1. 第1頁第1頁 Chapter 2 Analytic Function 9. Functions of a complex variable Let S be a set of complex numbers. A function defined on S is a rule that assigns to each z in S a complex number w. or S is the domain of definition of sometimes refer to the function itself, for simplicity. Both a domain of definition and a rule are needed in order for a function to be well defined.

2. 第2頁第2頁 Suppose is the value of a function at real-valued functions of real variables x, y or Ex. when v=0 is a real-valued function of a complex variable.

3. 第3頁第3頁 is a polynomial of degree n. : rational function, defined when For multiple-valued functions : usually assign one to get single-valued function Ex.

4. 第4頁第4頁 10. Mappings w= f (z) is not easy to graph as real functions are. One can display some information about the function by indicating pairs of corresponding points z=(x,y) and w=(u,v). (draw z and w planes separately). When a function f is thought of in this way. it is often refried to as a mapping, or transformation. zw inverse image of w image of z image of T T

5. 第5頁第5頁 Mapping can be translation, rotation, reflection. In such cases it is convenient to consider z and w planes to be the same. w=z+1translation+1 w=iz rotation w= reflection in real axis. Ex. image of curves y x real number a hyperbola is mapped in a one to one manner onto the line

6. 第6頁第6頁 2xy=C 2 Y X u=C 1 >1 V U V=C 2 >0 right hand branch x>0, u=C 1, left hand branch x<0 u=C 1, image

7. 第7頁第7頁 x y D B A 1 L1L1 L2L2 C x1x1 Ex 2. u v A’ L2’L2’ L1’L1’ B’ D’C’ When, point moves up a vertical half line, L 1, as y increases from y = 0. a parabola with vertex at half line CD is mapped of half line C’D’

8. 第8頁第8頁 Ex 3. x y u v one to one

9. 第9頁第9頁 11. Limits Let a function f be defined at all points z in some deleted neighborhood of z 0 means: the limit of as z approaches z 0 is w 0 can be made arbitrarily close to w 0 if we choose the point z close enough to z 0 but distinct from it. (1) means that, for each positive number,there is a positive number such that x y z z0z0 u v w w w0w0 0 (2)

10. 第 10 頁 Note: (2) requires that f be defined at all points in some deleted neighborhood of z 0 such a deleted neighborhood always exists when z 0 is an interior point of a region on which is defined. We can extend the definition of limit to the case in which z 0 is a boundary point of the region by agreeing that left of (2) be satisfied by only those points z that lie in both the region and the domain Example 1. show if

11. 第 11 頁 For any such z and any positive number whenever z y x v u

12. 第 12 頁 Then Let But Hence is a nonnegative constant, and can be chosen arbitrarily small. When a limit of a function exists at a point, it is unique. If not, suppose, and

13. 第 13 頁 Ex 2. If (4) then does not exist. show: since a limit is unique, limit of (4) does not exist. (2) provides a means of testing whether a given point w 0 is a limit, it does not directly provide a method for determining that limit.

14. 第 14 頁 12. Theorems on limits Thm 1. Suppose that Theniff pf : ” ”

15. 第 15 頁 since and “ ” But and

16. 第 16 頁 Thm 2. suppose that pf: utilize Thm 1. for (9). use Thm 1. and (7)

17. 第 17 頁 have the limits An immediate consequence of Thm. 1: by property (9) and math induction. (11) 利用 whenever

18. 第 18 頁 13. Limits involving the point at Infinity It is sometime convenient to include with the complex plane the point at infinity, denoted by, and to use limits involving it. Complex plane + infinity = extended complex plane. N P zO complex plane passing thru the equator of a unit sphere. To each point z in the plane there corresponds exactly one point P on the surface of the sphere. intersection of the line z-N with the surface. north pole To each point P on the surface of the sphere, other than the north pole N, there corresponds exactly one point z in the plane.

19. 第 19 頁 By letting the point N of the sphere correspond to the point at infinity, we obtain a one-to-one correspondence between the points of the sphere and the points of the extended complex plane. upper sphere exterior of unit circle points on the sphere close to N neighborhood of

20. 第 20 頁 Ex 2.

21. 第 21 頁 Ex 3.

22. 第 22 頁 14. Continuity A function f is continuous at a point z 0 if (1) (2) (3) if continuous at z 0, then are also continuous at z 0. ((3) implies (1)(2)) So is

23. 第 23 頁 A polynomial is continuous in the entire plane because of (11), section 12. p.37 A composition of continuous function is continuous. zw f g If a function f(z) is continuous and non zero at a point z 0, then throughout some neighborhood of that point. when let if there is a point z in the at which then, a contradiction.

24. 第 24 頁 Ex. The function is continuous everywhere in the complex plane since (i) are continuous (polynomial) (ii)cos, sin, cosh, sinh are continuous (iii) real and imaginary component are continuous complex function is continuous. From Thm 1., sec12. a function f of a complex variable is continuous at a point iff its component functions u and v are continuous there.

25. 第 25 頁 15. Derivatives Let f be a function whose domain of definition contain a neighborhood of a point z 0. The derivative of f at z 0, written, is provided this limit exists. f is said to be differentiable at z 0.

26. 第 26 頁 Ex1. Suppose at any point z since a polynomial in. Ex2. when thru on the real axis, Hence if the limit of exists, its value = when thru on the imaginary axis., limit = if it exists.

27. 第 27 頁 since limits are unique,, if is to exist. observe thatwhen exists only at, its value = 0 Example 2 shows that a function can be differentiable at a certain point but nowhere else in any neighborhood of that point. Reare continuous, partially Im differentiable at a point. but may not be differentiable there.

28. 第 28 頁 existence of derivative continuity. continuity derivative exists. not necessarily is continuous at each point in the plane since its components are continuous at each point. ( 前一節 )

29. 第 29 頁 16. Differentiation Formulas n a positive integer.

30. 第 30 頁 (F continuous at z)

31. 第 31 頁 f has a derivative at z 0 g has a derivative at f (z 0 ) F(z)=g[ f (z)] has a derivative at z 0 and chain rule (6) pf of (6) choose a z 0 at which f ’(z 0 ) exists. let w 0 = f (z 0 ) and assume g’(w 0 ) exists. Then, there is of w 0 such that we can define a function, with and (7) Hence is continuous at w 0

32. 第 32 頁 valid even when since exists and therefore f is continuous at z 0, then we can have f (z) lies in substitute w by f (z) in (9) when z in (9) becomes since f is continuous at z 0, is continuous at is continuous at z 0, and since so (10) becomes

33. 第 33 頁 17. Cauchy-Riemann Equations Suppose that writing Then by Thm. 1 in sec 12 where

34. 第 34 頁 Let tend to (0,0) horizontally through. i.e., Let tend to (0,0) vertically thru, i.e., then (6)= (7) Cauchy-Riemann Equations.

35. 第 35 頁 Thm : suppose exists at a point Then Ex 1. Cauchy-Riemann equations are Necessary conditions for the existence of the derivative of a function f at z 0. Can be used to locate points at which f does not have a derivative.

36. 第 36 頁 Ex 2. does not exist at any nonzero point. The above Thm does not ensure the existence of f ’(z 0 ) (say)

37. 第 37 頁 18. Sufficient Conditions For Differentiability but not Let be defined throughout some neighborhood of a point suppose exist everywhere in the neighborhood and are continuous at. Thm. Then, if

38. 第 38 頁 Thus where Now in view of the continuity of the first-order partial derivatives of u and v at the point

39. 第 39 頁 where and tend to 0 as in the -plane. (3) assuming that the Cauchy-Riemann equations are satisfied at, we can replace in (3), and divide thru by to get also tends to 0, as The last term in(4) tends to 0 as

40. 第 40 頁 Ex 1. everywhere, and continuous. exists everywhere, and Ex 2. has a derivative at z=0. can not have derivative at any nonzero point.

41. 第 41 頁 19. Polar Coordinates Suppose that exist everywhere in some neighborhood of a given non-zero point z 0 and are continuous at that point. also have these properties, and ( by chain rule ) Similarly,

42. 第 42 頁 from (2) (5), Thm. p53…

43. 第 43 頁

44. 第 44 頁 at any non-zero point exists Ex : Consider

45. 第 45 頁 A function f of the complex variable z is analytic in an open set if it has a derivative at each point in that set. f is analytic at a point z 0 if it is analytic in a neighborhood of z 0. 20. Analytic Functions Note: is analytic at each non-zero point in the finite plane is not analytic at any point since its derivative exists only at z = 0 and not throughout any neighborhood. - An entire function is a function that is analytic at each point in the entire finite plane. Ex: every polynomial. - If a function f fails to be analytic at a point z 0 but is analytic at some point in every neighborhood of z 0, then z 0 is called a singular point, or singularity, of f.

46. 第 46 頁 Ex: z=0 is a singular point of has no singular point since it is no where analytic. Sufficient conditions for analyticity continuity of f throughout D. satisfication of Cauchy-Riemann equation. Sum, Product of analytic functions is/are analytic. Quotient of analytic functions is/are analytic. if denominator A composition of two analytic function is analytic

47. 第 47 頁 Thm : If everywhere in a domain D, then must be constant throughout D. pf : write in D at each point in D. are x and y components of vector grad u. grad u is always the zero vector. u is constant along any line segment lying entirely in D.

48. 第 48 頁 21. Reflection Principle predicting when the reflection of f (z) in the real axis corresponds to the reflection of z. Thm: f analytic in domain D which contains a segment of the x axis and is symmetric to that axis. Then for each point z in the domain iff f (x) is real for each point x on the segment. pf: “ ” f (x) real on the segment Let write

49. 第 49 頁 Now, since f (x+ i t) is an analytic function of x + i t. From (4), From (5), Similarly, From (5), since, continuous F(z) is analytic in D.

50. 第 50 頁 Since f (x) is real on the segment of the real axis lying in D, on the segment This is at each point z = x on the segment. (6) From Chap 6 (sec.58) A function f that is analytic in D is uniquely determined by its value along any line segment lying in D. (6) holds throughout D. “ ”

51. 第 51 頁 expand (7): for (x,0) on real axis is real on the segment of real axis lying in D. Ex: for all z. since are real when x is real. do not have reflection property since are not real when x is real.

52. 第 52 頁 22. Haronic Functions A real-valued function H of two real variables x and y is said to be harmonic in a given domain of the xy plane if, throughput that domain, it has continuous partial derivatives of the first and second order and satisfies the partial differential equation known as Laplace's equation. Applications: Temperatures T(x,y) in thin plate. Electrostatic potential in the interior of a region of 3-D space. Ex 1. is harmonic in any domain of the xy plane. Thm 1. If a function is analytic in a domain D, then its component functions u and v are harmonic in D.

53. 第 53 頁 Pf: Assume f analytic in D. (then its real and imaginary components have continuous partial derivatives of all orders in D) “proved in chap 4” the continuity of partial derivatives of u and v ensures this (theorem in calculus) Ex 2. is entire. in Ex 1. must be harmonic. Hence

54. 第 54 頁 is harmonic. If u, v are harmonic in a domain D, and their first-order partial derivatives satisfy Cauchy-Riemann equation throughout D, v is said to be harmonic conjugate of u. (u is not necessary harmonic conjugate of v ) Thm 2 v is a harmonic conjugate of u. Ex 4: and are real & imag. parts of is a harmonic conjugate of. But is not analytic. Ex 3.is entire. so is f (z)g(z). ( f (z) in Ex 2 )

55. 第 55 頁 It can be shown that [Ex 11(b).] if two functions u and v are to be harmonic conjugates of each other, then both u and v must be constant functions. In chap 9, shall show that a function u which is harmonic in a domain of certain type always has a harmonic conjugate. ─ Thus, in such domains, every harmonic function is the real part of an analytic function. ─ A harmonic conjugate, when it exists, is unique except for an additive constant. If v is a harmonic conjugate of u in a domain D, then -u is a harmonic conjugate of v in D.

56. 第 56 頁 harmonic Ex 5. To obtain a harmonic conjugate of a given harmonic function