A New Numerical Procedure to Compute the Residues of a Complex Functions

Received on: 27/08/2006 Accepted on: 16/11/2006 ABSTRACT In this paper, we are going to deal with computations of Residues and Poles for the complex functions . We are also going to investigate a new numerical procedure theoretically and its implementation numerically to compute the residue of complex analytic functions with high order poles. The paper needs the knowledge of computing the complex improper integrations.


1-Introduction :
A final bit of theory in this paper is devoted to Cauchy's residue theorem and its applications .Recall that Cauchy's theorem stated that ) so long as f(z) was analytic every where inside C.What if f(z) is not analytic within C?
The answer is provided by Cauchy's integral formula .Here we have proposed a new procedure to compute the complex improper integrations by calculating the residues of the function .
To compute the residues of a complex function of the form ( ) such that the function f(z) has a pole at , we are going to use the (shortcut -method) to estimate the residues [9].
If R is the real field and C is the complex field then consider the following definitions and theorems:

1-2 Analytic functions :[9]
If the derivative ) ( / z f exists at all points (z) of a region R ,then ) (z f is said to be analytic in R.

1-3 Taylor's theorem :[3]
Let f(z) be analytic inside and on a simple closed curve C .Let (a) and (a+h) be two points inside C , then The above series is called Taylor expansion for f(z).

1-4 Cauchy integral formula :[2]
If f(z) is analytic inside and on a simple closed curve C and (a) is any point inside C then


Where C is traversed in the positive sense also the nth derivative of (z=a) is given by n=1,2 ,3,…..

1-5 Lemma (1) :[9]
Let ) ( o Z be a pole of order (m) of a function f(z) then the residue of the function at a pole given by the form : The formula namely (shortcutmethod ) .
2-A New method to compute the residues of a complex function: We know that : We find the value of the above integration by calculating the residues by this procedure and it can be found in [5].

2.1)
then the residue is given by the following form : then the residue is given by the following form : See [5] for the details of these procedures.

3-In this paper we are going to follow a procedure for a general function of a pole of order (m): 3.1) Let us start with order (m=3):
then the residue is given by the following form : )) ( ( is a pole of order (m=3) then by (short-cut-method) we get : We expand the analytic function q(z) into Taylar series valid in disk Then from (6) we get : (10) Then Substituting the above values in (10) in equation (9) we get the desired proof .
then the residue is given by the following form : is a pole of order (m=4) then by (shortcut -method) we get : We expand the analytic function q(z) into Taylor series valid in disk Then from (14) we get : Then substituting the above values in (19) in equation (18) we get the desired proof.

3.3) In this approach we can find the residue for (m=5):
then the residue is given by the following form : )) ( (

3.4) In this approach we can find the residue for (m=6):
then the residue is given by the following form : then the residue is given by the following form: )) ( ( z q z p z q z q z q z p z q z q z p z q z q z q z p z q z q z p z q z q z q z q z q z p )) ( ( )) ( ( is a pole of order (m=7) then by (shortcut -method) we get : We expand the analytic function q(z) into Taylor series valid in disk Then substituting the above values in (33) in equation (32) we get the desired proof .

3.6)
In the above manner the procedure can be easily extended for any pole of order (m) .

Solution:
the function f(z) has a pole Z=2i of order (m=6) .

Solution:
Now for this type of improper integrations we not can find the result by Cauchy integral formula.The only way to evaluate is the new procedure with general order (m).