Integrals of Complex Functions - Complex Functions - University Mathematics Handbook

University Mathematics Handbook (2015)

XII. Complex Functions

Chapter 5. Integrals of Complex Functions

5.1  Line Integral in the Complex Plane

Let be a continuous function in domain . Suppose is a continuous curve entirely in , starting at and ending at . Let us define integral over directed from to :

when the right side integrals are real line integrals (see IX, 9.1).

5.2  Properties of the Integral

a.  , when is directed from to .

b.  

c.  , is a complex constant.

d.  , when is a curve in the opposite direction of , that is, from to .

e.  If curve consists of several curves , then

f.  If, along the line of , function is bounded, that is ( constant) and is the length of (see formula in IX.5), then

5.3  CauchyTheorem and its Applications

a.  If function is analytic in a simply-connected (multi-connected) closed and bounded domain with boundary , then

b.  If function is analytic in multi-connected closed domain , then the integral along the interior boundary paths equals the sum of integrals by the inner boundary paths of domain , when all boundary paths are directed counterclockwise.

c.  Let be analytic function in a simply-connected domain . Let and be points belonging to domain . Then, the integral is independent of the path connecting and , directed from to , which is entirely in .

d.  If function is analytic in simply-connected domain , then function , is also analytic in , and . is called the anti-derivative of .

e.  Newton-Leibniz formula: If is the anti derivative of , then

5.4  Cauchy Integral Formula and its Applications

a.  If function is analytic in simply-connected domain and is a continuous closed curve contained in , then, for every inner point z0 of there holds

b.  If function is analytic in closed domain with boundary , then it is infinitely differentiable at every point in and its -th derivative equals to

c.  Morera's theorem: If function is continuous on and for every closed path in , then is analytic in .

d.  Maximum (absolute value) module principle: If non-constant function is analytic in bounded domain and continuous in , then the module of attains its maximum value on the boundary of .

e.  Liouville’s theorem: If function is analytic over the whole (entire) plane and bounded over it, that is, there exists a positive number such that for every there holds , then is a constant function, that is, for every .

f.  Fundamental theorem of algebra: Every equation in the form of

with complex coefficients has at last one root, that is, there is a least one complex such that .