What are analytic functions in complex variables

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As analytically In mathematics, one denotes a function that is given locally by a convergent series of powers. Due to the differences between real and complex analysis, one often speaks explicitly of real-analytic or complex-analytical Functions. In the complex are the properties analytically and holomorphic equivalent to. If a function is defined and analytical in the entire complex level, it is called all.

Table of Contents

definition


Let \ ({\ displaystyle \ mathbb {K} = \ mathbb {R}} \) or \ ({\ displaystyle \ mathbb {K} = \ mathbb {C}} \). Let \ ({\ displaystyle D \ subseteq \ mathbb {K}} \) be an open subset. A function \ ({\ displaystyle f \ colon D \ to \ mathbb {K}} \) is called analytically in the point \ ({\ displaystyle x_ {0} \ in D,} \) if there is a power series

\ ({\ displaystyle \ sum _ {n = 0} ^ {\ infty} a_ {n} (x-x_ {0}) ^ {n}} \)

that converges on a neighborhood of \ ({\ displaystyle x_ {0}} \) to \ ({\ displaystyle f (x)} \). If \ ({\ displaystyle f} \) is analytic in every point of \ ({\ displaystyle D} \), then \ ({\ displaystyle f} \) analytically.

properties


  • An analytic function can be differentiated any number of times. The reverse does not apply, see examples below.
  • The local power series representation of an analytic function \ ({\ displaystyle f} \) is its Taylor series. So it applies
\ ({\ displaystyle a_ {n} = {\ frac {f ^ {(n)} (x_ {0})} {n!}}} \).
  • Sums, differences, products, quotients (provided the denominator has no zeros) and concatenations of analytical functions are analytic.
  • Is \ ({\ displaystyle D} \) connected and the set of zeros of an analytic function \ ({\ displaystyle f \ colon D \ to \ mathbb {K}} \) has a cluster point in \ ({\ displaystyle D} \ ), then \ ({\ displaystyle f} \) is the null function. Are accordingly \ ({\ displaystyle f, g \ colon D \ to \ mathbb {K}} \) two functions that match on a set that has an accumulation point in \ ({\ displaystyle D} \), e.g. B. on an open subset, they are identical.

Real functions


Examples of analytical functions

Many common functions in real analysis, such as polynomial functions, exponential and logarithmic functions, trigonometric functions, and rational expressions in these functions are analytical. The set of all real-analytic functions on an open set \ ({\ displaystyle D} \) is denoted by \ ({\ displaystyle C ^ {\ omega} (D)} \).

Exponential function

A well-known analytical function is the exponential function

\ ({\ displaystyle \ exp (x) = \ sum _ {k = 0} ^ {\ infty} {\ frac {1} {k!}} x ^ {k} = 1 + x + {\ frac {1} {2}} x ^ {2} + {\ frac {1} {6}} x ^ {3} + {\ frac {1} {24}} x ^ {4} + \ dotsb} \),

which converges on the whole \ ({\ displaystyle \ mathbb {R}} \).

Trigonometric function

The trigonometric functions sine, cosine, tangent, cotangent and their arc functions are also analytic. However, the example of the arctangent shows

\ ({\ displaystyle \ arctan (x) = \ sum _ {k = 0} ^ {\ infty} {\ frac {(-1) ^ {k}} {2k + 1}} x ^ {2k + 1} = x - {\ frac {1} {3}} x ^ {3} + {\ frac {1} {5}} x ^ {5} - {\ frac {1} {7}} x ^ {7} + \ dotsb} \),

that a completely \ ({\ displaystyle \ mathbb {R}} \) analytic function can have a series expansion with a finite radius of convergence.

Special functions

Many special functions such as Euler's gamma function, Euler's beta function or Riemann's zeta function are also analytical.

Examples of non-analytical functions

The following examples of non-analytical functions count as smooth functions: They can be differentiated as often as desired on their domain, but there is no power series expansion at individual points. The following function

\ ({\ displaystyle f (x) = {\ begin {cases} \ exp \ left (- {\ frac {1} {x ^ {2}}} \ right) & \ mathrm {f {\ ddot {u} } r} \ x \ neq 0 \ 0 & \ mathrm {f {\ ddot {u}} r} \ x = 0 \ end {cases}}} \)

is at all points \ ({\ displaystyle x \ in \ mathbb {R}} \), thus also at the point 0, differentiable as often as desired. From \ ({\ displaystyle f ^ {(n)} \ left (0 \ right) = 0} \) for all \ ({\ displaystyle n} \) follows the Taylor series of \ ({\ displaystyle f} \ ),

\ ({\ displaystyle \ sum _ {n = 0} ^ {\ infty} {\ frac {0} {n!}} x ^ {n} = 0} \),

which, except in the point \ ({\ displaystyle x = 0} \), does not match \ ({\ displaystyle f \ left (x \ right)} \). Thus \ ({\ displaystyle f} \) at point 0 is not analytical.

The function too

\ ({\ displaystyle g (x) = {\ begin {cases} \ exp \ left (- {\ frac {1} {x ^ {2}}} \ right) & \ mathrm {f {\ ddot {u} } r} \ x> 0 \ 0 & \ mathrm {f {\ ddot {u}} r} \ x \ leq 0 \ end {cases}}} \)

can be differentiated any number of times, because all right-hand derivatives at the zero point are just as equal to 0 as all left-hand derivatives.

There is an important class of non-analytical functions that functions with compact carrier. The carrier of a function is the completion of the set of points at which a function does not vanish:

\ ({\ displaystyle {\ overline {\ {x \ mid f (x) \ not = 0 \}}}} \).

If the carrier is compact, one speaks of a function with a compact carrier (or of a test function). These functions play a major role in the theory of partial differential equations. For functions that are completely defined \ ({\ displaystyle \ mathbb {R}} \), this condition is equivalent to the fact that there is a number \ ({\ displaystyle C> 0} \) such that \ ({ \ displaystyle f (x) = 0} \) applies to all \ ({\ displaystyle x} \) with \ ({\ displaystyle | x |> C} \). A function with a compact support therefore agrees with the null function for large \ ({\ displaystyle x} \). If the function were now also analytic, it would already coincide completely \ ({\ displaystyle \ mathbb {R}} \) with the null function according to the above properties of analytical functions. In other words, the only analytical function with compact support is the null function.

The function

\ ({\ displaystyle h (x) = g (x) g (1-x) = {\ begin {cases} \ exp \ left (- {\ frac {1} {x ^ {2}}} - {\ frac {1} {(1-x) ^ {2}}} \ right) & \ mathrm {f {\ ddot {u}} r} \ 0

is an arbitrarily often differentiable function with compact support \ ({\ displaystyle [0,1]} \).

In the previous examples one can prove that the Taylor series has a positive radius of convergence at every point, but does not converge to the function everywhere. But there are also non-analytical functions for which the Taylor series has a radius of convergence of zero, for example the function is

\ ({\ displaystyle f (x) = \ int _ {0} ^ {\ infty} {\ frac {\ mathrm {e} ^ {- t}} {1 + x ^ {2} t}} \, \ mathrm {d} t} \)

differentiable over the whole \ ({\ displaystyle \ mathbb {R}} \) any number of times, but its Taylor series in \ ({\ displaystyle x_ {0} = 0} \)

\ ({\ displaystyle \ sum _ {k = 0} ^ {\ infty} (- 1) ^ {k} k! \ cdot x ^ {2k} = 1-x ^ {2} + 2x ^ {4} - 6x ^ {6} + 24x ^ {8} - \ dotsb} \)

is only convergent for \ ({\ displaystyle x = 0} \).[1]

More generally one can show that any formal power series occurs as a Taylor series of a smooth function.

Complex functions


In function theory it is shown that a function \ ({\ displaystyle f} \) of a complex variable, which is complexly differentiable in an open circular disk \ ({\ displaystyle D} \), in \ ({\ displaystyle D} \) even as often as you like is complex differentiable, and that the power series around the center \ ({\ displaystyle c} \) of the circular disk,

\ ({\ displaystyle \ sum _ {n = 0} ^ {\ infty} {\ frac {f ^ {(n)} (c)} {n!}} (z-c) ^ {n}} \),

for every point \ ({\ displaystyle z} \) from \ ({\ displaystyle D} \) converges to \ ({\ displaystyle f (z)} \). This is an important aspect under which functions in the complex plane are easier to handle than functions of a real variable. In fact, one uses the attributes in function theory analytically, holomorphic and regular synonym. Their equivalence is not immediately apparent from the original definitions of these terms; it was only proven later. Complex analytic functions that only take real values ​​are constant. A consequence of the Cauchy-Riemann differential equations is that the real part of an analytic function determines the imaginary part up to a constant and vice versa.

The following important relationship between real-analytic functions and complex-analytic functions applies:

Every real-analytic function \ ({\ displaystyle \ mathbb {R} \ to \ mathbb {R}} \) can lead to a complex-analytic, i.e. holomorphic function on a neighborhood of \ ({\ displaystyle \ mathbb {R} \ subset \ mathbb {C}} \).

Conversely, every holomorphic function becomes a real-analytic function if it is first restricted to \ ({\ displaystyle \ mathbb {R}} \) and then only the real part (or only the imaginary part) is considered. This is the reason why many properties of the real-analytic functions are most easily proven with the help of complex function theory.

Several variables


Even with functions \ ({\ displaystyle f} \) that depend on several variables \ ({\ displaystyle x_ {1}, \ dotsc, x_ {n}} \), a Taylor series expansion at the point \ ({ \ displaystyle x = (x_ {1}, \ dotsc, x_ {n})} \) define:

\ ({\ displaystyle \ sum _ {\ alpha \ in \ mathbb {N} _ {0} ^ {n}} {\ frac {\ partial ^ {\ alpha} f (x)} {\ alpha!}} ( \ xi -x) ^ {\ alpha}.} \)

The multi-index notation was used, the sum extends over all multi-indexes \ ({\ displaystyle \ alpha = (\ alpha _ {1}, \ dotsc, \ alpha _ {n}) \ in \ mathbb {N} _ { 0} ^ {n}} \) of length \ ({\ displaystyle n} \). In analogy to the case of a variable discussed above, a function is called analytical if the Taylor series expansion has a positive radius of convergence for every point of the domain and represents the function within the domain of convergence, that is, that

\ ({\ displaystyle f (\ xi) = \ sum _ {\ alpha \ in \ mathbb {N} _ {0} ^ {n}} {\ frac {\ partial ^ {\ alpha} f (x)} { \ alpha!}} (\ xi -x) ^ {\ alpha}} \)

for all \ ({\ displaystyle \ xi = (\ xi _ {1}, \ dotsc, \ xi _ {n})} \) from a neighborhood of \ ({\ displaystyle x = (x_ {1}, \ dotsc , x_ {n})} \) holds. In the case of complex variables, one speaks of holomorphic functions even in the case of several variables. Such functions are examined by function theory in several complex variables.

literature


  • Konrad Koenigsberger: Analysis. Volume 2. 3rd revised edition. Springer-Verlag, Berlin and others 2000, ISBN 3-540-66902-7.
  • Eberhard Freitag, Rolf Busam: Function theory 1. Springer-Verlag, Berlin, ISBN 3-540-67641-4.

Individual evidence


  1. ^ Archive of evidence (analysis / differential calculus): Taylor series with a radius of convergence of zero









Categories:Analysis | Analytical function




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