Ch.4:Complex Integration Ch.4:Complex lntegration L4.2 Contour Mntegrals L4.2 Contour Integrals Riemann Sum Integral of a Complex Function f along a Directed Smooth Curve y(Cont'd) Partition Pn is a finite number of points {zo,21,...,n on y such that zo=a,zn =B Theorem Riemann sum for the function f corresponding to the If f is continuous on the directed smooth curve,then f is partition Pn: integrable along s(Pn):=f(g)(a-20)+f(c2)(22-z1）+…+fcn)(2n-2n-1) On writing Zk-Zk-1 =Azk,this becomes This theorem is of great theatrical importance,but is gives us no information of how to compute the integral f(z)dz s(Pn)=∑f(cx)(2k-2k-1)=∑fc)△2 =1 Since we are ready skilled in evaluating the definite integral of With the concept of Riemann Sum,we can generalize the calculus,it would certainly be advantageous if we could definition of definite integral given in calculus express the complex integral in terms of real integrals Ch.:Complex Integration Ch.4:Complex lategration L4.2 Contour Integrals L4.2 Contour Integrals Integral of a Complex Function f along a Directed Smooth Contour Integrals Along a Directed Smooth Curve Curve y First consider the special case when y is the real line segment Definition a,b directed from left to right Let f be a complex-valued function defined on the directed smooth Notice that if f happened to be a real-valued function defined curve We say that f is integrable along y if there exists a on [a,b.the definition of complex integral reduces to the complex number L that is the limit of every sequence of Riemann integralf(t)given in calculus sums S(P1).S(P2),....S(Pn)....corresponding to any sequence When f is a complex-valued function continuous on [a,b],we of partitions of y satisfying lim u(Pn)=0:i.e. can write f(t)=u(t)+iu(t).where u and v are each li S(P)=L whenever lim u(P)=0 real-valued and continuous on [a,b.then we have The constant L is called the integral of f along 7.and we write (2) L=s含oa=ke地=/ this expresses the complex integral in terms of two real integralsCh.4: Complex Integration 4.2 Contour Integrals Riemann Sum Partition Pn is a finite number of points {z0, z1,...,zn} on γ such that z0 = α, zn = β Riemann sum for the function f corresponding to the partition Pn: S(Pn) := f(c1)(z1−z0)+f(c2)(z2−z1)+···+f(cn)(zn−zn−1) On writing zk − zk−1 = zk, this becomes S(Pn) =  n k=1 f(ck)(zk − zk−1) =  n k=1 f(ck)zk With the concept of Riemann Sum, we can generalize the definition of definite integral given in calculus Ch.4: Complex Integration 4.2 Contour Integrals Integral of a Complex Function f along a Directed Smooth Curve γ Definition Let f be a complex-valued function defined on the directed smooth curve γ. We say that f is integrable along γ if there exists a complex number L that is the limit of every sequence of Riemann sums S(P1), S(P2), ..., S(Pn), ... corresponding to any sequence of partitions of γ satisfying lim n→∞ μ(Pn)=0; i.e. lim n→∞ S(Pn) = L whenever lim n→∞ μ(Pn)=0 The constant L is called the integral of f along γ, and we write L = lim n→∞  n k=1 f(ck)Δzk = γ f(z)dz = γ f Ch.4: Complex Integration 4.2 Contour Integrals Integral of a Complex Function f along a Directed Smooth Curve γ (Cont’d) Theorem If f is continuous on the directed smooth curve γ, then f is integrable along γ This theorem is of great theatrical importance, but is gives us no information of how to compute the integral γ f(z)dz Since we are ready skilled in evaluating the definite integral of calculus, it would certainly be advantageous if we could express the complex integral in terms of real integrals Ch.4: Complex Integration 4.2 Contour Integrals Contour Integrals Along a Directed Smooth Curve First consider the special case when γ is the real line segment [a, b] directed from left to right Notice that if f happened to be a real-valued function defined on [a, b], the definition of complex integral reduces to the integral  ba f(t) given in calculus When f is a complex-valued function continuous on [a, b], we can write f(t) = u(t) + iv(t), where u and v are each real-valued and continuous on [a, b], then we have  b a f(t)dt =  b a u(t)dt + i  b a v(t)dt (2) this expresses the complex integral in terms of two real integrals