Complex-valued function

In mathematics, a complex-valued function is a function whose function values ​​are complex numbers . Closely related to this is the term complex function , which is not clearly used in the literature. Complex functions are investigated in analysis and in function theory and have a wide range of applications such as in physics and electrical engineering , where they are used, for example, to describe vibrations .

definition

Complex-valued function

A complex valued function is a function

${\displaystyle f\colon D\to \mathbb {C} }$,

where the target set is the set of complex numbers . There are no requirements for the definition set. ${\displaystyle D}$

Complex function

As with real-valued and real functions , the use of the term complex function in the literature is ambiguous. Sometimes it is used synonymously with a complex-valued function, sometimes it is only used for complex-valued functions of a complex variable, i.e. functions

${\displaystyle f\colon D\to \mathbb {C} }$,

where is. ${\displaystyle D\subseteq \mathbb {C} }$

Special cases

Sometimes an addition is appended to the complex-valued function to specify the structure of the definition set. This is the name of a function, for example ${\displaystyle f\colon D\to \mathbb {C} }$

• complex-valued function of a real variable, if is, ${\displaystyle D\subseteq \mathbb {R} }$
• complex-valued function of several real variables, if with is, ${\displaystyle D\subseteq \mathbb {R} ^{n}}$ ${\displaystyle n>1}$
• complex-valued function of a complex variable if is, ${\displaystyle D\subseteq \mathbb {C} }$
• complex-valued function of several complex variables, if with is. ${\displaystyle D\subseteq \mathbb {C} ^{n}}$ ${\displaystyle n>1}$

If is a subset of a complex vector space , then a function is also called a (complex-valued) functional . ${\displaystyle D}$ ${\displaystyle f\colon D\to \mathbb {C} }$

Examples

• The function defined by ${\displaystyle f\colon \mathbb {R} \to \mathbb {C} }$

${\displaystyle f(x)=\exp(\mathrm {i} x)=\cos(x)+\mathrm {i} \sin(x)}$

is a complex-valued function of a real variable. It is exactly Euler's formula .

• With is the exponential function ${\displaystyle z=x+\mathrm {i} y}$

${\displaystyle \exp(z)=\exp(x+\mathrm {i} y)=\exp(x)\left(\cos(y)+\mathrm {i} \sin(y)\right)}$

a complex-valued function of a complex variable.

• The function defined by ${\displaystyle f\colon \mathbb {R} ^{2}\to \mathbb {C} }$

${\displaystyle f(x_{1},x_{2})=x_{1}+\mathrm {i} x_{2}}$

is a complex-valued function of two real variables.

• Because the real numbers are embedded in the complex numbers, all real-valued functions can also be viewed as complex-valued functions.

properties

Algebraic properties

The set of all complex-valued functions over a given set forms a complex vector space , which is denoted by , or . The sum of two complex-valued functions and is defined by ${\displaystyle D}$ ${\displaystyle F(D,\mathbb {C} )}$ ${\displaystyle \operatorname {Abb} (D,\mathbb {C} )}$ ${\displaystyle \mathbb {C} ^{D}}$ ${\displaystyle f}$ ${\displaystyle g}$

${\displaystyle (f+g)(x)=f(x)+g(x)}$

for all and the product of a complex-valued function of a complex number by ${\displaystyle x\in D}$ ${\displaystyle f}$ ${\displaystyle c\in \mathbb {C} }$

${\displaystyle (c\cdot f)(x)=c\cdot f(x)}$

for everyone . These vector spaces are called complex function spaces . They play an important role in linear algebra and calculus . With addition and point-wise multiplication defined by ${\displaystyle x\in D}$

${\displaystyle (f\cdot g)(x)=f(x)\cdot g(x)}$

for all the complex-valued functions over the set form a commutative ring . With all three links, the complex-valued functions form a complex algebra . ${\displaystyle x\in D}$ ${\displaystyle D}$

Analytical properties

A complex-valued function is called bounded if there is a bound such that ${\displaystyle f\colon D\to \mathbb {C} }$ ${\displaystyle M}$

${\displaystyle |f(x)|\leq M}$

is for everyone . The set of bounded complex-valued functions forms with the supremum norm ${\displaystyle x\in D}$ ${\displaystyle B(D,\mathbb {C} )}$

${\displaystyle \|f\|_{\infty }:=\sup _{x\in D}|f(x)|}$

a standardized space . Since the complex numbers are complete , this is even a Banach space . A sequence of complex-valued functions with for is called uniformly bounded if every term in the sequence is a bounded function and the sequence ${\displaystyle (f_{1},f_{2},\ldots )}$ ${\displaystyle f_{n}\colon D\to \mathbb {C} }$ ${\displaystyle n=1,2,\ldots }$

${\displaystyle (\|f_{1}\|_{\infty },\|f_{2}\|_{\infty },\ldots )}$

is a bounded sequence of complex numbers. A sequence of complex-valued functions is called pointwise restricted if the complex number sequence for all ${\displaystyle x\in D}$

${\displaystyle (f_{1}(x),f_{2}(x),\ldots )}$

is limited. A uniformly bounded sequence of complex-valued functions is always bounded pointwise, but the converse does not have to apply. A sequence of complex-valued functions is said to converge uniformly to a complex-valued function if ${\displaystyle f\colon D\to \mathbb {C} }$

${\displaystyle \lim _{n\to \infty }\|f_{n}-f\|_{\infty }=0}$

applies. Correspondingly, a sequence of complex-valued functions is called pointwise convergent to a complex-valued function if for all ${\displaystyle f\colon D\to \mathbb {C} }$ ${\displaystyle x\in D}$

${\displaystyle \lim _{n\to \infty }(f_{n}(x)-f(x))=0}$

applies. Here, too, the uniform convergence results in point-wise convergence, but not the reverse. Further analytical properties, such as continuity , differentiability or integratability , require at least a topological , metric or mass-theoretical structure on the definition set .

Generalizations

The complex vector-valued functions form a generalization ; these are mapped in the . Even more general are vector-valued functions whose image space is an arbitrary vector space. ${\displaystyle \mathbb {C} ^{n}}$

literature

• Konrad Königsberger: Analysis 1 . 6th, revised edition. Springer-Verlag, Berlin Heidelberg New York 2004, ISBN 3-540-40371-X .
• Otto Forster: Analysis 1 . Differential and integral calculus of a variable. 11th, expanded edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-00316-6 , doi : 10.1007 / 978-3-658-00317-3 .

Web links

• Eric W. Weisstein : Complex Function . In: MathWorld (English).