Analytical function

In mathematics, analytical is a function that is given locally by a convergent power series . Due to the differences between real and complex analysis , one often speaks explicitly of real-analytic or complex-analytic functions for clarification purposes . In the complex, the properties are analytically and holomorphically equivalent. If a function is defined and analytical in the entire complex level, it is called whole .

definition

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

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

there that converges on a neighborhood of against . If every point is analytical, then it is called analytical . ${\ displaystyle x_ {0}}$${\ displaystyle f (x)}$${\ displaystyle f}$${\ displaystyle D}$${\ displaystyle f}$

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 is its Taylor series . So it applies${\ displaystyle f}$
${\ displaystyle a_ {n} = {\ frac {f ^ {(n)} (x_ {0})} {n!}}}$.
• Sums, differences, products, quotients (as long as the denominator has no zeros) and concatenations of analytical functions are analytic.
• If connected and if the set of zeros of an analytical function has an accumulation point in , then is the zero function . Are accordingly two functions that match on a set that has an accumulation point in , e.g. B. on an open subset, they are identical.${\ displaystyle D}$ ${\ displaystyle f \ colon D \ to \ mathbb {K}}$${\ displaystyle D}$${\ displaystyle f}$${\ displaystyle f, g \ colon D \ to \ mathbb {K}}$${\ displaystyle D}$

Real functions

Examples of analytical functions

Many common functions in real analysis, such as polynomials , 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 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 quite . ${\ displaystyle \ mathbb {R}}$

Trigonometric function

The trigonometric functions sine, cosine , tangent , cotangent and their arc functions are also analytical. 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 analytic function can have a series expansion with a finite radius of convergence . ${\ displaystyle \ mathbb {R}}$

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 graph of the function falls very quickly towards 0 in the vicinity of 0. Even the value can no longer be distinguished from 0 in the graph.${\ displaystyle f}$${\ displaystyle f (0 {,} 4) \ approx 0 {,} 0019 \ ldots}$

The following examples of non-analytical functions are among the smooth functions : They are infinitely differentiable on their domain of definition, but there is no power series expansion at individual points. The following function

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

can be differentiated as often as required for all , including at point 0. From for all follows the Taylor series of , ${\ displaystyle x \ in \ mathbb {R}}$${\ displaystyle f ^ {(n)} \ left (0 \ right) = 0}$${\ displaystyle n}$${\ displaystyle f}$

${\ displaystyle \ sum _ {n = 0} ^ {\ infty} {0 \ over n!} x ^ {n} = 0}$,

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

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. All derivatives of the two sub-functions at the zero point are 0, so they fit together.

There is an important class of non-analytical functions called compact-beam functions . 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 entirely defined on, this condition is equivalent to that there is a number such that with holds for all . A function with a compact carrier thus coincides with the null function for large ones. If the function were also analytical, it would, according to the above properties of analytical functions, completely agree with the null function. In other words, the only analytical function with compact support is the null function. ${\ displaystyle \ mathbb {R}}$${\ displaystyle C> 0}$${\ displaystyle f (x) = 0}$${\ displaystyle x}$${\ displaystyle | x |> C}$${\ displaystyle x}$${\ displaystyle \ mathbb {R}}$

At the maximum point is ${\ displaystyle h \ left ({\ tfrac {1} {2}} \ right) = e ^ {- 8} \ approx 0.000335 \ ldots}$

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 a compact carrier . ${\ displaystyle [0,1]}$

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. However, there are also non-analytical functions for which the Taylor series has a radius of convergence of zero, e.g. B.

It is and for${\ displaystyle f (0) = 1}$${\ displaystyle 0 ${\ displaystyle | x | \ to \ infty}$

The function

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

is differentiable to any number of times, but its Taylor series is in ${\ displaystyle \ mathbb {R}}$${\ displaystyle x_ {0} = 0}$

${\ displaystyle 1-x ^ {2} +2! \, x ^ {4} -3! \, x ^ {6} +4! \, x ^ {8} - \ dotsb}$

and thus only for convergent. ${\ displaystyle x = 0}$

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 of a complex variable, which is complexly differentiable in an open circular disk, can be complexly differentiated as often as desired in the same open environment , and that the power series around the center of the circular disk, ${\ displaystyle f}$${\ displaystyle D}$ ${\ displaystyle D}$${\ displaystyle c}$

${\ displaystyle \ sum _ {n = 0} ^ {\ infty} {f ^ {(n)} (c) \ over n!} (zc) ^ {n}}$,

for each point out against converges. This is an important aspect under which functions in the complex plane are easier to handle than functions of a real variable. In fact, in function theory, the attributes analytical , holomorphic and regular are used synonymously. From the original definitions of these terms, their equivalence is not immediately apparent; it was only proven later. Complex-analytic functions that only take real values ​​are constant. A conclusion from 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. ${\ displaystyle z}$${\ displaystyle D}$${\ displaystyle f (z)}$

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

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

Conversely, every holomorphic function becomes a real-analytic function if it is first restricted to 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 proved with the help of complex function theory. ${\ displaystyle \ mathbb {R}}$

Several variables

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

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

The multi-index notation was used, the sum extends over all multi-indexes of the length . 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 each point of the domain and represents the function within the domain of convergence, that is, that ${\ displaystyle \ alpha = (\ alpha _ {1}, \ dotsc, \ alpha _ {n}) \ in \ mathbb {N} _ {0} ^ {n}}$${\ displaystyle n}$

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

applies to everyone in a neighborhood of . In the case of complex variables, one speaks of holomorphic functions even in the case of several variables. Such functions are treated in function theory in several complex variables . ${\ displaystyle \ xi = (\ xi _ {1}, \ dotsc, \ xi _ {n})}$${\ displaystyle x = (x_ {1}, \ dotsc, x_ {n})}$