Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 16
Ali Hussein Mahmood Al-Obaidi
EX:- Find
i.
College of Education for Pure Sciences
Physics Department
Three Stage
ali.alobaidi81@yahoo.com
for all following:2
ii.
iii.
2.7 Analytic Function:Def:- We say that a function
is differentiable function at
of .
is analytic function at if
and all points in neighborhood
Def:- We said to be the function
analytic function in all points plane.
is entire function if it is
Def:- Let
is analytic function in some points of neighborhood of
except , then is called singular point.
Polynomial is entire function.
EX 1:-
is differentiable function at
EX 2:- Prove that
It isn’t analytic function at
. H.W
is differentiable function at
.
Sol:-
1
but
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 16
College of Education for Pure Sciences
Physics Department
Three Stage
Ali Hussein Mahmood Al-Obaidi
ali.alobaidi81@yahoo.com
=
If
, then
, we get
That is the derivative of function
at
is exist and equal .
We will prove that the derivative of this function isn’t exist at
neighborhood of .
If
and
then
by put it in , we get
Thus
And if
, then
by put it in ,we get
Thus
Since
, then the limit of derivative of function
isn't unique,
we get the function
isn't differentiable function at
neighborhood of
, so that
isn't analytic function at
.
2
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 16
College of Education for Pure Sciences
Physics Department
Three Stage
Ali Hussein Mahmood Al-Obaidi
ali.alobaidi81@yahoo.com
Theorem: - If the function
function in region , then
equations
and
is analytic
are satisfied Cauchy-Riemann
. If partial derivative in C-R equations
are continuous function in region
are sufficient to make the function
.
,then C-R equations
is analytic function in region
EX 1:- Prove that
is entire ?
SOL:-
Thus
The C-R equations are satisfied, we see the partial derivative of
are
continuous functions. So that
is an entire function by theorem C-R.
EX(2):- The function
isn't analytic function , prove that?
SOL:-
3
Ministry of Higher Education
and Scientifics research
University of Babylon
Complex Functions
Lecture 16
Ali Hussein Mahmood Al-Obaidi
College of Education for Pure Sciences
Physics Department
Three Stage
ali.alobaidi81@yahoo.com
By C-R equations, we get
Thus
The C-R equations aren’t satisfied, so that
isn't analytic function.
EX(3):- Prove that the function
C-R equations at
is satisfied
but it isn’t analytic function at
.
SOL:-since
Put
then
The function is satisfied C-R equation at
but we known from prior
example that
isn't analytic function.
4