Complex Integration  f(z)dz C Complex integrals are basically line integrals. The integrand function f(z) is integrated over a given curve C in the complex plane, called the path of integration. Such a curve is given by a parametric representation: z (t) = x(t) + i y(t), a  t  b. Assumptions: All paths of integration are assumed to be piece-wise smooth (i.e. continuous and with non-zero derivative dz/dt). The function f(z) is assumed to be continuous. The notation for a complex line integral is:  f(z)dz C

Definition and Existence The complex integral is defined as the limit of a sum in a way analogous to that used for defining line integrals in the real xy-plane. Partitioning the interval a  t  b, we get a subdivision of the arc C by points, and we can then take the value of f in the subintervals so formed. We thus get a sum of the form: Sn =  f (k) zk If we write f(z) = u(x,y) + iv(x,y), then continuity of f implies continuity of u and v. The above sum can be expressed as a sum in terms of u and v, which converges to a sum of real line integrals as the mesh of the partition goes to zero (under the assumptions). Hence the existence of the complex integral is assured under the given assumptions.

Complex Integrals - Calculation Definition: Let f(z) be a continuous function on a piece-wise smooth path C represented by z (t) = x(t) + i y(t), a  t  b. Then: b  f(z)dz =  f(z(t))z’(t) dt C a The above definition is motivated by the analogous rule for real line integrals. If f(z) = u(x,y) + iv(x,y), then the above can be re-stated as: b b  (udx – vdy) + i  (vdx + udy) a a

Properties of Complex Integrals Proposition 6: For integrable functions f(z) and g(z): Constant Multiples:  kf(z) dz = k f(z) dz for any complex constant k C C Sums and differences:  ( f(z)  g(z) )dz =  f(z) dz   g(z) dz C C C Additivity: if the arc C is the sum of the arcs A and B, then  f(z)dz +  f(z) dz =  f(z) dz A B C Reversal of Arc: If C is the arc obtained by traversing the arc C in the reverse direction, then:  f(z) dz =   f(z) dz C C

Bound for Absolute Value of Complex Integrals Proposition 7 (ML-Inequality): The absolute value of integral of f(z) over a path C is bounded by ML, where L is the length of C and M is a constant such that |f(z)|  M on C. NB: The upper bound M needs to apply on the path C only.

Indefinite Integral Proposition 8: Let f(z) be continuous in a simply connected domain D. If any one of the above statements is true, then so are the others: The integrals of f(z) along contours lying entirely in D and extending from any fixed point z0 to any fixed point z1 all have the same value (i.e. path independence) f(z) has an anti-derivative F(z) in D (i.e. there exists a function F(z) such that F’(z) = f(z) on the domain D), and for all paths in D joining two points z0 and z1, we have: z1  f(z) dz = F(z1 ) – F(z0) z0 The integrals of f(z) around closed contours lying entirely in D all have value zero. The above result is analogous to the Fundamental Theorem of Calculus. However, unlike in that case it does not necessarily apply to all continuous functions.

Complex Integrals  f(z) dz = 0 Theorem 2: Cauchy’s Integral Theorem: If f(z) is analytic in a simply connected domain D, then for every simple, closed path C in D:  f(z) dz = 0 C Corollary 1: Independence of Path: If f(z) is analytic in a simply connected domain D, then the integral of f(z) is independent of path in D. Corollary 2: If f(z) is analytic in a domain D, then it possesses an anti-derivative in the domain. Remark: Both of these follow from Proposition 8. The second corollary provides a simpler way of calculating integrals for analytic functions, i.e. we use the anti- derivative F(z) and the result that the integral = F(z1 ) – F(z0), instead of having to parametrize the curve and evaluate the integral using the definition of integral given earlier.