Chapter 4 Integrals Complex integral is extremely important

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Chapter 4 Integrals Complex integral is extremely important, mathematically elegant. 30. Complex-Valued Functions w(t)

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. tch-prob 1

Various other rules for real-valued functions of t apply here. However, not every rule

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

Example: Suppose w(t) is continuous on The “mean value theorem” for derivatives no longer

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 3

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

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

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

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

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

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

31. Contours Integrals of complex-valued functions of a complex variable are defined on curves

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 7

It is convenient to describe the points of arc C by The arc C

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 8

tch-prob 9

tch-prob 9

Suppose that x’(t) and y’(t) exist and are continuous throughout C is called a

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 10

L is invariant under certain changes in the parametric representation for C To be

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 11

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

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 12

For a smooth arc A contour, or piecewise smooth arc, is an arc consisting

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 13

32. Contour Integrals of complex valued functions f of the complex variable z: Such

32. Contour 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 tch-prob 14

Suppose that represents a contour C, extending from z 1=z(a) to z 2=z(b). Let

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 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 15

From section 30 Associated with contour C is the contour –C From z 2

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) tch-prob 16

order of C follows (t increasing) order of –C must also follow increasing parameter

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 17

After a change of variable, Definite integrals in calculus can be interpreted as areas,

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 18

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

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

Ex 2. tch-prob 20

Ex 2. tch-prob 20

Ex 3 Let C denote an arbitrary smooth arc z=z(t), Want to evaluate Note

Ex 3 Let C denote an arbitrary smooth arc z=z(t), 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 21

Ex 4. Semicircular path 起點 終點 Although the branch (sec. 26) p. 77. of

Ex 4. 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. tch-prob 22

34. Antiderivatives -There are certain functions whose integrals from z 1 to z 2

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. tch-prob 23

Theorem: Suppose f is continuous on a domain D. The following three statements are

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. tch-prob 24

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

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

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

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

35. Examples tch-prob 27

35. Examples tch-prob 27

For any contour from z 1 to z 2 that does not pass through

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) (In particular , if a ray from the origin is used to form the branch cut, F'(z) fails to exist at the point wheree the ray intersect the circle C. C does not lie in a domain throughout which tch-prob 28

Ex 3. The principal branch Log z of the logarithmic function serves as an

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 29

C 1 is any contour from z=-3 to z=3, that lies above the x-axis.

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

1. The integrand is piecewise continuous on C 1, and the integral therefore exists.

1. The integrand is piecewise continuous on C 1, 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. 4. 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). tch-prob 31

Since an antiderivative of F 1(z) is We can write (cf. p. 100, Ex

Since an antiderivative of F 1(z) is We can write (cf. p. 100, Ex 4) Replace the integrand by the branch tch-prob 32

tch-prob 33

tch-prob 33

36. Cauchy-Goursat Theorem We present a theorem giving other conditions on a function f

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. tch-prob 34

then Goursat was the first to prove that the condition of continuity on f’

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 35

37. Proof: Omit 38. Simply and Multiply Connected Domains A simply connected domain D

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 36

Corollary 1. A function f which is analytic throughout a simply connected domain D

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 Ck, ( which is interior to C, ) then C: simple closed contour, counter clockwise Ck: Simple closed contour, clockwise tch-prob 37

Corollary 2. Let C 1 and C 2 denote positively oriented simple closed contours,

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. tch-prob 38

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

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

39. Cauchy Integral Formula Thm. Let f be analytic everywhere within and on a

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 (2) tch-prob (Values of f interior to C are completely determined by the values of f on C) 40

Pf. of theorem: since is analytic in the closed region consisting of C and

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, tch-prob 41

Non-negative constant arbitrary tch-prob 42

Non-negative constant arbitrary tch-prob 42

40. Derivatives of Analytic Functions To prove : f analytic at a point its

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. tch-prob 43

tch-prob 44

tch-prob 44

Thm 1. If f is analytic at a point, then its derivatives of all

Thm 1. 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 45

tch-prob 46

tch-prob 46

41. Liouville’s Theorem and the Fundamental Theorem of Algebra Let z 0 be a

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 MR denote the Maximum value of tch-prob 47

Thm 1 (Liouville’s theorem): If f is entire and bounded in the complex plane,

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 48

Suppose that P(z) is not zero for any value of z. Then is clearly

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 49

tch-prob 50

tch-prob 50

From the (F. T. 0. A) theorem any polynomial P(z) of degree n can

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 tch-prob 51

42. Maximum Moduli of Functions Lemma. Suppose that f(z) is analytic throughout a neighborhood

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. tch-prob 52

From (3) and (5) tch-prob 53

From (3) and (5) tch-prob 53

Thm. (maximum modulus principle) If a function f is analytic and not constant in

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. tch-prob 54

tch-prob 55

tch-prob 55

Assume f(z) has a max value in D at z 0. f(z) also has

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. tch-prob 56

If a function f that is analytic at each point in the interior of

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 Maximum at the boundary. 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. tch-prob 57