西安电子科技大学：《Complex Analysis》课程教学资源（讲义）Chapter 2 Analytic Functions.pdf_大学文库

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 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

Ch.2: Analytic Functions 2.3 Analyticity Definition of Analyticity Definition A complex-valued function f(z) is said to be analyticity on an open set G if it has a derivative at every point of G Here we emphasize that analyticity is a property defined over open sets, while differentiability could conceivably hold at one point only A point where f is not analytic but which is the limit of points where f is analytic is known as a singular point or singularity If f(z) is analytic on the whole plane, then it is said to be entire Ch.2: Analytic Functions 2.3 Analyticity Comments to Analyticity As we will see in the next few chapters, analyticity is the criterion that we have been seeking, for functions to respect the complex structure of the variable z We will demonstrate later that all analytic fucntions can be written in terms of z alone (not x, y, or z) When a function is given in terms of real and imaginary parts as u(x, y) + iv(x, y), it may be very tedious to apply the definition to determine if f is analytic The next section will provide a test that is easier to use Ch.2: Analytic Functions 2.4 The Cauchy-Riemann Equations The Relationship Between u(x, y) and v(x, y) The property of analyticity for a function indicates some type of connection between its real and imaginary parts If f(z) = u(x, y) + iv(x, y) is differentiable at z0 = x0 + iy0, then the limit f(z0) = lim z→0 f(z0 + z) − f(z0) z can be computed by allowing z = x + i y to approach zero from any convenient direction in the complex plane Ch.2: Analytic Functions 2.4 The Cauchy-Riemann Equations The Relationship Between u(x, y) and v(x, y) (Cont’d) If it approaches horizontally, then z = x, we obtain f(z0) = lim x→0 u(x0+x,y0)+iv(x0+x,y0)−u(x0,y0)−iv(x0,y0)) x = lim x→0 u(x0+x,y0)−u(x0,y0) x + i lim x→0 v(x0+x,y0)−v(x0,y0) x Since the limits of the bracketed expressions are just the first partial derivatives of u and v with respect to x, we deduce that f(z0) = ∂u∂x(x0, y0) + i ∂v∂x(x0, y0) (1)