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A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as a domain and the complex numbers as a codomain. Complex functions are generally assumed to have a domain that contains a nonempty open subset of the complex plane.
This formula distinguishes the complex number i from any real number, since the square of any (negative or positive) real number is always a non-negative real number. With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers.
Suppose a punctured disk D = {z : 0 < |z − c| < R} in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res( f , c ) of f at c is the coefficient a −1 of ( z − c ) −1 in the Laurent series expansion of f around c .
The generalization of the Riemann integral to functions of a complex variable is done in complete analogy to its definition for functions from the real numbers. The partition of a directed smooth curve γ {\displaystyle \gamma } is defined as a finite, ordered set of points on γ {\displaystyle \gamma } .
A complex-valued function of several real variables may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values. If f(x 1, …, x n) is such a complex valued function, it may be decomposed as
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space, that is, n-tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space ), which the Mathematics Subject ...
The complex logarithm is needed to define exponentiation in which the base is a complex number. Namely, if a {\displaystyle a} and b {\displaystyle b} are complex numbers with a ≠ 0 {\displaystyle a\not =0} , one can use the principal value to define a b = e b Log a {\displaystyle a^{b}=e^{b\operatorname {Log} a}} .
Figure 1. This Argand diagram represents the complex number lying on a plane.For each point on the plane, arg is the function which returns the angle . In mathematics (particularly in complex analysis), the argument of a complex number z, denoted arg(z), is the angle between the positive real axis and the line joining the origin and z, represented as a point in the complex plane, shown as in ...