Holomorphic function

Holomorphic

Holomorphie (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 ${\textstyle f\colon U\rightarrow \mathbb {C} }$ with a open set ${\textstyle U\subseteq \mathbb {C} }$ is called holomorphic if $f$ is differentiable at each point of ${\textstyle U}$ .

Reelle and complex differentiation

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 realm. For example, each holomorphic function can be differentiated as often as desired (sttig) and can be developed locally at each point into a power series.

Definition: Complex differentiation

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

\lim _{h\to 0}{\frac {f(z_{0}+h)-f(z_{0})}{h}}

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

Definition: Holomorphy in one point

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

Definition: Full function

If ${\textstyle f:\mathbb {C} \to \mathbb {C} }$ is complexly differentiable to the whole ${\textstyle \mathbb {C} }$ , then ${\textstyle f}$ is called a ganze Funktion.

Explanatory notes

Link between complex and real differentiation

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

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

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

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

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

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 ${\textstyle C{\begin{pmatrix}y_{1}\\y_{2}\end{pmatrix}}:=y_{1}+iy_{2}}$ .

Jacobi Matrix

Main article: Cauchy-Riemann equations

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

{\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

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

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

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

are fulfilled.

Equivalent properties of holomorphic functions of a variable

In a neighbourhood of a complex number, the following properties of complex functions are equivalent:

(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 Wegintegral of the function disappears via any closed zusammenziehbaren path.
(H6) The functional values in the interior of a Kreisscheibe can be determined from the functional values at the edge using the cauchyschen Integralformel.
(H7) f can be differentiated and it applies
${\textstyle \quad {\frac {\partial f}{\partial {\bar {z}}}}=0,}$
where ${\textstyle {\tfrac {\partial }{\partial {\bar {z}}}}}$ the Cauchy-Riemann operator defined by ${\tfrac {\partial }{\partial {\bar {z}}}}:={\tfrac {1}{2}}\left({\tfrac {\partial }{\partial x}}+i{\tfrac {\partial }{\partial y}}\right)$

Examples

Entire Functions

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

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

Holomorphic, non-gant functions

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 ${\textstyle \log }$ can be developed at all points from ${\textstyle \mathbb {C} \setminus {]{-\infty },0]}}$ into a power series and is thus holomorphic on the set ${\textstyle \mathbb {C} \setminus {]{-\infty },0]}}$ .

Functions - not holomorphic at any point

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

the absolute value function ${\textstyle z\mapsto |z|}$ ,
the projections on the real part ${\textstyle z\mapsto \mathrm {Re} (z)}$ or on the imaginary part ${\textstyle z\mapsto \mathrm {Im} (z)}$ ,
complex conjugation ${\textstyle z\mapsto {\overline {z}}}$ .

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

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

If one identifies $\mathbb {C}$ with the real plane $\textstyle \mathbb {R} ^{2}$ , 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

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

Cauchy's integral theorem

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

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

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

Cauchy's integral formula

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 $U\subset \mathbb {C}$ is a complex domain, $f\colon U\to \mathbb {C}$ is a holomorphic function and the closed disk $D\equiv \{z:|z-z_{0}|\leq r\}$ is completely contained in $U$ . Let $\gamma$ be the circle forming the boundary of $D$ . Then for every $a$ in the interior of $D$ :

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

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

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 $a$ , and

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 $\gamma$ around $a$ .

Conformal map

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

Every 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 neighbourhood of $a$ . In fact, $f$ coincides with its Taylor series at $a$ in any disk centred at that point and lying within the domain of the function.

Functions as complex vector space

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 $U$ is an integral domain if and only if the open set $U$ 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

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

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

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

Choice of Path - Independency

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

Biholomorphic functions

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

In the n-dimensional complex space

Let ${\textstyle D\subseteq \mathbb {C} ^{n}}$ a complex open subset. An illustration ${\textstyle f\colon D\to \mathbb {C} ^{m}}$ is called holomorph if ${\textstyle f=(f_{1},\dotsc ,f_{m})}$ is holomorphous in each subfunction and each variable. With the Wirtinger-Kalkül ${\textstyle \textstyle {\frac {\partial }{\partial z^{j}}}}$ and ${\textstyle \textstyle {\frac {\partial }{\partial {\overline {z}}^{j}}}}$ 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 cauchyian integral set does not apply ${\textstyle f\colon D\to \mathbb {C} }$ and the Identitätssatz is only valid in a weakened version. For these functions, however, the integral formula of Cauchy can be generalized by Induktion to ${\textstyle n}$ dimensions. Salomon Bochner even proved in 1944 a generalization of the ${\textstyle n}$ -dimensional cauchyian integral formula. This bears the name Bochner-Martinelli-Formel.

In complex geometry

Holomorphic images are also considered in the komplexen Geometrie. Thus, holomorphic images can be defined between riemannschen Flächen and between komplexen Mannigfaltigkeiten analogously to differentiable functions between glatten Mannigfaltigkeiten. In addition, there is an important counterpart to the glatten Differentialformen for integration theory, called holomorphe Differentialform.

Literature

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

↑ 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.
↑ Markushevich, A.I. (1965). Theory of Functions of a Complex Variable. Prentice-Hall - in three volumes.
↑ Evans, L.C. (1998). Partial Differential Equations. American Mathematical Society.
↑ ^4.0 ^4.1 ^4.2 Lang, Serge (2003). Complex Analysis. Springer Verlag GTM. Springer Verlag.
↑ Rudin, Walter (1987). Real and Complex Analysis (3rd ed.). New York: McGraw–Hill Book Co.. ISBN 978-0-07-054234-1.
↑ 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

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[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.

[Mark-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.

[Lang-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.

[Gunning-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

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