International Science and Technology Journal

A complex variable is a quantity that contains both a real part and an imaginary part. Complex analysis is an application of mathematics that analyses the features of functions of complex variables. Physics, engineering, and computer science are just a few of the scientific disciplines that benefit from complex analysis. In order to tackle issues that are challenging or impossible to resolve using only real variables, complex analysis is crucial. ( Chanyu Xie، 2024) This research provides a definition of the most important theories in complex analysis, which is called the residue theory، which has many applications and uses in calculating path integrals because the residue theory can only be applied to path integrals, and therefore the real integral must be converted to a complex integral so that we can benefit from the theory. The residue theorem can be applied to many-connected curves as well as to an infinite number of isolated anomalous points, It can also be applied to the defective integral ، as it is called the Cauchy basis value، provided that the limit is on the right side of the integer . As well as the irregular integrals involving trigonometric functions, the residue theory is useful in calculating these integrals in the form , the defective integral is in the form where two real polynomials do not have common factors and do not have real zeros. The value of the integral can be easily calculated by finding the basic Cauchy value of the integral using the residue theorem. And the integrals of multi-valued functions when using the residue theorem، we must choose a region within which the integration function is single-valued، and also apply the residue theorem to calculate real integrals that include bifurcation points and secant branches and are in the form ، and this integration is of special importance in studying a function Gamma. I recommend completing this research to reach new applications of this theory due to its importance in the fields of mathematics and other sciences, as well as studying its applications to semi-analytical functions...................... Key words:.......... The Residue Theorem, Defective real integrals, Complex variable, Cauchy basis value, Lorant's Series.