# Complex Functions

```Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 11
Ali Hussein Mahmood Al-Obaidi
College of Education for Pure Sciences
Physics Department
Three Stage
ali.alobaidi81@yahoo.com
2.1 Function of a complex variables: Complex Variable:- A symbol , such as , which can stand for any one
of a set complex numbers is called a complex variable. is called
domain of variation.
 Complex function:- If to each value of a complex variable which can
assume there corresponds one or more values of a complex variable ,
we say that is a function of and write
or
. The
variable is sometimes called an independent variable, while is
called a dependent variable. The value of a function at
is often
written
. Thus if
, then
. is
called domain of the function and the set of value of a function is
called range of the function.
 Single and Multiple-valued function:- If only one value of
corresponds to each value of , we say that is a single-valued
function. If more than one value of corresponds to each value of ,
we say that is a multiple-valued or many-valued function of .
Ex:1) The function
is single-valued function.
2) The function
is multiple-valued function because
1
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 11
College of Education for Pure Sciences
Physics Department
Three Stage
Ali Hussein Mahmood Al-Obaidi
ali.alobaidi81@yahoo.com
 Inverse function:- If
then we can also consider as a
function of , written
. The function
is often
called the inverse function corresponding to Thus
and
one inverse functions of each other.
EX:- Find inverse function corresponding to
Sol:Thus
, where
where
, where
 One-one function:- Let
called one-one function if
.
is a function, then a function
is
2.2 Transformations:If
function of
(where and are real function) is a single-valued
(where and are real numbers). we can write
. By equating real and imaginary parts this is seen to
be equivalent to
,
……….
Thus given a point
in the plan , such as in (fig.1) below , there
corresponds a point (
in the plane ,say
in (fig.2) below .The
set of equations [or the equivalent ,
]is called a transformation.
We say that point is transformed
Q
P
into point
by means of the
Transformed and call
the image
of .
Fig.1
2
Fig.2
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 11
Ali Hussein Mahmood Al-Obaidi
Ex:- If
College of Education for Pure Sciences
Physics Department
Three Stage
ali.alobaidi81@yahoo.com
then
Then the transformation is
. The image of a point
(1,2) in the plane is the point (-3,4) in the plane .
In general, under a transformation, a set of points such as these on
curve
of (fig.1) is mapped into corresponding set of points , called
the image , such as those on curve
in(fig.2). The particular
characteristic of the image depend of course on the type of function
, which is sometimes called a mapping function. If
is multiplevalued, a point (or curve) in the plane is mapping in general in to
more than one point (or curve) in the plane.
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