Chapter2复变函数与积分变换(英文版)课件

1、Chapter 2 Analytic Functions2.1 The concept of the analytic functions1. Derivative of complex functions Def 1. Let where is a domain.Then is said to be differentiable in the complex sense at if (1)exists. This limit is denoted by , or sometimes by . called the derivative of at . Thus is a complex nu

2、mber. 第1页，共45页。 By expressing the variable in definition (1) in terms of the new complex variable we can write that definition as (2)or (3) EX. 1. Suppose that . At any point , ,since is a polynomial in . Hence , or .第2页，共45页。 EX. 2. Show that is not differentiable on . Solution： The limit does not

3、exist.Figure. 2.1第3页，共45页。EX. 3. Consider not the function . Here We conclude that exists only an , its value there being 0.If exists, then is continuous at .第4页，共45页。iv. Any polynomial is differentiable on with derivative . iii. If for all , then is differentiable at and Suppose that and are differ

4、entiable at . Then i. is differentiable at and for any complex numbers and .ii. is differentiable at and第5页，共45页。 v. (Chain Rule) Suppose that has a derivative at and that has a derivative at the point . Then the function has a derivative at , and .2. The concept of an analytic function A function o

5、f the complex variable is analytic in a domain if it has a derivative at each point in . In particular, is analytic at a point if it is analytic throughout some neighborhood of .第6页，共45页。 is analytic at each nonzero point in the finite plane. But is not analytic at any point since its derivative exi

6、sts only at and not throughout any neighborhood. If a function fails to be analytic at a point but is analytic at some point in every neighborhood of , then is called a singular point, or singularity, of . The point is a singular point of the function . The function , has no singular points since it

7、 is nowhere analytic.第7页，共45页。2.2 A necessary and sufficient condition for differentiability EX1. Consider now the function Solution: So is not differentiable in . But their partial derivatives exist and are continuous. 第8页，共45页。 Let in a domain , if is differentiable at , then Let us take the speci

8、al case that . Then As , the left side of the equation converges to the limit . 第9页，共45页。 Thus both real and imaginary parts of the right side must converge to a limit. From the definition of the partial derivatives, this limit is .Thus . Next let . Then we similarly have 第10页，共45页。As , we get Thus,

9、 since exists and has the same value regardless of how approaching , we get By comparing real and imaginary parts of these equations, we drive and called the Cauchy Riemann equations.第11页，共45页。Let Here 第12页，共45页。 Expanding the right side of the equation and using the C.-R equations, and by comparing

10、 real and imaginary parts of it, we derive thatSince and we derive that and are differentiable at .第13页，共45页。Next we will prove that they are also sufficient.Let and are differentiable at .We havehere So that第14页，共45页。i.e. .So we have Theorem 2.2.1 Let on a domain , is differentiable at Bath and are

11、 differentiable at and satisfy that and at 第15页，共45页。 Theorem 2.2.2 Let is analytic on Both and are differentiable on and satisfy that and on . Thus, if , , and exist, are continuous on , and satisfy the C.-R equations, then is analytic on . Corollary: If does exist, then 第16页，共45页。 EX. 1. Show that

12、 satisfies the C.-R equations but is not differentiable at .Solution: Then So satisfies the C.-R equations. But 第17页，共45页。 EX. 2. Determine whether is analytic on . Solution: So They are continuous on but they satisfy the C.-R equations only at . So that is differentiable at , is not analytic on .第1

13、8页，共45页。 We can also express the C.-R equations in terms of polar coordinates, but care must be exercised because the change of coordinates defined by and is differentiable change only if is restricted to the open interval or any other open interval of length and if the origin is omitted. Without su

14、ch a restriction is discontinuous because it jumps by on crossing the x-axis. 第19页，共45页。Solution EX.3 Using , , we easily see that the Cauchy-Riemann equations are equivalent to saying that 第20页，共45页。 EX. 4. Define the symbol . Show that the C.-R equations are equivalent to (It is sometimes said, be

15、cause of this result, that analytic functions are not functions of but of alone. This statement should be taken only as a rough guide. Since is not really a derivative of with to but merely a shorthand notation for ).called complex derivative.第21页，共45页。Solution: EX. 5. Let is analytic on the domain

16、, and , Then is a constant on .Proof: So that both and are constant .第22页，共45页。2.4 Elementary Functions The trigonometric functions sine and cosine, as well as the exponential function and the logarithmic function, are covered in elementary calculus. Let us recall that the trigonometric functions ar

17、e definded in terms of the ratios of sides of a right-angled triangle. The definition of “angle may be extended to include any real value, and thus and become real-valued functions of the real variable . It is a basic mathematical fact that and are differentiable with derivatives given by and . Alte

18、rnatively, and can be defined by their power series: 第23页，共45页。 Convergence must also be proved : such a proof can be found in any calculus text. Alternatively, can be defined as the unique solution to the differential equation , and satisfying ;and can be defined as the unique solution to . The exp

19、onential function, denoted , may be defined as the unique solution to the differential equation , subject to the initial condition that . The exponential function, can also be defined by its power series:第24页，共45页。 In this section these functions will be extended to the complex plane. The extension

20、should be natural in that the familiar properties of , , and are retained.1. Exponential Function We know from elementary calculus that for real , can be represented by its Maclaurin series: Thus it would be most natural to define by for 第25页，共45页。of course, this definition is not quite legitimate a

21、s convergence of series in has not yet been discussed. Chapter 4 will show that this series does indeed represent a well-defined complex number for each given , but for the moment the series is used informally as the basis for the definition that follows, which will be precise. A slight rearrangemen

22、t of the series shows that But we recognize this as being simply . So we define 第26页，共45页。 If , we define Note that if is real (that is, if ), this definition agrees with the usual exponential function .Some of the important properties of : 1) for all . Let and . Then, by our definition of ,第27页，共45

23、页。 3) is periodic; the period for is . Suppose that for all . Setting we get . If . Hence any period is of the form , . Suppose that , that is . Then , and so for some integer . 2) is never zero For any , we have since we know that the usual exponential satisfies . Thus can never be zero, because if

24、 it were, then would be zero, which is not true.第28页，共45页。 4) is analytic on and . By definition, , So , and Thus and , so the C.-R equations hold, and and are continuous hence is analytic. Also, since 第29页，共45页。 5) Let denote the set of complex numberssuch that ; symbolically, Then maps in a one-to

25、-one manner onto the set . In fact, If , then , and so for some integer . But because and both lie in where the difference between the imaginary parts of any points is less that , we have . This argument shows that is one-to-one. Let with , We claim the equation has a solution in . The equation 第30页

26、，共45页。 is then equivalent to the two equations and , , . This is merely .Figure. 2.2 Particular, is one-to-one manner onto the set .第31页，共45页。2. The trigonometric functions Using Euler formulation We get and Since is now defined for any , we are led to formulate the following definition: for any com

27、plex number .Figure. 2.3第32页，共45页。 Again if is real, these definitions agree with the usual definitions of sine and cosine. By the definitions, we have(1) and are analytic on (2) and is periodic; the period is .(3) is an even function; and is an odd function.(4) Their formulas hold on 第33页，共45页。(5)

28、As As (6) and do not hold for some . EX. Note that in exponential form, the polar representation of a complex number becomes ,which is sometimes abbreviated to . EX. 1. Find the real and imaginary parts of 第34页，共45页。 Solution: Let ; then . Thus . Hence and .3. The logarithmic function The logarithmi

29、c function is the inverse function of the exponential function. . satisfies , is called the logarithmic of , noted . Because is periodic, must be multiple valued.第35页，共45页。Let So We derive that and So i.e. Where is the argument of . is its principal value. If we fix a , we will get a branch. We chos

30、e the principal branch as follow第36页，共45页。 is analytic on and4. Power function Let and , we define “ raised to the power ” . has distinct values, these values differ by factors of the form .So if , we haveIt shows has infinitely values.第37页，共45页。 If . 1) As is single valued. 2) As is in its lowest t

31、erms As it has distinct values, then it has exactly distinct values. 3) As is irrational. If (is irrational). It has infinity values.第38页，共45页。 If is an integer we know that is entire (with derivative ) namely is analytic on . But in general, is analytic only on the domain of . Namely. EX. 1. 第39页，共45页。Find all th