Complex integration

UNIT - IV

COMPLEX INTEGRATION

Line integral - Cauchy's integral theorem - Cauchy's integral formula - Taylor's and Laurent's series - Singularities - Residues - Residue theorem Application of residue theorem for evaluation of real integrals - Use of circular contour and semicircular contour.

 Line integral - Cauchy's integral theorem - Cauchy's integral formula

In case of real variable, the path of integration of  is always along the x-axis, from x = a to x = b. But, in case of a complex function f(z) the path of the definite integral  can be along any curve from z = a to z = b. Its value depends upon the path (curve) of integration. But, the value of integral from a to b remains the same, if the different curves from a to b are regular curves.

CAUCHY'S THEOREM Statement and Application

(a) Connected Region :

A connected region is one in which any two points in it can be connected by a curve which lies entirely within the region.

(b) Simply connected region.

A curve which does not cross itself is called a simple closed curve.

A curve is called multiple curve, if it crosses itself.

A region in which every closed curve in it, encloses points of the region only is called a simply connected region.

i.e., A region which has no holes is called simply connected region. Otherwise, it is said to be multiply connected.

A multi connected region can be converted into simple connected region by taking one or more cuts.

Region between two concentric circles is an example of multi connected region.

Note (1): Cauchy's Integral theorem is applicable only for a simply connected region R enclosed by a simple curve C.

(c) Contour Integral

An integral along a simple closed curve is called a contour integral.

Note (2): In case of closed paths, the positive direction is anti-clockwise.

Note (3): If the direction of C is reversed (Clockwise), the integral changes its sign (i.e.,) 

Note (4): If the contour is divided into two parts i.e., C₁ and C₂, then

The evaluation of the line integral of the complex function is reduced to the evaluation of two line integrals of real functions.

Note (5): If C is a point on the arc joining a and b, then