Chapter 4 Integrals Complex integral is extremely important, mathematically elegant.

1. Chapter 4 IntegralsComplex 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. tch-prob

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

3. 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 tch-prob

4. Definite Integral of w(t) over when exists Can verify that tch-prob

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

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

7. 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. tch-prob

8. 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. tch-prob

9. tch-prob

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 tch-prob

11. 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 tch-prob

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

13. 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 tch-prob

14. 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=z1 to a point z=z2in 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 tch-prob

15. represents a contour C, extending from z1=z(a) to z2=z(b). Let f(z) be piecewise continuous on C. Or f [z(t)] is piecewise continuous on Suppose that The contour integral of f along C is defined a t的變化　define contour C Since C is a contour, z’(t) is piecewise continuous on Section 31 So the existence of integral (2) is ensured. tch-prob

16. From section 30 Associated with contour C is the contour –C From z2 to z1 Parametric representation of -C z2=z(b) z1=z(a) tch-prob

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

18. 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. tch-prob

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

20. Ex2. tch-prob

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

22. 起點 終點 Ex4. Semicircular path Although the branch (sec. 26) p.77. of the multiple-valued function z1/2 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. tch-prob

23. 34. Antiderivatives －There are certain functions whose integrals from z1 to z2are 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. tch-prob

24. 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 z1 to any fixed point z2 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. tch-prob

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

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

27. 35. Examples tch-prob

28. Note that: can not be evaluated in a similar way (In particular , if a ray from the origin is used to form the branch cut, F'(z) fails to exist at the point where e the ray \ intersect the circle C. C does not lie in a domain throughou t which For any contour from z1 to z2 that does not pass through the origin . 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) tch-prob

29. Ex 3 . 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) tch-prob

30. C1 is any contour from z=-3 to z=3, that lies above the x-axis. (Except end points) tch-prob

31. The integrand is piecewise continuous on C1, and the integral therefore exists. • 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 • But another branch. • is defined and continuous everywhere on C1. • 4. The values of F1(z) at all points on C1 except z=3 coincide with those of our integrand (2); so the integrand can be replaced by F1(z). tch-prob

32. Since an antiderivative of F1(z) is We can write (cf. p. 100, Ex4) Replace the integrand by the branch tch-prob

33. tch-prob

34. 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. 36. Cauchy-Goursat Theorem tch-prob

35. 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 tch-prob

36. 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. tch-prob

37. Theorem 2. If f is analytic within C and on C except for points interior to any Ck, ( which is interior to C, ) then 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 C: simple closed contour, counter clockwise Ck: Simple closed contour, clockwise tch-prob

38. Corollary 2. Let C1 and C2 denote positively oriented simple closed contours, where C2 is interior to C1. If f is analytic in the closed region consisting of those contours and all points between them, then Principle of deformation of paths. tch-prob

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

40. 39. Cauchy Integral Formula Thm. Let f be analytic everywhere within and on a simple closed contour C, taken in the positive sense. If z0 is any point interior to C then, • Cauchy integral formula • (Values of f interior to C are completely determined by the values of f on C) tch-prob

41. Pf. of theorem: since is analytic in the closed region consisting of C and C0 and all points between them, from corollary 2, section 38, tch-prob

42. Non-negative constant arbitrary tch-prob

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

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

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47. 41. Liouville’s Theorem and the Fundamental Theorem of Algebra Let z0be a fixed complex number. If f is analytic within and on a circle Let MR denote the Maximum value of tch-prob

48. 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 tch-prob

49. 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 tch-prob

50. tch-prob