Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 13
College of Education for Pure Sciences
Physics Department
Three Stage
Ali Hussein Mahmood Al-Obaidi
ali.alobaidi81@yahoo.com
 Theorems of limits:Theorem 1:- (Uniqueness of limit)
If a function has a limit at
That is we can't find
.
then the limit of function
such that
Theorem 2:- Let
if and only if (iff)
where
at
is unique.
&
, then
,
.
Theorem(3):- Let
and
&
are two functions, such that
, then
a)
b)
c)
Theorem(4):a) The limit of a constant function is constant.
Let
, then
b) Let
,then
c) Let
, then
d) If
function with coefficient
is polynomial
are complex numbers, then
1
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 13
College of Education for Pure Sciences
Physics Department
Three Stage
Ali Hussein Mahmood Al-Obaidi
ali.alobaidi81@yahoo.com
e) If
, then
EX:-Find
1.
2.
3.
H.W.
2.4 Continuity:Def:- Let
be defined and single-valued function in a neighborhood of
. The function
is said to be continuous at if
is
exist .
Note that this implies three conditions which must be met in order that
be continuous at :1)
2)
3)
must exist.
must exist , i.e.
.
Equivalently, if
suggestive form
is defined at
is continuous at
.
2
.
, we can write this in the
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 13
Ali Hussein Mahmood Al-Obaidi
College of Education for Pure Sciences
Physics Department
Three Stage
ali.alobaidi81@yahoo.com
Alternative to above definition of continuity, we can defined
as continuous at
if for any
we can find
such that
wherever
. Note that this is simply the
definition of limit with
and removal of restriction
.
3