1. tch-prob1 Chapter 4 Integrals Complex integral is extremely important, mathematically elegant. 30. Complex-Valued Functions w(t) First consider derivatives and definite integrals of complex-valued functions w of a real variable t. Real function of t Provided they exist.

2. tch-prob2 Various other rules for real-valued functions of t apply here. However, not every rule carries over.

3. tch-prob3 Example: Suppose w(t) is continuous on The “mean value theorem” for derivatives no longer applies. There is a number c in a<t<b such that

4. tch-prob4 Definite Integral of w(t) over whenexists Can verify that

5. tch-prob5 Anti derivative (Fundamental theorem of calculus)

6. tch-prob6 real must be real Real part of real number is itself

7. tch-prob7 31. Contours Integrals of complex-valued functions of a complex variable are defined on curves in the complex plane, rather than on just intervals of the real line. A set of points z=(x, y) in the complex plane is said to be an arc if where x(t) and y(t) are continuous functions of real t. 非任意的組合 This definition establishes a continuous mapping of interval into the xy, or z, plane; and the image points are ordered according to increasing values of t.

8. tch-prob8 It is convenient to describe the points of arc C by The arc C is a simple arc, or a Jordan arc, if it does not cross itself. When the arc C is simple except that z(b)=z(a), we say that C is a simple closed curve, or a Jordan Curve.

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10. 10 Suppose that x’(t) and y’(t) exist and are continuous throughout C is called a differentiable arc The length of the arc is defined as

11. tch-prob11 L is invariant under certain changes in the parametric representation for C To be specific, Suppose that where is a real valued function mapping the interval onto the interval.We assume that is continuous with a continuous derivative We also assume that

12. tch-prob12 Exercise 6(b) Exercise 10 Then the unit tangent vector ( 不代表水平，而是在此處停頓 長度不增加） is well defined for all t in that open interval. Such an arc is said to be smooth.

13. tch-prob13 For a smooth arc A contour, or piecewise smooth arc, is an arc consisting of a finite number of smooth arcs joined end to end. If z=z(t) is a contour, z(t) is continuous, Whereas z’(t) is piecewise continuous. When only initial and final values of z(t) are the same, a contour is called a simple closed contour

14. tch-prob14 32. Contour Integrals Integrals of complex valued functions f of the complex variable z: Such an integral is defined in terms of the values f(z) along a given contour C, extending from a point z=z 1 to a point z=z 2 in the complex plane. (a line integral) Its value depends on contour C as well as the functions f. Written as When value of integral is independent of the choice of the contour. Choose to define it in terms of

15. tch-prob15 Suppose that represents a contour C, extending from z 1 =z(a) to z 2 =z(b). Let f(z) be piecewise continuous on C. Or f [z(t)] is piecewise continuous on The contour integral of f along C is defined a Since C is a contour, z’(t) is piecewise continuous on t 的變化 define contour C Section 31 So the existence of integral (2) is ensured.

16. tch-prob16 From section 30 Associated with contour C is the contour –C From z 2 to z 1 Parametric representation of -C z 2 =z(b) z 1 =z(a)

17. tch-prob17 order of –C must also follow increasing parameter value order of C follows (t increasing) Thus where z’(-t) denotes the derivative of z(t) with respect to t, evaluated at –t.

18. tch-prob18 After a change of variable, Definite integrals in calculus can be interpreted as areas, and they have other interpretations as well. Except in special cases, no corresponding helpful interpretation, geometric or physical, is available for integrals in the complex plane.

19. tch-prob19 33. Examples Ex. 1 By def. Note:on the circle for z

20. tch-prob20 Ex2.

21. tch-prob21 Ex3 Want to evaluate Note that dep. on end points only. indep. of the arc. Let C denote an arbitrary smooth arc z=z(t), Integral of z around a closed contour in the plane is zero

22. tch-prob22 Ex4. Semicircular path 起點終點 Although the branch (sec. 26) p.77. of the multiple-valued function z 1/2 is not defined at the initial point z=3 of the contour C, the integral of that branch nevertheless exists. For the integrand is piecewise continuous on C.

23. tch-prob23 34. Antiderivatives － There are certain functions whose integrals from z 1 to z 2 are independent of path. The theorem below is useful in determining when integration is independent of path and, moreover, when an integral around a closed path has value zero. － Antiderivative of a continuous function f : a function F such that F’(z)=f(z) for all z in a domain D. － note that F is an analytic function.

24. tch-prob24 Theorem: Suppose f is continuous on a domain D. The following three statements are equivalent. (a) f has an antiderivative F in D. (b) The integrals of f(z) along contours lying entirely in D and extending from any fixed point z 1 to any fixed point z 2 all have the same value. (c) The integrals of f(z) around closed contours lying entirely in D all have value zero. Note: The theorem does not claim that any of these statements is true for a given f in a given domain D.

25. tch-prob25 Pf: (b) (c) (b)

26. tch-prob26 (c) -->(a)

27. tch-prob27 35. Examples

28. tch-prob28 For any contour from z 1 to z 2 that does not pass through the origin. Note that:can not be evaluated in a similar way though derivative of any branch F(z) of log z is, F(z) is not differentiable, or even defined, along its branch cut. (p.77) which t throughoudomaina in lienot does C C. circle theintersect ray theepoint where at theexist tofails cut, branch the from the origin is used to form raya if,particular (In F'(z) 

29. tch-prob29.Ex3 The principal branch Log z of the logarithmic function serves as an antiderivative of the continuous function 1/z throughout D. Hence when the path is the arc (compare with p.98)

30. tch-prob30 C 1 is any contour from z=-3 to z=3, that lies above the x-axis. (Except end points)

31. tch-prob31 1.The integrand is piecewise continuous on C1, and the integral therefore exists. 2.The branch (2) of z 1/2 is not defined on the ray in particular at the point z=3. F(z) 不可積 3.But another branch. is defined and continuous everywhere on C 1. 4.The values of F 1 (z) at all points on C 1 except z=3 coincide with those of our integrand (2); so the integrand can be replaced by F 1 (z).

32. tch-prob32 Since an antiderivative of F 1 (z) is We can write (cf. p. 100, Ex4) Replace the integrand by the branch

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34. tch-prob34 36. Cauchy-Goursat Theorem We present a theorem giving other conditions on a function f ensuring that the value of the integral of f(z) around a simple closed contour is zero. Let C denote a simple closed contour z=z(t) described in the positive sense (counter clockwise). Assume f is analytic at each point interior to and on C.

35. tch-prob35 then Goursat was the first to prove that the condition of continuity on f’ can be omitted. Cauchy-Goursat Theorem: If f is analytic at all points interior to and on a simple closed contour C, then

36. tch-prob36 37. Proof: Omit 38. Simply and Multiply Connected Domains A simply connected domain D is a domain such that every simple closed contour within it encloses only points of D. Multiply connected domain : not simply connected. Can extend Cauchy-Goursat theorem to: Thm 1: If a function f is analytic throughout a simply connected domain D, then for every closed contour C lying in D. Not just simple closed contour as before.

37. tch-prob37 Corollary 1. A function f which is analytic throughout a simply connected domain D must have an antiderivative in D. Extend cauchy-goursat theorem to boundary of multiply connected domain Theorem 2. If f is analytic within C and on C except for points interior to any C k, ( which is interior to C, ) then C: simple closed contour, counter clockwise C k : Simple closed contour, clockwise

38. tch-prob38 Corollary 2. Let C 1 and C 2 denote positively oriented simple closed contours, where C 2 is interior to C 1. If f is analytic in the closed region consisting of those contours and all points between them, then Principle of deformation of paths.

39. tch-prob39 Example: C is any positively oriented simple closed contour surrounding the origin.

40. tch-prob40 39. Cauchy Integral Formula Thm. Let f be analytic everywhere within and on a simple closed contour C, taken in the positive sense. If z 0 is any point interior to C then, (1)Cauchy integral formula (Values of f interior to C are completely determined by the values of f on C)

41. tch-prob41 Pf. of theorem: since is analytic in the closed region consisting of C and C 0 and all points between them, from corollary 2, section 38,

42. tch-prob42 Non-negative constantarbitrary

43. tch-prob43 40. Derivatives of Analytic Functions To prove : f analytic at a point its derivatives of all orders exist at that point and are themselves analytic there.

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45. tch-prob45 Thm1. If f is analytic at a point, then its derivatives of all orders are also analytic functions at that pint. In particular, when

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47. tch-prob47 41. Liouville’s Theorem and the Fundamental Theorem of Algebra Let z 0 be a fixed complex number. If f is analytic within and on a circle Let M R denote the Maximum value of

48. tch-prob48 Thm 1 (Liouville’s theorem): If f is entire and bounded in the complex plane, then f(z) is constant throughout the plane. finite 可以 Arbitrarily large Thm 2 (Fundamental theorem of algebra): Any polynomial Pf. by contradiction

49. tch-prob49 Suppose that P(z) is not zero for any value of z. Then is clearly entire and it is also bounded in the complex plane. To show that it is bounded, first write Can find a sufficiently large positive R such that Generalized triangle inequality

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51. tch-prob51 From the (F. T. 0. A) theorem any polynomial P(z) of degree n can be expressed as Polynomial of degree n-1 Polynomial of degree n-2

52. tch-prob52 42. Maximum Moduli of Functions Lemma. Suppose that f(z) is analytic throughout a neighborhood for each point z in that neighborhood, then f(z) has the constant value f(z 0 ) throughout the neighborhood. f’s value at the center is the arithmetic mean of its values on the circle. ~ Gauss’s mean value theorem.

53. tch-prob53 From (3) and (5)

54. tch-prob54 Thm. (maximum modulus principle) If a function f is analytic and not constant in a given domain D, then has no maximum value in D. That is, there is no point z 0 in the domain such that for all points z in it.

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56. tch-prob56 Assume f(z) has a max value in D at z 0. f(z) also has a max value in N 0 at z 0. From Lemma, f(z) has constant value f(z 0 ) throughout N 0.

57. tch-prob57 If a function f that is analytic at each point in the interior of a closed bounded region R is also continuous throughout R, then the modulus has a maximum value somewhere in R. (sec 14) p.41 Corollary: Suppose f is continuous in a closed bounded region R. and that it is analytic and not constant in the interior of R. Then Maximum value of in R occurs somewhere on the boundary R and never in the interior.  Maximum at the boundary.