ x ∂ v ∂ y ) . {\displaystyle {\begin{pmatrix}{\frac {\partial u}{\partial x}}&{\frac {\partial u}{\partial y}}\\\\{\frac {\partial v}{\partial x}}&{\frac {\partial v}{\partial y}}\end{pmatrix}}.}

### Cauchy-Riemannian differential equations

[edit | edit source]

Consequently, the function  f {\textstyle f} is total differentiable precisely when it can be differentiated relatively and for  u , v {\textstyle u,v} the Cauchy-Riemann equations

∂ u ∂ x = ∂ v ∂ y {\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}}

∂ u ∂ y = − ∂ v ∂ x {\displaystyle \displaystyle {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}}

are fulfilled.

## Equivalent properties of holomorphic functions of a variableIn neighbourhood of a complex number, the following properties of complex functions are equivalent:

[edit | edit source]

* (H1) The function can be differentiated in a complex manner.
* (H2) The function can be varied as often as desired.
* (H3) Real and imaginary parts meet the Cauchy-Riemann equations and can be continuously differentiated.
* (H4) The function can be developed into a complex power series.
* (H5) The function is steady and the path integral of the function disappears via any closed contractible path.
* (H6) The functional values in the interior of a circular disk can be determined from the functional values at the edge using the Cauchy Integral formula.
* (H7) f can be differentiated and it applies
∂ f ∂ z ¯ = 0 , {\textstyle \quad {\frac {\partial f}{\partial {\bar {z}}}}=0,}
where  ∂ ∂ z ¯ {\textstyle {\tfrac {\partial }{\partial {\bar {z}}}}} the Cauchy-Riemann operator defined by  ∂ ∂ z ¯ := 1 2 ( ∂ ∂ x + i ∂ ∂ y ) {\displaystyle {\tfrac {\partial }{\partial {\bar {z}}}}:={\tfrac {1}{2}}\left({\tfrac {\partial }{\partial x}}+i{\tfrac {\partial }{\partial y}}\right)}

## Examples

[edit | edit source]

### Entire Functions

[edit | edit source]

An entire function is holomorphic on the whole  C {\textstyle \mathbb {C} } . Examples are:

* each polynomial  z ↦ ∑ j = 0 n a j z j {\displaystyle \displaystyle z\mapsto \sum _{j=0}^{n}a_{j}z^{j}} with coefficients  a j ∈ C {\textstyle a_{j}\in \mathbb {C} } ,
* the Exponential Function exp {\textstyle \exp } ,
* the trigonometric functions sin {\textstyle \sin } and  cos {\textstyle \cos } ,
* the hyperbolic functions sinh {\textstyle \sinh } and  cosh {\textstyle \cosh } .

### Holomorphic, non-gant functions

[edit | edit source]

* Rational functions are holomorphic apart from the zero points of polynomial in the denominator. Then rational function has isolated singularities (e.g. poles. Rational functions are examples for meromorphic functions.
* The Logarithm function log {\textstyle \log } can be developed at all points from  C ∖ ] − ∞ , 0 ] {\textstyle \mathbb {C} \setminus {]{-\infty },0]}} into a power series and is thus holomorphic on the set  C ∖ ] − ∞ , 0 ] {\textstyle \mathbb {C} \setminus {]{-\infty },0]}} .

### Functions - not holomorphic at any point

[edit | edit source]

The following functions are not holomorphic in any  z ∈ C {\textstyle z\in \mathbb {C} } . Examples are:

* the absolute value function  z ↦ | z | {\textstyle z\mapsto |z|} ,
* the projections on the real part  z ↦ R e ( z ) {\textstyle z\mapsto \mathrm {Re} (z)} or on the imaginary part  z ↦ I m ( z ) {\textstyle z\mapsto \mathrm {Im} (z)} ,
* complex conjugation  z ↦ z ¯ {\textstyle z\mapsto {\overline {z}}} .

The function  z ↦ | z | 2 {\textstyle z\mapsto |z|^{2}} is complex differentiable only at the point  z o = 0 {\textstyle z_{o}=0} , but the function is not' holomorphic in  z o {\displaystyle z_{o}} , since it is not complex differentiable in a neighborhood of  0 {\textstyle 0} .

## Properties

[edit | edit source]

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.[1] That is, if functions  f {\displaystyle f} and  g {\displaystyle g} are holomorphic in a domain  U {\displaystyle U} , then so are  f + g {\displaystyle f+g} ,  f − g {\displaystyle f-g} ,  f g {\displaystyle fg} , and  f ∘ g {\displaystyle f\circ g} . Furthermore,  f / g {\displaystyle f/g} is holomorphic if  g {\displaystyle g} has no zeros in  U {\displaystyle U} ; otherwise it is meromorphic.

If one identifies  C {\displaystyle \mathbb {C} } with the real plane R 2 {\displaystyle \textstyle \mathbb {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 equations.[2]

### Functions for real and imaginary parts

[edit | edit source]

Every holomorphic function can be separated into its real and imaginary parts  1 = f ( x + i y ) = u ( x , y ) + i v ( x , y ) {\displaystyle 1=f(x+iy)=u(x,y)+i\,v(x,y)} , and each of these is a harmonic function on  R 2 {\displaystyle \textstyle \mathbb {R} ^{2}} (each satisfies Laplace's equation 1 = ∇ 2 u = ∇ 2 v = 0 {\displaystyle 1=\textstyle \nabla ^{2}u=\nabla ^{2}v=0} ), with  v {\displaystyle v} the harmonic conjugate of  u {\displaystyle u} .[3] Conversely, every harmonic function  u ( x , y ) {\displaystyle u(x,y)} on a simply connected domain  Ω ⊂ R 2 {\displaystyle \textstyle \Omega \subset \mathbb {R} ^{2}} is the real part of a holomorphic function: If  v {\displaystyle v} is the harmonic conjugate of  u {\displaystyle u} , unique up to a constant, then  1 = f ( x + i y ) = u ( x , y ) + i v ( x , y ) {\displaystyle 1=f(x+iy)=u(x,y)+i\,v(x,y)} is holomorphic.

### Cauchy's integral theorem

[edit | edit source]

Cauchy's integral theorem implies that the contour integral of every holomorphic function along a loop vanishes:[4]

∮ γ f ( z ) d z = 0. {\displaystyle \oint _{\gamma }f(z)\,\mathrm {d} z=0.}

Here  γ {\displaystyle \gamma } is a rectifiable path in a simply connected complex domain U ⊂ C {\displaystyle U\subset \mathbb {C} } whose start point is equal to its end point, and  f : U -> C {\displaystyle f\colon U\to \mathbb {C} } is a holomorphic function.

### Cauchy's integral formula

[edit | edit source]

Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary.[4] Furthermore: Suppose  U ⊂ C {\displaystyle U\subset \mathbb {C} } is a complex domain,  f : U -> C {\displaystyle f\colon U\to \mathbb {C} } is a holomorphic function and the closed disk  D ≡ { z : | z − z 0 | ≤ r } {\displaystyle D\equiv \\{z:|z-z_{0}|\leq r\\}} is completely contained in  U {\displaystyle U} . Let  γ {\displaystyle \gamma } be the circle forming the boundary of  D {\displaystyle D} . Then for every  a {\displaystyle a} in the interior of  D {\displaystyle D} :

f ( a ) = 1 2 π i ∮ γ f ( z ) z − a d z {\displaystyle f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,\mathrm {d} z}

where the contour integral is taken counter-clockwise.

### Cauchy's differentiation formula

[edit | edit source]

The derivative  f ′ ( a ) {\displaystyle {f'}(a)} can be written as a contour integral[4] using Cauchy's differentiation formula:

f ′ ( a ) = 1 2 π i ∮ γ f ( z ) ( z − a ) 2 d z , {\displaystyle f'\\!(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-a)^{2}}}\,\mathrm {d} z,}

for any simple loop positively winding once around  a {\displaystyle a} , and

f ′ ( a ) = lim γ -> a i 2 A ( γ ) ∮ γ f ( z ) d z ¯ , {\displaystyle f'\\!(a)=\lim \limits _{\gamma \to a}{\frac {i}{2{\mathcal {A}}(\gamma )}}\oint _{\gamma }f(z)\,\mathrm {d} {\bar {z}},}

for infinitesimal positive loops  γ {\displaystyle \gamma } around  a {\displaystyle a} .

### Conformal map

[edit | edit source]

In regions where the first derivative is not zero, holomorphic functions are conformal: they preserve angles and the shape (but not size) of small figures.[5]

### Analytic - Taylor series

[edit | edit source]

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

### Functions as complex vector space

[edit | edit source]

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  U {\displaystyle U} is an integral domain if and only if the open set  U {\displaystyle U} is connected. [6] In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.

### Geometric perspective - infinitely differentiable

[edit | edit source]

From a geometric perspective, a function  f {\displaystyle f} is holomorphic at  z 0 {\displaystyle z_{0}} if and only if its exterior derivative d f {\displaystyle \mathrm {d} f} in a neighbourhood  U {\displaystyle U} of  z 0 {\displaystyle z_{0}} is equal to  f ′ ( z ) d z {\displaystyle f'(z)\,\mathrm {d} z} for some continuous function  f ′ {\displaystyle f'} . It follows from

0 = d 2 f = d ( f ′ d z ) = d f ′ ∧ d z {\displaystyle 0=\mathrm {d} ^{2}f=\mathrm {d} (f'\,\mathrm {d} z)=\mathrm {d} f'\wedge \mathrm {d} z}

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

### Choice of Path - Independency

[edit | edit source]

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

## Biholomorphic functions

[edit | edit source]

A function which is holomorphous bijective and whose reverse function is holomorph is called biholomorph. In the case of a complex change, the equivalent is that the image is bijective and conformal. From the Implicit Function Theorem it implies for holomorphic functions of a single variable that a bijective, holomorphic function always has a holomorphic inverse function.

## Holomorphy of several variable

[edit | edit source]

### In the n-dimensional complex space

[edit | edit source]

Let D ⊆ C n {\textstyle D\subseteq \mathbb {C} ^{n}} a complex open subset. An illustration  f : D -> C m {\textstyle f\colon D\to \mathbb {C} ^{m}} is called holomorph if  f = ( f 1 , … , f m ) {\textstyle f=(f_{1},\dotsc ,f_{m})} is holomorphous in each sub-function and each variable. With the Wittgenstein-calculus ∂ ∂ z j {\textstyle \textstyle {\frac {\partial }{\partial z^{j}}}} and  ∂ ∂ z ¯ j {\textstyle \textstyle {\frac {\partial }{\partial {\overline {z}}^{j}}}} a calculus is available, with which it is easier to manage the partial derivations of a complex function. However, holomorphic functions of several changers no longer have so many beautiful properties.For instance, the Cauchy integral set does not apply  f : D -> C {\textstyle f\colon D\to \mathbb {C} } and the Identity law is only valid in a weakened version. For these functions, however, the integral formula of Cauchy can be generalized by Induction to  n {\textstyle n} dimensions. Salomon Bochner even proved in 1944 a generalization of the  n {\textstyle n} -dimensional Cauchy integral formula. This bears the name Bochner-Martinelli-Formel.

### In complex geometry

[edit | edit source]

Holomorphic images are also considered in the Complex Geometry. Thus, holomorphic images can be defined between Riemann surface and between Complex Manifolds analogously to differentiable functions between smooth maifolds. In addition, there is an important counterpart to the smooth Differential forms for integration theory, called holomorphic Differential form.

## Literature

[edit | edit source]

* Bak, J., Newman, D. J., & Newman, D. J. (2010). Complex analysis (Vol. 8). New York: Springer.
* Lang, S. (2013). Complex analysis (Vol. 103). Springer Science & Business Media.

## References

[edit | edit source]

1. ↑ Henrici, Peter (1993). Applied and Computational Complex Analysis. Wiley Classics Library. 3 (Reprint ed.). New York - Chichester - Brisbane - Toronto - Singapore: John Wiley & Sons. ISBN 0-471-58986-1. https://books.google.com/books?id=vKZPsjaXuF4C.
2. ↑ Markushevich, A.I. (1965). Theory of Functions of a Complex Variable. Prentice-Hall - in three volumes.
3. ↑ Evans, L.C. (1998). Partial Differential Equations. American Mathematical Society.
4. ↑ 4.0 4.1 4.2 Lang, Serge (2003). Complex Analysis. Springer Verlag GTM. Springer Verlag.
5. ↑ Rudin, Walter (1987). Real and Complex Analysis (3rd ed.). New York: McGraw–Hill Book Co.. ISBN 978-0-07-054234-1.
6. ↑ Gunning, Robert C.; Rossi, Hugo (1965). Analytic Functions of Several Complex Variables. Modern Analysis. Englewood Cliffs, NJ: Prentice-Hall. ISBN 9780821869536. MR 0180696. Zbl 0141.08601

## See also

[edit | edit source]

* Holomorphic Criteria
* Holomorph Group
* Complex Analysis
* holomorphic function

## Page Information

[edit | edit source]

You can display this page as Wiki2Reveal slides

### Wiki2Reveal

[edit | edit source]

The Wiki2Reveal slides were created for the Complex Analysis' and the Link for the Wiki2Reveal Slides was created with the link generator.

* This page is designed as a PanDocElectron-SLIDE document type.
* Source: Wikiversity https://en.wikiversity.org/wiki/Holomorphic function
* see Wiki2Reveal for the functionality of Wiki2Reveal.

### Translation and Version Control

[edit | edit source]

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

* Source: Holomorphie URL: https://de.wikiversity.org/wiki/Holomorphie
* Date: 11/4/2024