Ch.2:Analytic Functions Ch.2:Analytic Functions LOutline 2.1 Functions of a Complex Variable Chapter 2:Analytic Functions 2.2 Limits and Continuity Li,Yongzhao State Key Laboratory of Integrated Services Networks,Xidian University 2.3 Analyticity September 28,2010 2.4 The Cauchy-Riemann Equations +口·0+t。年之，220C 4日10。+之+1生，意0G Ch.2:Analytic Functions Ch.2:Analytic Functions 21 Functions of a Complex Variable -2.1 Functions of a Complex Variable Introduction Review of Functions of a Real Variable Ch.1 is focused on the algebraic operation of a complex number z If f assigns the value y to the element z in A,we write From this chapter,we shall study function f(z)defined on these complex variables y=f() Our objective is to mimic the concepts,theorems,and and call y the image of z under f mathematical structure of calculus;such as differentiating The set A is the domain of definition of f,and the set of all and integrating the function f(z) images f(r)is the range of f The notation of a derivative is far more subtle in the complex Sometimes we refer to f as a mapping of A into B case because of the intrinsically two-dimensional nature of the complex variables
Ch.2: Analytic Functions Chapter 2: Analytic Functions Li, Yongzhao State Key Laboratory of Integrated Services Networks, Xidian University September 28, 2010 Ch.2: Analytic Functions Outline 2.1 Functions of a Complex Variable 2.2 Limits and Continuity 2.3 Analyticity 2.4 The Cauchy-Riemann Equations Ch.2: Analytic Functions 2.1 Functions of a Complex Variable Introduction Ch. 1 is focused on the algebraic operation of a complex number z From this chapter, we shall study function f(z) defined on these complex variables Our objective is to mimic the concepts, theorems, and mathematical structure of calculus; such as differentiating and integrating the function f(z) The notation of a derivative is far more subtle in the complex case because of the intrinsically two-dimensional nature of the complex variables Ch.2: Analytic Functions 2.1 Functions of a Complex Variable Review of Functions of a Real Variable If f assigns the value y to the element x in A, we write y = f(x) and call y the image of x under f The set A is the domain of definition of f, and the set of all images f(x) is the range of f Sometimes we refer to f as a mapping of A into B
Ch.2:Analytic Functions Ch.2:Analytic Functions L2.1 Functions of a Complex Variabie L21 Functions of a Complex Variable Functions of a Complex Variable Functions of a Complex Variable (cont'd) Denote w as the value of the function f(z)at point z.Then Now we consider the complex-valued functions of a complex we write w=f(z) variable Just as z decomposes into real and imaginary parts as The domains of definition and the ranges are subset of the z=x+iy,the real and imaginary parts of w are each(real) complex numbers functions of z or,equivalently,of z and y.and so we See an example: customarily write 22-1 f()=2+1 w=u(r,y)+iv(I,y) where u and v denoting the real and imaginary parts, We take the domain of f to be the set of all z for which the respectively,of w formula is well defined (hence,ti are excluded) Thus a complex valued function of a complex variable is a pair functions of two real variables (Example 1 on page 54) Ch.2:Analytic Functions Ch.2:Analytic Functions 21 Functions of a Complex Variable L2.2 Limits and Continuity Functions of a Complex Variable(cont'd) Limit of a Sequence of Complex Numbers Unfortunately,it is generally impossible to draw the graph of a The definition of absolute value can be used to designate the complex function;to display two real functions of two real distance between two complex numbers variables graphically would require four dimensions Having a concept of distance,we can proceed to introduce Instead,we can visualize some of the properties of a complex the notions of limit and continuity function w=f(z)by sketching of domain of definition in the z-plane and its range in the u-plane (Examples 2 and 3) When we have an infinite sequences 21,22,23,..of complex numbers,we say that the number zo is the limit of the The function f(z)=1/z is called the inversion mapping.It is an example of a one-to-one function because it maps distinct sequence if the zn eventually (i.e.,for large enough n)stay arbitrarily close to zo points to distinct points,i.e.,if zz,then f(z)f(z2)
Ch.2: Analytic Functions 2.1 Functions of a Complex Variable Functions of a Complex Variable Now we consider the complex-valued functions of a complex variable The domains of definition and the ranges are subset of the complex numbers See an example: f(z) = z2 − 1 z2 + 1 We take the domain of f to be the set of all z for which the formula is well defined (hence, ±i are excluded) Ch.2: Analytic Functions 2.1 Functions of a Complex Variable Functions of a Complex Variable (cont’d) Denote w as the value of the function f(z) at point z. Then we write w = f(z) Just as z decomposes into real and imaginary parts as z = x + iy, the real and imaginary parts of w are each (real) functions of z or, equivalently, of x and y, and so we customarily write w = u(x, y) + iv(x, y) where u and v denoting the real and imaginary parts, respectively, of w Thus a complex valued function of a complex variable is a pair functions of two real variables (Example 1 on page 54) Ch.2: Analytic Functions 2.1 Functions of a Complex Variable Functions of a Complex Variable (cont’d) Unfortunately, it is generally impossible to draw the graph of a complex function; to display two real functions of two real variables graphically would require four dimensions Instead, we can visualize some of the properties of a complex function w = f(z) by sketching of domain of definition in the z-plane and its range in the w-plane (Examples 2 and 3) The function f(z)=1/z is called the inversion mapping. It is an example of a one-to-one function because it maps distinct points to distinct points, i.e., if z1 = z2, then f(zz) = f(z2) Ch.2: Analytic Functions 2.2 Limits and Continuity Limit of a Sequence of Complex Numbers The definition of absolute value can be used to designate the distance between two complex numbers Having a concept of distance, we can proceed to introduce the notions of limit and continuity When we have an infinite sequences z1, z2, z3, ··· of complex numbers, we say that the number z0 is the limit of the sequence if the zn eventually (i.e., for large enough n) stay arbitrarily close to z0
Ch.2:Analytic Functions Ch.2:Analytic Functions L2.2 Limits and Continuity L22 Limits and Continuity Definition of Limit of a Sequence of Complex Numbers Definition of Limit of a Complex-Valued Function Definition Definition A sequence of complex numbers {n is said to have the limit zo Let f be a function defined in some neighborhood of zo,with the or to converge to zo and we write possible exception of the point zo itself.We say that the limit of f(z)as z approaches zo is the number wo and write 细2n=0 n。f2)=w or equivalently, or equivalently, 2m+20a5n→00 f(z）→0asz→20 if for any >0,there exists an integer N such that zn-zo< for all nN (see Fig.2.3 on page 59) if for any >0,there exists a positive number 6 such that If(z)-wol E whenever 0<lz-zol<6 Ch.2:Analytic Functions Ch.2:Analytic Functions L2.2 Limits and Continuity L2.2 Limits and Continuity Relation between the limit of a function and the limit of a Condition of Continuity sequence Definition Let f be a function defined in a neighborhood of zo.Then f is If lim f(z)=wo.then for every sequence {n continuous at zo if converging to zo(zn zo)the sequence {f(zn)converges 1limf(z)=f（2o) to wo,and vice versa The definitions of this section are direct analogous of concepts introduced in elementary calculus.Hence,many of the In other words,for f to be continuous at zo.it must have a familiar theorems on real sequences,limits,and continuity limiting value at zo,and this limiting value must be f(zo) remain valid in the complex case A function f is said to be continuous on a set S if it is continuous at each point of S
Ch.2: Analytic Functions 2.2 Limits and Continuity Definition of Limit of a Sequence of Complex Numbers Definition A sequence of complex numbers {zn}∞1 is said to have the limit z0 or to converge to z0 and we write lim n→∞ zn = z0 or equivalently, zn → z0 as n → ∞ if for any ε > 0, there exists an integer N such that |zn − z0| < ε for all n>N (see Fig. 2.3 on page 59) Ch.2: Analytic Functions 2.2 Limits and Continuity Definition of Limit of a Complex-Valued Function Definition Let f be a function defined in some neighborhood of z0, with the possible exception of the point z0 itself. We say that the limit of f(z) as z approaches z0 is the number w0 and write lim z→ z0 f(z) = w0 or equivalently, f(z) → w0 as z → z0 if for any ε > 0, there exists a positive number δ such that |f(z) − w0| < ε whenever 0 < |z − z0| < δ Ch.2: Analytic Functions 2.2 Limits and Continuity Relation between the limit of a function and the limit of a sequence If limz→z0 f(z) = w0, then for every sequence {zn}∞1 converging to z0(zn = z0) the sequence {f(zn)}∞1 converges to w0, and vice versa The definitions of this section are direct analogous of concepts introduced in elementary calculus. Hence, many of the familiar theorems on real sequences, limits, and continuity remain valid in the complex case Ch.2: Analytic Functions 2.2 Limits and Continuity Condition of Continuity Definition Let f be a function defined in a neighborhood of z0. Then f is continuous at z0 if lim z→z0 f(z) = f(z0) In other words, for f to be continuous at z0, it must have a limiting value at z0, and this limiting value must be f(z0) A function f is said to be continuous on a set S if it is continuous at each point of S
Ch.2:Analytic Functions Ch.2:Analytic Functions L2.2 Limits and Continuity L22 Limits and Continuity Some Comments Limits Involving Infinity One can show that f(z)approaches a limit precisely when its real and imaginary parts approach limits We say "znoo"if,for each positive number M(no matter Theorems 1 and 2 on page 61 are also derived from the how large).there is an integer N such that n>M familiar theorems on real sequences whenever n>M It is easy to show that the constant function f(z)=z is Similarly."limo f(z)=oo"means that for each positive continuous on the whole plane C.Then,we can deduce that: number M(no matter how large),there is a >0 such that The polynomial function ao +az+a2z2+...+anz"is If(z)>M whenever 0<z-zol<6 continuous on the whole plane The rational function a0+a12+a222+..+an2n Essentially,we are saying that complex numbers approach is o+b12+a222+..+bm2m infinity when their magnitudes approach infinity continuous at each point where the denominator does not vanish Ch.2:Analytic Functions Ch.2:Analytic Functions L2.2 Limits and Continuity L2.3 Analyticity Distinction Between the Concepts of Real and Complex Introduction Cases The theory of analytic functions is the main topic of this course.Before we discuss this topic,we will give an informal There is an important distinction between the concepts of preview of what it is we want to achieve. limit in the (one-dimensional)real and(two-dimensional) In the real calculus,we don't deal with the function that looks complex case like 3+4v2. For latter situation,a sequence {n may approach a limit This is because wewe treat this number as an indivisible zo from any direction in the plane,or even along a spiral module and we don't ever perform separate algebraic Thus,the manner in which a sequence of numbers approaches operations on integer part and a different operation on the v. its limit can be much more complicated in the complex case part We seek to classify the complex functions that behave this same way with regard to their complex argument
Ch.2: Analytic Functions 2.2 Limits and Continuity Some Comments One can show that f(z) approaches a limit precisely when its real and imaginary parts approach limits Theorems 1 and 2 on page 61 are also derived from the familiar theorems on real sequences It is easy to show that the constant function f(z) = z is continuous on the whole plane C. Then, we can deduce that: The polynomial function a0 + a1z + a2z2 + ... + anzn is continuous on the whole plane The rational function a0 + a1z + a2z2 + ... + anzn b0 + b1z + a2z2 + ... + bmzm is continuous at each point where the denominator does not vanish Ch.2: Analytic Functions 2.2 Limits and Continuity Limits Involving Infinity We say ”zn → ∞” if, for each positive number M (no matter how large), there is an integer N such that |zn| > M whenever n>M Similarly,”limzn→z0 f(z) = ∞” means that for each positive number M (no matter how large), there is a δ > 0 such that |f(z)| > M whenever 0 < |z − z0| < δ Essentially, we are saying that complex numbers approach infinity when their magnitudes approach infinity Ch.2: Analytic Functions 2.2 Limits and Continuity Distinction Between the Concepts of Real and Complex Cases There is an important distinction between the concepts of limit in the (one-dimensional) real and (two-dimensional) complex case For latter situation, a sequence {zn}∞1 may approach a limit z0 from any direction in the plane, or even along a spiral Thus, the manner in which a sequence of numbers approaches its limit can be much more complicated in the complex case Ch.2: Analytic Functions 2.3 Analyticity Introduction The theory of analytic functions is the main topic of this course. Before we discuss this topic, we will give an informal preview of what it is we want to achieve. In the real calculus, we don’t deal with the function that looks like 3+4√2. This is because we we treat this number as an indivisible module and we don’t ever perform separate algebraic operations on integer part and a different operation on the √· part We seek to classify the complex functions that behave this same way with regard to their complex argument
Ch.2:Analytic Functions Ch.2:Analytic Functions L2.3 Analyticity L2.3 Analyticity Introduction(Cont'd) Introduction (Cont'd) We want to admit the functions such as:2.22,23,and 1/z. We treat the complex functions in the same way,i.e.,we treat and there basic arithmetic combinations(sums,products, the complex variable z as a single quantity and don't perform quotients,powers,and roots) different algebraic operations on the z and a different one on But we want to ban such functions as Rz=,Sz=y,and the y. 12-y2+i3ry Actually,we did in the same way in finding the limit of a Zis also banned because if we admit it we will open the gate complex sequence or function to =(z+)/2 and y=(z-)/2i For a complex variable z=x+iy,the complex function f(z) Similarly,admitting would be a mistake as well,since also has its real and imaginary parts u and v: =la2/2 f(z)=u+iv=u(x,y)+iv(I,y) The function e=is more vexing.We suspect it to be admissible but postpone the official verification until the next section Ch.2:Analytic Functions Ch.2:Analytic Functions L2.3 Analyticity 2.3 Analyticity Derivative of a Complex Function Derivative of a Complex Function(Cont'd) The catch here is that Az is a complex number,so it can In the following chapters,we will see that the criterion of approach zero in many different ways(from the right,from analyticity we are seeking can be expressed simply in terms of below,along a spiral,etc.):but the difference quotient must differentiability tend to a unique limit f(zo)independent of the manner in which△z+0 Definition Let f be a complex-valued function defined in a neighborhood of Example 2 on page 67-68 shows why analyticity disqualifies zo.Then the derivative of f at zo is given by Theorem 3 on page 69 is corresponding to the rules of elementary calculus 票o=fa)-+g-儿e It should be also noted that differentiability implies continuity. A安0 △z as in the real case provided this limit exists.(Such as f is said to be differentiable at We see then that for purpose of differentiation,polynomial 20) and rational functions in z can be treated as if z were a real variable
Ch.2: Analytic Functions 2.3 Analyticity Introduction (Cont’d) We treat the complex functions in the same way, i.e., we treat the complex variable z as a single quantity and don’t perform different algebraic operations on the x and a different one on the y. Actually, we did in the same way in finding the limit of a complex sequence or function For a complex variable z = x + iy, the complex function f(z) also has its real and imaginary parts u and v: f(z) = u + iv = u(x, y) + iv(x, y) Ch.2: Analytic Functions 2.3 Analyticity Introduction (Cont’d) We want to admit the functions such as: z, z2, z3, and 1/z, and there basic arithmetic combinations (sums, products, quotients, powers, and roots) But we want to ban such functions as z = x, z = y, and x2 − y2 + i3xy z is also banned because if we admit it we will open the gate to x = (z + z)/2 and y = (z − z)/2i Similarly, admitting |z| would be a mistake as well, since z = |z|2/z The function ez is more vexing. We suspect it to be admissible but postpone the official verification until the next section Ch.2: Analytic Functions 2.3 Analyticity Derivative of a Complex Function In the following chapters, we will see that the criterion of analyticity we are seeking can be expressed simply in terms of differentiability Definition Let f be a complex-valued function defined in a neighborhood of z0. Then the derivative of f at z0 is given by df dz (z0) ≡ f(z0) := lim z→0 f(z0 + z) − f(z0)) z provided this limit exists. (Such as f is said to be differentiable at z0) Ch.2: Analytic Functions 2.3 Analyticity Derivative of a Complex Function (Cont’d) The catch here is that z is a complex number, so it can approach zero in many different ways (from the right, from below, along a spiral, etc.); but the difference quotient must tend to a unique limit f(z0) independent of the manner in which z → 0 Example 2 on page 67-68 shows why analyticity disqualifies z Theorem 3 on page 69 is corresponding to the rules of elementary calculus It should be also noted that differentiability implies continuity, as in the real case We see then that for purpose of differentiation, polynomial and rational functions in z can be treated as if z were a real variable