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 Functions
Ch.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
Ch.4:Complex Integration Ch.4:Complex Integration L4.1 Contours L4.1 Contours LCurves Parametrization of a Curve Smooth Curves(Cont'd) To study the complex integration in a plane,the first problem is finding a mathematical explication of our intuitive concept of a curve in the ry-plane (or called z-plane) Definition Although most of the applications described in this book A point set y is called a smooth closed curve if it is the range of involve only two simple types of curves-line segments and some continuous function z =z(t).a <t<b,satisfying conditions arc of circles-it will be necessary for proving theorems to nail i and ii and the following: down the definition of more general curves (iii')z(t)is one-to-one on the half open a,b),but A curve y can be constituted by the points z(t)=x(t)+iy(t) z(b)=z(a)and 2'(b)=2'(a) over an interval of time a <t<b.Then the curve y is the range of z(t)as t varies between a and b In such a case,z(t)is called the parametrization of y Ch.4:Complex Integration Ch.4:Complex lntegration L4.1 Contours 4.1 Contours LCurves Smooth Curves Smooth Curves(Cont'd) The phrase is a smooth curve"means that y is either a Definition smooth arc or a smooth closed curve A point set y in the complex plane is said to be a smooth arc if it The conditions of the definition imply that smooth curve is the range of some continuous complex-valued function z=z(t). possesses a unique tangent at every point and the tangent a<t<b,that satisfies the following conditions: direction varies continuous along the curve.Consequently a (i)z(t)has a continuous derivative on [a,b) smooth curve has no corners or cusps (ii)z'(t)never vanishes on [a,b] To show that a set of points in the complex plane is a 2'(t)must exist (no corners) smooth curve,we have to exhibit a parametrization function '(t)is nonzero (no cusps) z(t)whose range is y,and is"admissible"in the sense that it meets the criteria of the definition (iii)z(t)is one-to-one on [a,b(no self-intersections) A given smooth curve will have many different admissible parameterizations,but we need produce only one admissible parametrization in order to show that a given curve is smooth
Ch.4: Complex Integration 4.1 Contours Curves Parametrization of a Curve To study the complex integration in a plane, the first problem is finding a mathematical explication of our intuitive concept of a curve in the xy-plane (or called z-plane) Although most of the applications described in this book involve only two simple types of curves – line segments and arc of circles – it will be necessary for proving theorems to nail down the definition of more general curves A curve γ can be constituted by the points z(t) = x(t) + iy(t) over an interval of time a ≤ t ≤ b. Then the curve γ is the range of z(t) as t varies between a and b In such a case, z(t) is called the parametrization of γ Ch.4: Complex Integration 4.1 Contours Curves Smooth Curves Definition A point set γ in the complex plane is said to be a smooth arc if it is the range of some continuous complex-valued function z = z(t), a ≤ t ≤ b, that satisfies the following conditions: (i) z(t) has a continuous derivative on [a, b] (ii) z(t) never vanishes on [a, b] z(t) must exist (no corners) z(t) is nonzero (no cusps) (iii) z(t) is one-to-one on [a, b] (no self-intersections) Ch.4: Complex Integration 4.1 Contours Curves Smooth Curves (Cont’d) Definition A point set γ is called a smooth closed curve if it is the range of some continuous function z = z(t), a ≤ t ≤ b, satisfying conditions i and ii and the following: (iii’) z(t) is one-to-one on the half open [a, b), but z(b) = z(a) and z(b) = z(a) Ch.4: Complex Integration 4.1 Contours Curves Smooth Curves (Cont’d) The phrase ”γ is a smooth curve” means that γ is either a smooth arc or a smooth closed curve The conditions of the definition imply that smooth curve possesses a unique tangent at every point and the tangent direction varies continuous along the curve. Consequently a smooth curve has no corners or cusps To show that a set of points γ in the complex plane is a smooth curve, we have to exhibit a parametrization function z(t) whose range is γ, and is ”admissible” in the sense that it meets the criteria of the definition A given smooth curve γ will have many different admissible parameterizations, but we need produce only one admissible parametrization in order to show that a given curve is smooth
Ch.4:Complex Integration Ch.4:Complex Integration L4.1 Contours L4.1 Contours LCurves Directed Smooth Arcs Concept of a Contour A smooth arc,together with a specific ordering of its points, The general curves are formed by joining directed smooth is called a directed smooth arc.The ordering can be curves together end-to-end;this allows self-intersection,cusps, indicated by an arrow and corners The point z(t1)will precede z(t2)whenever t1<t2.Since It will be convenient to include single isolated points as there are only two possible ordering,any admissible members of this class parametrization must fall into one the two categories, Definition according to the particular ordering it respects A contour I is either a single point zo or a finite sequence of If z=z(t),a<t<b,is an admissible parametrization directed smooth curves (1,72,...,Yn)such that the terminal consistent with one of the ordering,then point of y coincides with the initial point of+for each z =z(-t),-b<t <-a,always corresponds to the opposite =1,2,...,n-1.In this case one can write ordering T=m+2+.+m Ch.4:Complex Integration Ch.4:Complex lategration L4.1 Contours 4.1 Contours LCurves Contours Directed Smooth Arcs(Cont'd) Concept of a Contour(Cont'd) The theory of contour is easier to express in terms of contour The points of a smooth closed curve have been ordered when parameterizations (i)a designation of the initial point is made and (ii)one of the two "directions of transit"from this point is selected One can say that z=z(t),a<t<b,is a parametrization of the contour I=(1,72,...,n)if there is a subdivision of If this parametrization is given by z=z(t),a <t <b,then (i) [a,b]into n subintervals [ro,],[1,T2],...,[Tn-1,Tn],where the initial point must be z(a)and (ii)the point z(t1)precedes a=TO<TI<...Tn-1 Tn =b,such that on each the point z(t2)whenever a<ti<t2<b subinterval 1,T]the function z(t)is an admissible The phrase directed smooth curve will be used to mean either parametrization of the smooth curve y,consistent with the a directed smooth arc or a directed smooth closed curve direction on Next,we are ready to specify the more general kinds of curves Since the endpoints of consecutive 's are properly that will be used in the theory of integration connected,z(t)must be continuous on [a,b].However z'(t) may have jump discontinuities at the points y
Ch.4: Complex Integration 4.1 Contours Curves Directed Smooth Arcs A smooth arc, together with a specific ordering of its points, is called a directed smooth arc. The ordering can be indicated by an arrow The point z(t1) will precede z(t2) whenever t1 < t2. Since there are only two possible ordering, any admissible parametrization must fall into one the two categories, according to the particular ordering it respects If z = z(t), a ≤ t ≤ b, is an admissible parametrization consistent with one of the ordering, then z = z(−t), −b ≤ t ≤ −a, always corresponds to the opposite ordering Ch.4: Complex Integration 4.1 Contours Curves Directed Smooth Arcs (Cont’d) The points of a smooth closed curve have been ordered when (i) a designation of the initial point is made and (ii) one of the two ”directions of transit” from this point is selected If this parametrization is given by z = z(t), a ≤ t ≤ b, then (i) the initial point must be z(a) and (ii) the point z(t1) precedes the point z(t2) whenever a<t1 < t2 < b The phrase directed smooth curve will be used to mean either a directed smooth arc or a directed smooth closed curve Next, we are ready to specify the more general kinds of curves that will be used in the theory of integration Ch.4: Complex Integration 4.1 Contours Contours Concept of a Contour The general curves are formed by joining directed smooth curves together end-to-end; this allows self-intersection, cusps, and corners It will be convenient to include single isolated points as members of this class Definition A contour Γ is either a single point z0 or a finite sequence of directed smooth curves (γ1, γ2,...,γn) such that the terminal point of γk coincides with the initial point of γk+1 for each k = 1, 2,...,n − 1. In this case one can write Γ = γ1 + γ2 + ... + γn Ch.4: Complex Integration 4.1 Contours Contours Concept of a Contour (Cont’d) The theory of contour is easier to express in terms of contour parameterizations One can say that z = z(t), a ≤ t ≤ b, is a parametrization of the contour Γ=(γ1, γ2,...,γn) if there is a subdivision of [a, b] into n subintervals [τ0, τ1], [τ1, τ2],..., [τn−1, τn], where a = τ0 < τ1 <...< τn−1 < τn = b, such that on each subinterval [τk−1, τk] the function z(t) is an admissible parametrization of the smooth curve γk, consistent with the direction on γk Since the endpoints of consecutive γk’s are properly connected, z(t) must be continuous on [a, b]. However z(t) may have jump discontinuities at the points γk
Ch.4:Complex Integration Ch.4:Complex Integration L4.1 Contours L4.1 Contours LJordan Curve Theorem Parametrization of a Contour Jordan Curve Theorem When we have admissible parameterizations of the Theorem components y of a contour T.We can piece these together Any simple closed contour separates the plane into two domains, to get a contour parametrization for I by simply rescaling each having the curves as its boundary.One of these domains, and shifting the parameter intervals for t(Example 2 on page called the interior,is bounded;the other,called the exterior,is 156) unbounded The (undirected)point set underlying a contour is known as a piecewise smooth curve When the interior domain lies to the left,we say that I is We shall use the symbol I ambiguously to refer to both the positively oriented.Otherwise I is said to be oriented contour and its underlying curve,allowing the context to negatively. provide the proper interpretation A positive orientation generalizes the concept of The opposite contour is denoted by-T counterclockwise motion Ch.4:Complex Integration Ch.4:Complex lntegration L4.1 Contours 4.1 Contours LContours LThe Length of a Contou Closed Contour The Length of a Contour If one admissible parametrization for curve y is T is said to be a closed contour or a loop if its initial and z(t)=x(t)+iy(t),a <t <b,let s(t)be the length of the arc terminal points coincide of traversed in going from the point z(a)to the point z(b). A simple closed contour is closed contour with no multiple As shown in elementary calculus,we have points other than its initial-terminal point;in other words,if ds dr\ dz z=z(t),a<t<b,is a parametrization of the closed ()+()= contour,then z(t)is one-to-one on the half-open interval Consequently,the length of the smooth curve is given by a,b)(no self-intersections) the important integral formula There is an alternative way of specifying the direction along a curve if the curve happens to be a simple closed contour h）=lengh of=人= (1) 口
Ch.4: Complex Integration 4.1 Contours Contours Parametrization of a Contour When we have admissible parameterizations of the components γk of a contour Γ. We can piece these together to get a contour parametrization for Γ by simply rescaling and shifting the parameter intervals for t (Example 2 on page 156) The (undirected) point set underlying a contour is known as a piecewise smooth curve We shall use the symbol Γ ambiguously to refer to both the contour and its underlying curve, allowing the context to provide the proper interpretation The opposite contour is denoted by −Γ Ch.4: Complex Integration 4.1 Contours Contours Closed Contour Γ is said to be a closed contour or a loop if its initial and terminal points coincide A simple closed contour is closed contour with no multiple points other than its initial-terminal point; in other words, if z = z(t), a ≤ t ≤ b, is a parametrization of the closed contour, then z(t) is one-to-one on the half-open interval [a, b) (no self-intersections) There is an alternative way of specifying the direction along a curve if the curve happens to be a simple closed contour Ch.4: Complex Integration 4.1 Contours Jordan Curve Theorem Jordan Curve Theorem Theorem Any simple closed contour separates the plane into two domains, each having the curves as its boundary. One of these domains, called the interior, is bounded; the other, called the exterior, is unbounded When the interior domain lies to the left, we say that Γ is positively oriented. Otherwise Γ is said to be oriented negatively. A positive orientation generalizes the concept of counterclockwise motion Ch.4: Complex Integration 4.1 Contours The Length of a Contour The Length of a Contour If one admissible parametrization for curve γ is z(t) = x(t) + iy(t), a ≤ t ≤ b, let s(t) be the length of the arc of γ traversed in going from the point z(a) to the point z(b). As shown in elementary calculus, we have ds dt = dxdt 2 + dydt 2 = dzdt Consequently, the length of the smooth curve is given by the important integral formula l(γ) = length of γ = b a dsdt dt = b a dzdt dt (1)
Ch.4:Complex Integration Ch.4:Complex lntegration L4.1 Contours L4.2 Contour Integrals LThe Length of a Contour The Length of a Contour(Cont'd) Introduction (Cont'd) (is a geometric quantity that depends only on the point We will accomplish this by first defining the integral along a set y and is independent of the particular admissible single directed smooth curve and then defining integrals along parametrization used in the computation a contour in terms of the integrals along its smooth components The length of a contour is simply defined to be the sum of the length of its component curves Finally,we once again obtain simple rules for evaluating integrals in terms of antiderivatives +口·0+t。年之，220C 4日10。+之+1生，意0G Ch.4:Complex Integration Ch.4:Commplex lntegration L4.2 Contour Integrals L4.2 Contour Integrals Introduction Riemann Sum In calculus,the definite integral of a real-valued function f Ca over an interval [a,b]is defined as the limit of certain sums ∑k=1f(ck)△rk(called Riemann sums) However,the fundamental theorem of calculus lets us evaluate ”23 integrals more directly when an antiderivative is known The aim of this section is to use this notion of Riemann sums to define integral of a complex-valued function along a contour T in the z-plane Partitioned Curve +口·811定+1意1意00
Ch.4: Complex Integration 4.1 Contours The Length of a Contour The Length of a Contour (Cont’d) l(γ) is a geometric quantity that depends only on the point set γ and is independent of the particular admissible parametrization used in the computation The length of a contour is simply defined to be the sum of the length of its component curves Ch.4: Complex Integration 4.2 Contour Integrals Introduction In calculus, the definite integral of a real-valued function f over an interval [a, b] is defined as the limit of certain sums n k=1 f(ck)xk (called Riemann sums) However, the fundamental theorem of calculus lets us evaluate integrals more directly when an antiderivative is known The aim of this section is to use this notion of Riemann sums to define integral of a complex-valued function along a contour Γ in the z-plane Ch.4: Complex Integration 4.2 Contour Integrals Introduction (Cont’d) We will accomplish this by first defining the integral along a single directed smooth curve and then defining integrals along a contour in terms of the integrals along its smooth components Finally, we once again obtain simple rules for evaluating integrals in terms of antiderivatives Ch.4: Complex Integration 4.2 Contour Integrals Riemann Sum