Holomorphic function

Holomorphic Function

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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   with a open set   is called holomorphic if   is differentiable at each point of  .

Real and complex differentiation

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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

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It is   an open subset of the complex plane and   a point of this subset. A function   is called complex differentiable in point  , if the limit

 

with  . If the limit exists, then the limit is denoted with  .

Definition: Holomorphic in one point

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Let   be an open set and   a function.   is called holomorphic in point  , if a neighbourhood of   exists, in which   is complex differentiable.

Definition: Full function

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If   is complexly differentiable to the whole  , then   is called a entire function.

Explanatory notes

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  can be interpreted as a two-dimensional real vector space with the canonical base   and so one can examine a function   on an open set  . In multivariable calculus it is known that   total differentiable in   if there exists a  -linear mapping  , so that

 

where   is a function with the property

 

It can now be seen that the function   is complex differentiable in  , if   is total differentiable in   and   is even  -linear. The latter is a sufficient condition. It means that the transformation matrix   with respect to the canonical base   has the form

 

with  .

Jacobi Matrix

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Main article: Cauchy-Riemann equations

If a function   is decompose into functions of its real and imaginary parts with real-valued functions  , the total derivative   with tranformation matrix has the Jacobian matrix

 


Cauchy-Riemannian differential equations

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Consequently, the function   is total differentiable precisely when it can be differentiated relatively and for   the Cauchy-Riemann equations

 
 

are fulfilled.

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

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  • (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
     
    where   the Cauchy-Riemann operator defined by  

Examples

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Entire Functions

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An entire function is holomorphic on the whole  . Examples are:

  • each polynomial   with coefficients  ,
  • the Exponential Function  ,
  • the trigonometric functions   and  ,
  • the hyperbolic functions   and  .

Holomorphic, non-gant functions

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  • 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   can be developed at all points from   into a power series and is thus holomorphic on the set  .

Functions - not holomorphic at any point

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The following functions are not holomorphic in any  . Examples are:

  • the absolute value function  ,
  • the projections on the real part   or on the imaginary part  ,
  • complex conjugation  .

The function   is complex differentiable only at the point  , but the function is not' holomorphic in  , since it is not complex differentiable in a neighborhood of  .

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.[1] That is, if functions   and   are holomorphic in a domain   , then so are   ,   ,   , and   . Furthermore,   is holomorphic if   has no zeros in   ; otherwise it is meromorphic.

If one identifies   with the real plane   , 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

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Every holomorphic function can be separated into its real and imaginary parts   , and each of these is a harmonic function on   (each satisfies Laplace's equation   ), with   the harmonic conjugate of   .[3] Conversely, every harmonic function   on a simply connected domain   is the real part of a holomorphic function: If   is the harmonic conjugate of   , unique up to a constant, then   is holomorphic.

Cauchy's integral theorem

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

 

Here   is a rectifiable path in a simply connected complex domain   whose start point is equal to its end point, and   is a holomorphic function.

Cauchy's integral formula

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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   is a complex domain,   is a holomorphic function and the closed disk   is completely contained in   . Let   be the circle forming the boundary of   . Then for every   in the interior of   :

 

where the contour integral is taken counter-clockwise.

Cauchy's differentiation formula

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The derivative   can be written as a contour integral[4] using Cauchy's differentiation formula:

 

for any simple loop positively winding once around   , and

 

for infinitesimal positive loops   around   .

Conformal map

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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

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

Functions as complex vector space

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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. Additionally, the set of holomorphic functions in an open set   is an integral domain if and only if the open set   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

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From a geometric perspective, a function   is holomorphic at   if and only if its exterior derivative   in a neighbourhood   of   is equal to   for some continuous function   . It follows from

 

that   is also proportional to   , implying that the derivative   is itself holomorphic and thus that   is infinitely differentiable. Similarly,   implies that any function   that is holomorphic on the simply connected region   is also integrable on   .

Choice of Path - Independency

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For a path   from   to   lying entirely in   , define   ; in light of the Jordan curve theorem and the generalized Stokes' theorem,   is independent of the particular choice of path   , and thus   is a well-defined function on   having   or   .

Biholomorphic functions

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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

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In the n-dimensional complex space

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Let  a complex open subset. An illustration   is called holomorph if   is holomorphous in each sub-function and each variable. With the Wittgenstein-calculus   and   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   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   dimensions. Salomon Bochner even proved in 1944 a generalization of the  -dimensional Cauchy integral formula. This bears the name Bochner-Martinelli-Formel.

In complex geometry

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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

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  • 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

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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. 
  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

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