Complex Analysis ITL SEL

Table Of Contents 1.Complex Numbers 3 2 .Analytic Function 27 3. Elementary Functions 83 4 .Integrals 108 5. Series 165 6 . Residues and Poles 196 7 . Applications of Residues 231 8 . Mapping by Elementary Functions 277 9. Conformal Mapping 314

Chap 1. Complex Numbers. 1. Sums and Products. Complex numbers can be defined as ordered pairs (x,y) of real numbers that are to be interpreted as points in the complex plane, with rectangular coordinates x and y, just as real numbers x are thought of as points on the real line. (x,0) real number (0, y) pure imaginary number It is customary to denote a complex number (x,y) by z , so that x: real part of z Re z = x y: imaginary part of z Im z = y

z+0=z 0=(0,0) additive identity z 1=z 1=(1,0) multiplicative identity 2. Algebraic Properties commutative law associative law distributive law z+0=z 0=(0,0) additive identity z 1=z 1=(1,0) multiplicative identity For each z, there is a -z such that z+(-z)=0 additive Inverse

For any nonzero z=(x,y), multiplicative Inverse There is a such that less obvious than additive inverse Division by a non-zero complex number if

Other Identities Example.

In fact, we often refer to z as the point z or the vector z. 3. Moduli and conjugates It is natural to associate any nonzero complex number z=x+iy with the directed line segment, or vector, from the origin to the point (x,y) that represent z in complex plane. In fact, we often refer to z as the point z or the vector z. y (x,y) Z1 (-2,1) -2+i x+iy Z1+Z2 Z2 x -Z2 -Z2 Z1-Z2

distance between point z and 0 The modulus, or absolute value, of a complex number z=x+iy is defined as length of the vector z. distance between point z and 0 the distance between two points is Z0 z x+iy complex conjugate of z =x+iy is x-iy

If

4. Triangle Inequality geometrically

algebraically, Now

Example: z on unit circle 2 The triangle inequality can be generalized by mathematical induction to sums …

5. Polar coordinates and Euler’s Formula Let r, and be polar coordinates of the point (x,y) that corresponds to a non-zero complex number z=x+iy. since if z=0, the coordinate is undefined. the length of the radius vector for z. has an possible values. Each value of is called an argument of z. and the set of all such values is arg z The principal value of arg z, Arg z, is that unique , s.t. using Euler’s formula then

6. Product and Quotients in Exponential From Two non-zero complex numbers are equal iff 6. Product and Quotients in Exponential From If (1)

Moivre’s formula Ex. Find

If we know two of these, can find the third. Z1Z2 argument of product Z2 (7) Z1 If we know two of these, can find the third. A. If From Expression (1) is a value of B. If If we choose (7) is satisfied.

C. Similarly for Z1Z2 Z2 Z1 Then choose Finally Ex:

Suppose z is nth root of a nonzero number . 7. Roots of Complex Numbers Suppose z is nth root of a nonzero number . are the nth root of These roots are on the circle and are equally spaced every

All of the distinct roots are obtained when k = 0,1,2,…,n-1 Let denote these distinct roots and denote the set of nth roots of (і) if is a positive real number then denotes the entire set of roots. (іі) if in (1) is the principal value of arg is referred to as the principal root.

Ex. nth roots of unity 1 n = 4 ： 1

Ex.2. Find c1 c2 2 c0

Regions in the complex Plane closeness of points to one another -neighborhood or neighborhood of a given point Z Z0 Deleted neighborhood Interior point A point is said to be an interior point of a set S whenever there is some neighborhood of that contains only points of S. Exterior point when there exists a neighborhood of containing no points of S Z0 S

all of whose neighborhoods contain points is S and points not in S Boundary point all of whose neighborhoods contain points is S and points not in S Boundary = { all boundary points } Ex. is the boundary of and A set is open if it contains none of its boundary points A set is closed if it contains all of its boundary points. The closure of a set S is the closed set consisting of all points in S together with the boundary of S is open is closed and closure of - neither open nor closed.

The set of all complex number is both open and closed since it has no boundary points. An open set S is connected if each pair of points and in it can be joined by a polygonal line that lies entirely in S. An open set that is connected is called a domain. (any neighborhood is a domain ) A domain together with some, none, or all of its boundary points is a region. A set S is bounded if every point of S lies inside some circle ; otherwise it is unbounded. Z1 1 2 Z2 open, connected

A point is said to be an accumulation point of a set S if each deleted neighborhood of contains at least one point of S. If a set S is closed, then it contain s each of its accumulation points. pf: If an accumulation point were not in S, it would be a boundary point of S; (can not be exterior points) but this contradicts the fact that a closed set contains all of its boundary points. Ex: For the set the origin is the only accumulation point.

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.

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.

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

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. T z w image of z image of T inverse image of w

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

u=C1>1 V U V=C2>0 Y 2xy=C2 X image right hand branch x>0, u=C1, left hand branch x<0 u=C1,

x y D B A 1 L1 L2 C x1 Ex 2. u v A’ L2’ L1’ B’ D’ C’ When , point moves up a vertical half line, L1, as y increases from y = 0. a parabola with vertex at half line CD is mapped of half line C’D’

Ex 3. x y u v one to one

11. Limits Let a function f be defined at all points z in some deleted neighborhood of z0 means: the limit of as z approaches z0 is w0 can be made arbitrarily close to w0 if we choose the point z close enough to z0 but distinct from it. (1) means that, for each positive number ,there is a positive number such that (2) x y z z0 u v w w0

Note: (2) requires that f be defined at all points in some deleted neighborhood of z0 such a deleted neighborhood always exists when z0 is an interior point of a region on which is defined. We can extend the definition of limit to the case in which z0 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

For any such z and any positive number whenever y v z x u

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

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 w0 is a limit, it does not directly provide a method for determining that limit.

12. Theorems on limits Thm 1. Suppose that Then iff pf : ” ”

since and “ ” But

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

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

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 z O 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.

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

Ex 2.

Ex 3.

14. Continuity A function f is continuous at a point z0 if (1) (2) (3) ((3) implies (1)(2)) if continuous at z0, then are also continuous at z0. So is

A polynomial is continuous in the entire plane because of (11), section 12. p.37 A composition of continuous function is continuous. z w f g If a function f(z) is continuous and non zero at a point z0, then throughout some neighborhood of that point. when let if there is a point z in the at which then , a contradiction.

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. Ex. The function is continuous everywhere in the complex plane since are continuous (polynomial) cos, sin, cosh, sinh are continuous real and imaginary component are continuous complex function is continuous.

15. Derivatives Let f be a function whose domain of definition contain a neighborhood of a point z0. The derivative of f at z0, written , is provided this limit exists. f is said to be differentiable at z0.

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.

since limits are unique, , if is to exist. observe that when 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. Re are continuous, partially Im differentiable at a point. but may not be differentiable there.

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

Differentiation Formulas n a positive integer.

(F continuous at z)

f has a derivative at z0 g has a derivative at f(z0) F(z)=g[f (z)] has a derivative at z0 and chain rule (6) pf of (6) choose a z0 at which f’(z0) exists. let w0 = f (z0) and assume g’(w0) exists. Then, there is of w0 such that we can define a function , with and (7) Hence is continuous at w0

valid even when since exists and therefore f is continuous at z0, 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 z0 , is continuous at is continuous at z0 , and since so (10) becomes

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

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

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 z0. Can be used to locate points at which f does not have a derivative.

Ex 2. does not exist at any nonzero point. The above Thm does not ensure the existence of f ’(z0) (say)

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

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

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

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

19. Polar Coordinates Suppose that exist everywhere in some neighborhood of a given non-zero point z0 and are continuous at that point. also have these properties, and ( by chain rule ) Similarly,

from (2) (5), Thm. p53…

Ex : Consider at any non-zero point exists

20. Analytic Functions 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 z0 if it is analytic in a neighborhood of z0. 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 z0 but is analytic at some point in every neighborhood of z0 , then z0 is called a singular point, or singularity, of f .

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

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.

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

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.

Since f (x) is real on the segment of the real axis lying in D, 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. “ ”

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.

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.

Pf: Assume f analytic in D 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

Ex 3. is entire. so is f (z)g(z). ( f (z) in Ex 2 ) 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.

If v is a harmonic conjugate of u in a domain D, then -u is a harmonic conjugate of v in D. 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.

Ex 5. To obtain a harmonic conjugate of a given harmonic function

Chapter 3. Elementary Functions Consider elementary functions studied in calculus and define corresponding functions of a complex variable. To be specific, define analytic functions of a complex variable z that reduce to the elementary functions in calculus when z = x+i0 . 23. Exponential Function If f (z), is to reduce to when z=x i.e. for all real x, (1) It is natural to impose the following conditions: f is entire and for all z. (2) As shown in Ex.1 of sec.18 is differentiable everywhere in the complex plane and .

It can be shown that (Ex.15) this is the only function satisfying conditions and (2). And we write (3) when Euler’s Formula (5) since is positive for all x and since is always positive, for any complex number z.

can be used to verify the additive property

Ex : There are values of z such that

24. Trigonometric Functions By Euler’s formula It is natural to define These two functions are entire since are entire.

Ex:

when y is real. in Exercise 7. unbounded

A zero of a given function f (z) is a number z0 such that f (z0)=0 Since And there are no other zeros since from (15)

25. Hyperbolic Functions (3) (4)

Frequently used identities

From (4), sinhz and coshz are periodic with period

The Logarithmic Function and Its Branches To solve Thus if we write

Now, If z is a non-zero complex number, , then is any of , when Note that it is not always true that since has many values for a given z or , From (5),

The principal value of log z is obtained from (2) when n=0 and is denoted by

If we let denote any real number and restrict the values of in expression (4) to the interval then with components is single-valued and continuous in the domain. is also analytic,

27. Some Identities Involving Logarithms non-zero. complex numbers (1) Pf:

Example: (A) (B) also Then (1) is satisfied when is chosen. has n distinct values which are nth routs of z Pf: Let

28. Complex Exponents when , c is any complex number, is defined by where log z donates the multiple-valued log function. ( is already known to be valid when c=n and c=1/n ) Example 1: Powers of z are in general multi-valued. since

If and is any real number, the branch of the log function is single-valued and analytic in the indicated domain. when that branch is used, is singled-valued and analytic in the same domain.

Example 3. The principal value of It is analytic in the domain In (1) now define the exponential function with base C. when a value of logc is specified, is an entire function of z.

Inverse Trigonometric and Hyperbolic Functions write Solving for taking log on both sides.

Example: But since similarly,

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

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

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

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

Anti derivative (Fundamental theorem of calculus) tch-prob

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

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

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

tch-prob

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

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

( 不代表水平，而是在此處停頓 長度不增加） 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

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

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=z2 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

represents a contour C, extending from z1=z(a) to z2=z(b). Suppose that 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 The contour integral of f along C is defined a 　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

Associated with contour C is the contour –C 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

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

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

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

Ex2. tch-prob

Let C denote an arbitrary smooth arc z=z(t), Ex3 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

Although the branch (sec. 26) p.77. Ex4. Semicircular path 起點 終點 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

34. Antiderivatives －There are certain functions whose integrals from z1 to z2 　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

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

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

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

35. Examples tch-prob

For any contour from z1 to z2 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 throughou domain a in lie not does C C. circle the intersect ray e point where at the exist to fails cut, branch from the origin is used to form if , particular (In F'(z) \ tch-prob

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

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

is defined and continuous everywhere on C1. 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

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

tch-prob

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

Cauchy-Goursat Theorem: 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

38. Simply and Multiply Connected Domains 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

Theorem 2. If f is analytic within C and on C except for points 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

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

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

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

is analytic in the closed region consisting of C 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

Non-negative constant arbitrary tch-prob

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

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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

tch-prob

41. Liouville’s Theorem and the Fundamental Theorem of Algebra Let z0 be a fixed complex number. If f is analytic within and on a circle Let MR denote the Maximum value of tch-prob

Thm 2 (Fundamental theorem of algebra): Any polynomial 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

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

tch-prob

any polynomial P(z) of degree n can be expressed as 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

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(z0) 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

From (3) and (5) tch-prob

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 z0 in the domain such that for all points z in it. tch-prob

tch-prob

Assume f(z) has a max value in D at z0. f(z) also has a max value in N0 at z0. From Lemma, f(z) has constant value f(z0) throughout N0. tch-prob

Maximum at the boundary. 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

Series representations of analytic functions Chap 5. Series Series representations of analytic functions 43. Convergence of Sequences and Series An infinite sequence 數列 of complex numbers has a limit z if, for each positive , there exists a positive integer n0 such that tch-prob

The limit z is unique if it exists. (Exercise 6) The limit z is unique if it exists. (Exercise 6). When the limit exists, the sequence is said to converge to z. Otherwise, it diverges. Thm 1. tch-prob

An infinite series tch-prob

A necessary condition for the convergence of series (6) is that The terms of a convergent series of complex numbers are, therefore, bounded, Absolute convergence: Absolute convergence of a series of complex numbers implies convergence of that series. tch-prob

44. Taylor Series Thm. Suppose that a function f is analytic throughout an open disk Then at each point z in that disk, f(z) has the series representation That is, the power series here converges to f(z) tch-prob

This is the expansion of f(z) into a Taylor series about the point z0 Any function that is known to be analytic at a point z0 must have a Taylor series about that point. (For, if f is analytic at z0, it is analytic in some neighborhood  may serve as R0 is the statement of Taylor’s Theorem) ~ Maclaurin series. z0=0的case Positively oriented within and z is interior to it. tch-prob

The Cauchy integral formula applies: tch-prob

tch-prob

Suppose f is analytic when and note that the 主要 原因 (b) For arbitrary z0 Suppose f is analytic when and note that the composite function must be analytic when tch-prob

The analyticity of g(z) in the disk ensures the existence of a Maclaurin series representation: tch-prob

45 Examples Ex1. Since is entire It has a Maclaurin series representation which is valid for all z. tch-prob

Ex2. Find Maclaurin series representation of tch-prob

Ex4. tch-prob

Ex5. 為Laurent series 預告 tch-prob

Thm. Suppose that a function f is analytic in a domain 46. Laurent Series If a function f fails to be analytic at a point z0, we can not apply Taylor’s theorem at that point. However, we can find a series representation for f(z) involving both positive and negative powers of (z-z0). Thm. Suppose that a function f is analytic in a domain and let C denote any positively oriented simple closed contour around z0 and lying in that domain. Then at each z in the domain tch-prob

where Pf: see textbook. tch-prob

Alterative way to calculate 47. Examples The coefficients in a Laurent series are generally found by means other than by appealing directly to their integral representation. Ex1. Alterative way to calculate tch-prob

Ex2. tch-prob

has two singular points z=1 and z=2, and is analytic in the domains Ex3. has two singular points z=1 and z=2, and is analytic in the domains Recall that (a) f(z) in D1 tch-prob

(b) f(z) in D2 tch-prob

(c) f(z) in D3 tch-prob

48. Absolute and uniform convergence of power series Thm1. (1) tch-prob

The greatest circle centered at z0 such that series (1) converges at each point inside is called the circle of convergence of series (1). The series CANNOT converge at any point z2 outside that circle, according to the theorem; otherwise circle of convergence is bigger. tch-prob

When the choice of depends only on the value of and is independent of the point z taken in a specified region within the circle of convergence, the convergence is said to be uniform in that region. tch-prob

then that series is uniformly convergent in the closed disk Corollary. tch-prob

49. Integration and Differentiation of power series Have just seen that a power series represents continuous function at each point interior to its circle of convergence. We state in this section that the sum S(z) is actually analytic within the circle. Thm1. Let C denote any contour interior to the circle of convergence of the power series (1), and let g(z) be any function that is continuous on C. The series formed by multiplying each term of the power series by g(z) can be integrated term by term over C; that is, tch-prob

Corollary. The sum S(z) of power series (1) is analytic at each point z interior to the circle of convergence of that series. Ex1. is entire But series (4) clearly converges to f(0) when z=0. Hence f(z) is an entire function. tch-prob

Thm2. The power series (1) can be differentiated term by term. That is, at each point z interior to the circle of convergence of that series, Ex2. Diff. tch-prob

50. Uniqueness of series representation Thm 1. If a series at all points interior to some circle , then it is the Taylor series expansion for f in powers of . Thm 2. If a series converges to f(z) at all points in some annular domain about z0, then it is the Laurent series expansion for f in powers of for that domain. tch-prob

51. Multiplication and Division of Power Series Suppose then f(z) and g(z) are analytic functions in Their product has a Taylor series expansion tch-prob

The Maclaurin series for is valid in disk Ex1. The Maclaurin series for is valid in disk Ex2. Zero of the entire function sinh z tch-prob

Chap 6 Residues and Poles if f analytic. Cauchy-Goursat Theorem: What if f is not analytic at finite number of points interior to C Residues. 53. Residues z0 is called a singular point of a function f if f fails to be analytic at z0 but is analytic at some point in every neighborhood of z0. A singular point z0 is said to be isolated if, in addition, there is a deleted neighborhood of z0 throughout which f is analytic. tch-prob

The origin is a singular point of Log z, but is not isolated Ex1. Ex2. The origin is a singular point of Log z, but is not isolated Ex3. not isolated isolated When z0 is an isolated singular point of a function f, there is a R2 such that f is analytic in tch-prob

Consequently, f(z) is represented by a Laurent series and C is positively oriented simple closed contour When n=1, The complex number b1, which is the coefficient of in expansion (1) , is called the residue of f at the isolated singular point z0. A powerful tool for evaluating certain integrals. tch-prob

Ex4. 湊出z-2在分母 tch-prob

Ex5. tch-prob

54. Residue Theorems Thm1. Let C be a positively oriented simple closed contour. If f is analytic inside and on C except for a finite number of (isolated) singular points zk inside C, then Cauchy’s residue theorem tch-prob

Ex1. tch-prob

Thm2: If a function f is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour C, then Pf: tch-prob

Ex2. tch-prob

55. Three Types of Isolated Singular points If f has an isolated singular point z0, then f(z) can be represented by a Laurent series tch-prob

(i) Type 1. Ex1. tch-prob

is known as a removable singular point. Ex2. (ii) Type 2 bn=0, n=1, 2, 3,…… is known as a removable singular point. * Residue at a removable singular point is always zero. tch-prob

* If we redefine f at z0 so that f(z0)=a0 define Above expansion becomes valid throughout the entire disk * Since a power series always represents an analytic function Interior to its circle of convergence (sec. 49), f is analytic at z0 when it is assigned the value a0 there. The singularity at z0 is therefore removed. Ex3. tch-prob

Infinite number of bn is nonzero. (iii) Type 3: Infinite number of bn is nonzero. is said to be an essential singular point of f. In each neighborhood of an essential singular point, a function assumes every finite value, with one possible exception, an infinite number of times. ~ Picard’s theorem. tch-prob

has an essential singular point at where the residue Ex4. has an essential singular point at where the residue an infinite number of these points clearly lie in any given neighborhood of the origin. tch-prob

an infinite number of these points clearly lie in any given neighborhood of the origin. tch-prob

56. Residues at Poles identify poles and find its corresponding residues. Thm. An isolated singular point z0 of a function f is a pole of order m iff f(z) can be written as tch-prob

Pf: “<=“ tch-prob

“=>” tch-prob

Ex1. tch-prob

Need to write out the Laurent series for f(z) as in Ex 2. Sec. 55. tch-prob

Ex4. tch-prob

57. Zeros and Poles of order m Consider a function f that is analytic at a point z0. (From Sec. 40). Then f is said to have a zero of order m at z0. tch-prob

Thm. Functions p and q are analytic at z0, and Ex1. Thm. Functions p and q are analytic at z0, and If q has a zero of order m at z0, then has a pole of order m there. tch-prob

Corollary: Let two functions p and q be analytic at a point z0. Ex2. Corollary: Let two functions p and q be analytic at a point z0. Pf: Form Theorem in sec 56, tch-prob

The singularities of f(z) occur at zeros of q, or Ex3. The singularities of f(z) occur at zeros of q, or try tan z tch-prob

Ex4 tch-prob

58. Conditions under which Lemma : If f(z)=0 at each point z of a domain or arc containing a point z0, then in any neighborhood N0 of z0 throughout which f is analytic. That is, f(z)=0 at each point z in N0. Pf: Under the stated condition, For some neighborhood N of z0 f(z)=0 Otherwise from (Ex13, sec. 57) There would be a deleted neighborhood of z0 throughout which tch-prob

Since in N, an in the Taylor series for f(z) about z0 must be zero. Thus in neighborhood N0 since that Taylor series also represents f(z) in N0. 圖解 Theorem. If a function f is analytic throughout a domain D and f(z)=0 at each point z of a domain or arc contained in D, then in D. tch-prob

along real x-axis (an arc) Corollary: A function that is analytic in a domain D is uniquely determined over D by its values over a domain, or along an arc, contained in D. Example: along real x-axis (an arc) tch-prob

59. Behavior of f near Removable and Essential Singular Points Observation : A function f is always analytic and bounded in some deleted neighborhood of a removable singularity z0. tch-prob

Assume f is not analytic at z0. Thm 1: Suppose that a function f is analytic and bounded in some deleted neighborhood of a point z0. If f is not analytic at z0, then it has a removable singularity there. Pf: Assume f is not analytic at z0. The point z0 is an isolated singularity of f and f(z) is represented by a Laurent series If C denotes a positively oriented circle tch-prob

is satisfied at some point z in each deleted neighborhood Thm2. Suppose that z0 is an essential singularity of a function f, and let w0 be any complex number. Then, for any positive number , the inequality (a function assumes values arbitrarily close to any given number) (3) is satisfied at some point z in each deleted neighborhood tch-prob

Since z0 is an isolated singularity of f. There is a Pf: Since z0 is an isolated singularity of f. There is a throughout which f is analytic. Suppose (3) is not satisfied for any z there. Thus is bounded and analytic in According to Thm 1, z0 is a removable singularity of g. We let g be defined at z0 so that it is analytic there, becomes analytic at z0 if it is defined there as But this means that z0 is a removable singularity of f, not an essential one, and we have a contradiction. tch-prob

tch-prob

Chapter 7 Applications of Residues - evaluation of definite and improper integrals occurring in real analysis and applied math - finding inverse Laplace transform by the methods of summing residues. 60. Evaluation of Improper Integrals In calculus when the limit on the right exists, the improper integral is said to converge to that limit.

If f(x) is continuous for all x, its improper integral over the is defined by When both of the limits here exist, integral (2) converges to their sum. There is another value that is assigned to integral (2). i.e., The Cauchy principal value (P.V.) of integral (2). provided this single limit exists

If integral (2) converges its Cauchy principal value (3) exists. If is not, however, always true that integral (2) converges when its Cauchy P.V. exists. Example. (ex8, sec. 60)

(1) (3) if exist

To evaluate improper integral of p, q are polynomials with no factors in common. and q(x) has no real zeros. See example

Example has isolated singularities at 6th roots of –1. and is analytic everywhere else. Those roots are

61. Improper Integrals Involving sines and cosines To evaluate Previous method does not apply since sinhay (See p.70) However, we note that

Ex1. An even function And note that is analytic everywhere on and above the real axis except at

Take real part

It is sometimes necessary to use a result based on Jordan’s inequality to evaluate

Suppose f is analytic at all points

Example 2. Sol:

But from Jordan’s Lemma

62. Definite Integrals Involving Sines and Cosines

63. Indented Paths

Ex1. Consider a simple closed contour

Jordan’s Lemma

64. Integrating Along a Branch Cut (P.81, complex exponent)

Then

65. Argument Principle and Rouche’s Theorem A function f is said to be meromorphic in a domain D if it is analytic throughout D - except possibly for poles. Suppose f is meromorphic inside a positively oriented simple close contour C, and analytic and nonzero on C. The image of C under the transformation w = f(z), is a closed contour, not necessarily simple, in the w plane.

Positive: Negative:

The winding number can be determined from the number of zeros and poles of f interior to C. Number of poles zeros are finite (Ex 15, sec. 57) (Ex 4) Argument principle

Pf.

Rouche’s theorem Thm 2. Pf.

66. Inverse Laplace Transforms Suppose that a function F of complex variable s is analytic throughout the finite s plane except for a finite number of isolated singularities. Bromwich integral

Jordan’s inequality

67. Example Exercise 12 When t is real

Ex1.

Chap 8 Mapping by Elementary Functions 68. Linear Transformations

69. The Transformation mapping between nonzero points of z and w planes. a reflection in the real axis An inversion with respect to unit circle

Ex1.

Ex2. Ex3.

70. Linear Fractional Transformation is called a linear fractional transformation or Mobius transformation. bilinear transformation linear in z linear in w bilinear in z and w

Denominator=0

This makes T continuous on the extended z plane (Ex10, sec14). We enlarge the domain of definition, (5) is a one-to-one mapping of the extended z plane onto the extended w plane.

A linear fractional transformation

There is always a linear fractional transformation that maps three given distinct points, z1, z2 and z3 onto three specified distinct points w1, w2 and w3. Ex1.

Ex2:

71. An Implicit Form The equation

Ex1.

Ex2.

72. Mapping of the upper Half Plane Determine all 1inear fractional transformation T that

Ex1. Ex2.

73. Exponential and Logarithmic Transformations

Ex1

Ex2. any branch of log z , maps onto a strip

Ex3.

74. The transformation Ex1. (1-to-1)

Ex2. Ex3. Ex4.

75. Mapping by Branches of

Ex1 Ex2

76. Square roots of polynomials Ex1.

Ex.2

Chap 9. Conformal Mapping 不會在某一點 t 停頓 79. Preservation of Angles

Ex1.

Ex2. Consider two smooth arcs

Isogonal mapping : a mapping that preserves the magnitude of the angle bet not necessarily the sense. Ex3.

Ex4.

80. Further Properties We know length

Large region will be different in shape after transformation. Ex1.

Ex2.

81. Harmonic Conjugates Recall

P Q

Ex.

82. Transformations of Harmonic Functions Since A function that is harmonic in a simply connected domain always has a harmonic conjugate (sec.81), solutions of (boundary value) problems in such domains are the real or imaginary ports of analytic functions.

Ex. If we can identify a function as the real or imaginary part of an analytic function, then we know it is a harmonic function, But how ? not easy Other aid:

Thm. Suppose that an analytic function 藉由f and h的條件, 目的:show H(x,y) is a harmonic function Simple pf:

Ex2.

Ex3. arctan arctan

83. Transformations of Boundary Conditions Boundary condition: a function or its normal derivative (boundary value problem) have prescribed values along the bandore of a domain. we can transform a given boundary valne problem in the xy plane into a simpler one in the u v plane and then write the solution of the original problem in terms of the solution obtained from the simpler one.

Thm: suppose that a transformation

Pf:

Ex.