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[2. Mathematical elements.]
Complex functions of real variables.
Let X and Y be two sets and let f be a rule that assigns to every
element of X exactly one element of Y; f is called a function
from X to Y and the sets X and Y are called respectively the domain
and the codomain (or range) of the function f.
A function f from X to Y is represented by the notation
y = f (x) where y is the element of Y assigned
to the element x of X. The element x is called the argument
of the function f and the element y is called the
value of f for the argument x.
The elements x and y are also called variables;
x is the independent variable and y is the dependent variable.
If the domain and the codomain of a function f are equal to the set R
of real numbers (that is: X=Y=R) then the function f is called a real
function of a real variable. A well known function of this kind is the function
square ( y = x² ) which associates to
every real number x its square x².
Multiplication between two real numbers is a real function of two real
variables; it is an example of functions of the type
y = f (x_{1} , x_{2})
where x_{1} and x_{2} are real numbers.
From a formal point of view, a function like
y = f (x_{1} , x_{2})
with x_{1} and x_{2} real numbers is a function from RxR to R,
where RxR is the Cartesian product of R by itself: The domain of f is the
set of all ordered pairs of real numbers, that is the set whose elements are the
ordered pairs
< a, b > for every real number a and b.
Complex functions of a real variable are functions from the set R of real number
to the set C of complex numbers; a function of this kind associates to every real
number exactly one complex number.
An example is the square root function that assign to every real number its square
root: If the argument of this function is less than zero, then the value of the
function is a complex number.
A complex function of n real variables is a function whose arguments are
n real numbers and whose value is a complex number. It is a function
from R^{n} to C.
From this point, Greek letters will be used to denote complex functions or
complex numbers; Latin letters will be used to denote real functions or real
numbers. Here are some examples:
 φ (x) and Ψ (x)
denote complex functions of a real variable.
 φ (x, y) and Ψ (x, y, z)
denote complex functions of real variables (two and three real variables, respectively).
 φ and Ψ denote complex functions of a real variables,
without specifying the number of real variables.
 σ denotes a complex number.
 x y z denote real numbers.
