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
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 integrals
Ch.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
Ch.4:Complex Integration Ch.4:Complex lntegration L4.2 Contour lntegrals L4.2 Contour Integrals Contour Integrals Along a Directed Smooth Curve(Cont'd) Contour Integrals Along a Contour Theorem If the complex-valued function f is continuous on [a,b and Definition F'(t)=f(t)for all t in [a,b,then Suppose that I is a contour consisting of the directed smooth curves(,2.....),and let f be a function continuous on I. f(t)dt F(b)-F(a) Then the contour integral of f along I is denoted by the symbol f()dz and is defined by the equation Theorem 人fe=fed+ f)dz+..+ f(=)dz Let f be a function continuous on the directed smooth curve y. Then if z =z(t),a <t<b,is any admissible parametrization ofy If I consists of a single point,then for obvious reasons we set consistent with its direction,we have f(=)dz= f(z(t))z'(t)dt f(z)dz:=0 Ch.4:Complex Integration Ch.年Complex lategration L4.2 Contour Integrals L4.2 Contour Integrals Contour Integrals Along a Directed Smooth Curve(Cont'd) Contour Integrals Along a Contour(Cont'd) Since the integral of f along y was defined independently of If we have a parametrization z=z(t),a <t<b,for the whole any parametrization,we immediately deduce the following contour I=(1,72,...,n),we can get the following formula corollary rb f(z)dz f(z(t))z'(t)dt Corollary If f is continuous on the directed smooth curvey and if Using this formula it is not difficult to prove that integration z=2a(t),a≤t≤b,andz=22(t),c≤t≤d,are any two around simple closed contour is independent of the choice of admissible parameterizations ofy consistent with its direction,then the initial-terminal point In problems dealing with integrals along such contours,we n()sc-ex need only specify the direction of transit,not the starting point
Ch.4: Complex Integration 4.2 Contour Integrals Contour Integrals Along a Directed Smooth Curve (Cont’d) Theorem If the complex-valued function f is continuous on [a, b] and F(t) = f(t) for all t in [a, b], then b a f(t)dt = F(b) − F(a) Theorem Let f be a function continuous on the directed smooth curve γ. Then if z = z(t), a ≤ t ≤ b, is any admissible parametrization of γ consistent with its direction, we have γ f(z)dz = b a f(z(t))z(t)dt Ch.4: Complex Integration 4.2 Contour Integrals Contour Integrals Along a Directed Smooth Curve (Cont’d) Since the integral of f along γ was defined independently of any parametrization, we immediately deduce the following corollary Corollary If f is continuous on the directed smooth curve γ and if z = z1(t), a ≤ t ≤ b, and z = z2(t), c ≤ t ≤ d, are any two admissible parameterizations of γ consistent with its direction, then b a f(z1(t))z1(t)dt = dc f(z2(t))z2(t)dt Ch.4: Complex Integration 4.2 Contour Integrals Contour Integrals Along a Contour Definition Suppose that Γ is a contour consisting of the directed smooth curves (γ1, γ2,...,γn), and let f be a function continuous on Γ. Then the contour integral of f along Γ is denoted by the symbol Γ f(z)dz and is defined by the equation Γ f(z)dz := γ1 f(z)dz + γ2 f(z)dz + ... + γn f(z)dz If Γ consists of a single point, then for obvious reasons we set Γ f(z)dz := 0 Ch.4: Complex Integration 4.2 Contour Integrals Contour Integrals Along a Contour (Cont’d) If we have a parametrization z = z(t), a ≤ t ≤ b, for the whole contour Γ=(γ1, γ2,...,γn), we can get the following formula Γ f(z)dz = b a f(z(t))z(t)dt Using this formula it is not difficult to prove that integration around simple closed contour is independent of the choice of the initial-terminal point In problems dealing with integrals along such contours, we need only specify the direction of transit, not the starting point
Ch.4:Complex Integration Ch.4:Complex lntegration L4.2 Contour lntegrals L4.3 Independence of Path Upper Bound of the Magnitude of a Contour Integral Introduction Theorem One of the important results in the theory of complex analysis If f is continuous on the contour T and if f(z)<M for all z on is the extension of the Fundamental Theorem of Calculus to T,then contour integrals It implies that in certain situations,the integral of a function Rs An) is independent of the particular path joining the initial and terminal points,in fact,it completely characterize the where (T)denotes the length ofT.In particular,we have conditions under which this property holds In this section,we will explore this phenomenon in detail.We 人eydas解fel-o)四 will begin with the Fundamental Theorem,which enables us to evaluate integrals without introducing parameterizations, provided that an antiderivative of the integrand is known Ch.4:Complex Integration Ch.4:Complex lntegration L4.2 Contour Integrals 4.3 Independence of Path Comments Independence of Path Theorem Suppose that the function f(z)is continuous in a domain D and Although the real definite integral can be interpreted,among has an antiderivative F(z)throughout D:i.e.,dF(z)/dz =f(z) other things,as an area,no corresponding geometric for each a in D.Then for any contour I lying in D,with initial visualization is available for contour integrals point zr and terminal point zr,we have Nevertheless,the latter integrals are extremely useful in applied problems,as we shall see in subsequent chapters f(z)dz=F(2r）-F(2) Note that the conditions of the theorem imply that F(z)is analytic and hence continuous in D
Ch.4: Complex Integration 4.2 Contour Integrals Upper Bound of the Magnitude of a Contour Integral Theorem If f is continuous on the contour Γ and if |f(z)| ≤ M for all z on Γ, then Γ f(z)dz ≤ Ml(Γ) where l(Γ) denotes the length of Γ. In particular, we have Γ f(z)dz ≤ max z on Γ |f(z)| · l(Γ) Ch.4: Complex Integration 4.2 Contour Integrals Comments Although the real definite integral can be interpreted, among other things, as an area, no corresponding geometric visualization is available for contour integrals Nevertheless, the latter integrals are extremely useful in applied problems, as we shall see in subsequent chapters Ch.4: Complex Integration 4.3 Independence of Path Introduction One of the important results in the theory of complex analysis is the extension of the Fundamental Theorem of Calculus to contour integrals It implies that in certain situations, the integral of a function is independent of the particular path joining the initial and terminal points, in fact, it completely characterize the conditions under which this property holds In this section, we will explore this phenomenon in detail. We will begin with the Fundamental Theorem, which enables us to evaluate integrals without introducing parameterizations, provided that an antiderivative of the integrand is known Ch.4: Complex Integration 4.3 Independence of Path Independence of Path Theorem Suppose that the function f(z) is continuous in a domain D and has an antiderivative F(z) throughout D; i.e., dF(z)/dz = f(z) for each a in D. Then for any contour Γ lying in D, with initial point zI and terminal point zT , we have Γ f(z)dz = F(zT ) − F(zI ) Note that the conditions of the theorem imply that F(z) is analytic and hence continuous in D
Ch.4:Complex Integration Ch.4:Complex Integration L4.3 Independence of Path L4.4 Cauchy's Integral Theorem Independence of Path(Cont'd) Cauchy's Integral Theorem Since the endpoints of a loop,i.e.,a closed contour,are equal, Theorem we have the following immediate consequence of the theorem If f is analytic in a simple connected domain D and I is any loop Corollary (closed contour)in D,then If f is continuous in a domain D and has an antiderivative throughout D.then f(z)dz =0 for all loops I lying in D f(z)dz=0 Another important conclusion that can be drawn from the theorem is that when a function f has an antiderivative Theorem throughout a domain D,its integral along a contour in D In a simple connected domain,an analytic function has an depends only on the endpoints zr and zr:i.e..the integral is antiderivative,its contour integral are independent of path,and its independent of the path I joining these two points loop integrals vanish Ch.4:Complex Integration Ch.4:Complex lategration 4.3 Independence of Path 4.5 Cauchy's Integral Formula and Its Consequences Independence of Path(Cont'd) Introduction Theorem From Cauchy's theorem we know that,if f is analytic inside let f be continuous in a domain D.Then the following are and on the simple closed contour T.f(z)dz=0 equivalent: Now the question is how about the integral (i)f has antiderivative in D rf(z)/(z-z0)dz,where zo is a point in the interior of r (ii)Every loop integral of f in D vanishes i.e.,ifI is any loop in Obviously.there is no reason to expect that this integral is D.thenfrf(z)dz=] zero,because the integrand has a singularity inside the contour T (iii)The contour integral of f are independent of path in D [i.e., ifT and T2 are any two contours in D sharing the same In fact,as the primary result of this section,we shall show initial and terminal points,thenf()dz=f()dz] that for all zo inside I the value of the integral is proportional to f(zo)
Ch.4: Complex Integration 4.3 Independence of Path Independence of Path (Cont’d) Since the endpoints of a loop, i.e., a closed contour, are equal, we have the following immediate consequence of the theorem Corollary If f is continuous in a domain D and has an antiderivative throughout D, then Γ f(z)dz = 0 for all loops Γ lying in D Another important conclusion that can be drawn from the theorem is that when a function f has an antiderivative throughout a domain D, its integral along a contour in D depends only on the endpoints zI and zT ; i.e., the integral is independent of the path Γ joining these two points Ch.4: Complex Integration 4.3 Independence of Path Independence of Path (Cont’d) Theorem let f be continuous in a domain D. Then the following are equivalent: (i) f has antiderivative in D (ii) Every loop integral of f in D vanishes [i.e., if Γ is any loop in D, then Γ f(z)dz = 0] (iii) The contour integral of f are independent of path in D [i.e., if Γ1 and Γ2 are any two contours in D sharing the same initial and terminal points, then Γ1 f(z)dz = Γ2 f(z)dz ] Ch.4: Complex Integration 4.4 Cauchy’s Integral Theorem Cauchy’s Integral Theorem Theorem If f is analytic in a simple connected domain D and Γ is any loop (closed contour) in D, then Γ f(z)dz = 0 Theorem In a simple connected domain, an analytic function has an antiderivative, its contour integral are independent of path, and its loop integrals vanish Ch.4: Complex Integration 4.5 Cauchy’s Integral Formula and Its Consequences Introduction From Cauchy’s theorem we know that, if f is analytic inside and on the simple closed contour Γ, Γ f(z)dz = 0 Now the question is how about the integral Γ f(z)/(z − z0)dz, where z0 is a point in the interior of Γ Obviously, there is no reason to expect that this integral is zero, because the integrand has a singularity inside the contour Γ In fact, as the primary result of this section, we shall show that for all z0 inside Γ the value of the integral is proportional to f(z0)
Ch.4:Complex Integration Ch.4:Complex Integration L4.5 Cauchy's Integral Formula and Its Coesequences L45 Cauchy's Integral Formula and Its Consequences Cauchy's Integral Formula Cauchy's Integral Formula (Cont'd) Theorem let T be a simple closed positively oriented contour.If f is analytic Theorem in some simple connected domain D containing T and zo is any If f is continuous in a domain D and if point inside T,then e地=0 1 f(2o)= f()dz 2miJ外2-20 for every closed contour I in D,then f is analytic in D Theorem One remarkable consequence of Cauchy's formula is that by If f is analytic inside and on the simple closed oriented contour D merely knowing the values of the analytic function f on I we and if z is any point inside T,then can compute the above integral and hence all the values of f inside I.In other words,the behavior of a function analytic in fm-(o)=-m-业 f f(a) (2-0m4 a region is completely determined by its behavior on the 2πi boundary Ch.4:Complex Integration Ch.4:Complex lntegration 4.5 Cauchy's lntegral Formula and Its Co 4.6 Bounds for Analytic Functions Cauchy's Integral Formula(Cont'd) Introduction Theorem Many interesting facts about analytic functions are uncovered If f is analytic in a domain D,then all its derivatives f',f",.... when one considers upper bounds on their moduli f(m),...exist and are analytic in D We already have one result in this direction,namely,the integral estimate Theorem 5 of Sec 4.2 Theorem When this is judiciously applied to the Cauchy integral If f=u+iv is analytic in a domain D,then all partial derivatives formulas we obtain the Cauchy estimates for the derivatives of of u and v exist and are analytic in D an analytic function
Ch.4: Complex Integration 4.5 Cauchy’s Integral Formula and Its Consequences Cauchy’s Integral Formula Theorem let Γ be a simple closed positively oriented contour. If f is analytic in some simple connected domain D containing Γ and z0 is any point inside Γ, then f(z0) = 12πi Γ f(z) z − z0 dz One remarkable consequence of Cauchy’s formula is that by merely knowing the values of the analytic function f on Γ we can compute the above integral and hence all the values of f inside Γ. In other words, the behavior of a function analytic in a region is completely determined by its behavior on the boundary Ch.4: Complex Integration 4.5 Cauchy’s Integral Formula and Its Consequences Cauchy’s Integral Formula (Cont’d) Theorem If f is analytic in a domain D, then all its derivatives f, f, ..., f(n), ... exist and are analytic in D Theorem If f = u + iv is analytic in a domain D, then all partial derivatives of u and v exist and are analytic in D Ch.4: Complex Integration 4.5 Cauchy’s Integral Formula and Its Consequences Cauchy’s Integral Formula (Cont’d) Theorem If f is continuous in a domain D and if Γ f(z)dz = 0 for every closed contour Γ in D, then f is analytic in D Theorem If f is analytic inside and on the simple closed oriented contour Γ and if z is any point inside Γ, then f(m−1)(z0) = (m − 1)! 2πi Γ f(z) (z − z0)m dz Ch.4: Complex Integration 4.6 Bounds for Analytic Functions Introduction Many interesting facts about analytic functions are uncovered when one considers upper bounds on their moduli We already have one result in this direction, namely, the integral estimate Theorem 5 of Sec 4.2 When this is judiciously applied to the Cauchy integral formulas we obtain the Cauchy estimates for the derivatives of an analytic function