Residues of Complex Functions with Definite and Infinite Poles on X-axis

Abstract: Problem statement: One of the most popular areas in the mathematics is the computational complex analysis. In this study several computation l complex techniques were investigated and implemented numerically. Objective: This study produced new procedures to compute the residues of complex functions by changing their numerator from a constant number to either even or odd function. Approach: In this project we studied the functions that had f inite and infinite poles Z i, i greater than one of order greater or equal one, also we found ne w relation between residues at the poles Z i and residues at the poles -Z i, i greater than one and we had used these relation s to solve improper integrals of this type. The project needed the knowledge of c omputing the complex improper integrations. Results: Our numerical results in computing the residues fo r improper integrals of definite and infinite poles on the x-axis were well defined. Conclusion: In this study, we had concluded that the residues of the complex functions had definite and infinite pol es of higher order with constant numerator. A general form of residues of these functions of high orders were also investigated.


INTRODUCTION
The residue theorem is one of the main results of complex analysis. It includes Cauchy's theorem and Cauchy's integrals formula as special cases and leads quickly to important applications. In particular it becomes one of the most powerful tools of analysis for evaluation of definite and infinite integrals [3] .
If ℜ is the real field and C is the Complex field then consider the following Definitions: A function f (z) is said to be analytic in a domain D if it has a derivative at every point in the same domain D [4] . A point at which f(z) fails to be analytic is called a singular point of the function f(z) or singularities of the function f(z).There are three type of singular point, Isolated, Removable and Essential [2] .
If we can find a positive integer number (m) such that ( ) 0 m z z 0 lim (z z ) f (z) 0 → − ≠ then (z = z 0 ) is called a pole of order (m), as special case if (m = 1) is called simple pole [2] . 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 expansion: where, (Z = a+h) is called Taylor's series [5] .

MATERIALS AND METHODS
Lemma 1: Let (z 0 ) be a pole of order (m) of a function f(z) then the residues of the functions were given by the formula: This was called the (Short-Cut-Method) [3] .
Because Z = −Z 0 is harmonic conjugate of Z = z 0 and there are roots of the function q(Z) therefore: Then we get: Also If Z 0 is a pole of order m = 2 then the residue is given by the form: Then we evaluate the residue at a pole Z = −Z 0 by (Short-Cut-Method): We expanded the analytic function q(Z) in the bounded region 0 Z Z r + < by Taylor series expansion: Then substituting (9) into (8) we get: Then by deriving the function (9) for (Z) we get: Because p(Z) is constant .Substituting (13) into (12) yields: Then from Eq. 4 and 14 we have: Also If Z = -Z 0 is a pole of order (m = 3) then the residue at a pole (Z = Z 0 ) is given by the following formula: then by (short-cutmethod) we find the residue at a pole Z = −Z 0 of order (m = 3): We expand the analytic function q(Z) by Taylor's series valid in the disc 0 Z Z r + < : Where: Substituting (19) in Eq. 18 we get the following formula: Where: Then substituting the value of (23) into (22) we get the following formula: Because q (Z) is even with order (m = 3) then: In the above manner the procedure can be easily extended for any pole of order (m) [1] .
Then by the same way, we can generalize the procedure for any high order poles (m>3).  .
By addition (25) we get: Then: Then the relation is true for all (m>1).

RESULTS
For computing the residues for improper functions of definite poles on x-axis let us consider the following example.
Example 1: Evaluate the following integral: where, k is constant.

Solution:
We know that:
Example 2: Evaluate the following integral: where, k is a constant.
Solution: By C.P.V: On the real axis (x-axis).
By new relation we get:

DISCUSSION
Since there are a finite number of poles of that lie in the upper half plane, a real numbers can be found such that the poles all lie inside the contour C, which consists of the segment -R < x < R of the x axis together with the upper semicircle of radius R. In this study we have investigated and evaluated definite and indefinite integrals of higher order poles with some computationally complex techniques with an efficient numerical results.

CONCLUSION
In this study, we have concluded that the residues of complex functions which had definite and infinite poles of higher order with constant numerator and we have find a general form of these residues for functions when we have used these facts to evaluate improper integrals. Also we can able to change numerator of these complex functions from a constant number to either even or odd function.