Ch.4:Complex Integration Ch.4:Complex Integration L4.1 Contours Introduction The two-dimensional nature of the complex plane required us Chapter 4:Complex Integration to generalize our notion of a derivative because of the freedom of the variable to approach its limit along any of an infinite number of directions. Li,Yongzhao This two-dimensional aspect will have an effect on the theory of integration,necessitating the consideration of integrals State Key Laboratory of Integrated Services Networks,Xidian University along general curves in the plane not merely segments of the r-axis October 10,2010 Fortunately,such well-known techniques as using antiderivatives to evaluate integrals carry over to the complex case Ch.4:Complex Integration Ch.4:Complex lategration LOutline L4.1 Contours Introduction(Cont'd) 4.1 Contours Curves Contours Jordan Curve Theorem When the function under consideration is analytic the theory The Length of a Contour of integration becomes an instrument of profound significance in studying its behavior 4.2 Contour Integrals The main result is the theorem of Cauchy,which roughly 4.3 Independence of Path says that the integral of a function around a closed loop is zero if the function is analytic"inside and on"the loop 4.4 Cauchy's Integral Theorem Using this result,we shall derive the Cauchy integral formula,which explicitly displays many of the important 4.5 Cauchy's Integral Formula and Its Consequences properties of analytic function 4.6 Bounds for Analytic FunctionsCh.4: Complex Integration Chapter 4: Complex Integration Li, Yongzhao State Key Laboratory of Integrated Services Networks, Xidian University October 10, 2010 Ch.4: Complex Integration Outline 4.1 Contours Curves Contours Jordan Curve Theorem The Length of a Contour 4.2 Contour Integrals 4.3 Independence of Path 4.4 Cauchy’s Integral Theorem 4.5 Cauchy’s Integral Formula and Its Consequences 4.6 Bounds for Analytic Functions Ch.4: Complex Integration 4.1 Contours Introduction The two-dimensional nature of the complex plane required us to generalize our notion of a derivative because of the freedom of the variable to approach its limit along any of an infinite number of directions. This two-dimensional aspect will have an effect on the theory of integration, necessitating the consideration of integrals along general curves in the plane not merely segments of the x-axis Fortunately, such well-known techniques as using antiderivatives to evaluate integrals carry over to the complex case Ch.4: Complex Integration 4.1 Contours Introduction (Cont’d) When the function under consideration is analytic the theory of integration becomes an instrument of profound significance in studying its behavior The main result is the theorem of Cauchy, which roughly says that the integral of a function around a closed loop is zero if the function is analytic ”inside and on” the loop Using this result, we shall derive the Cauchy integral formula, which explicitly displays many of the important properties of analytic function