1. Complex Analysis
ITL SEL

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

3. 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

5. 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) additive identity
z 1=z =(1,0) multiplicative identity
For each z, there is a -z
such that z+(-z)= additive Inverse

6. 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

7. Other Identities
Example.

8. 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

9. 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

10. If

11. 4. Triangle Inequality
geometrically

12. algebraically,
Now

13. Example: z on unit circle2
The triangle inequality can be generalized by mathematical induction to sums …

14. 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

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

16. Moivre’s formula
Ex. Find

17. 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.

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

19. 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

20. 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.

21. Ex. nth roots of unity 1
n = 4 ： 1

22. Ex.2. Find c1 c2 2 c0

23. 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

24. 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.

25. 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

26. 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.

27. 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.

28. 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.

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

30. 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

31. 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

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

33. 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’

34. Ex 3. x y u v
one to one

35. 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

36. 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

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

38. 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.

39. 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.

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

41. since
and
“ ”
But

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

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

44. 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.

45. 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

46. Ex 2.

47. Ex 3.

48. 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

49. 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.

50. 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.

51. 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.

52. 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.

53. 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.

54. 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.

55. Differentiation Formulas
n a positive integer.

56. (F continuous at z)

57. 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

58. 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

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

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

61. 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.

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

63. 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

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

65. 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

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

67. 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,

68. from (2) (5),
Thm. p53…

70. Ex : Consider
at any non-zero point
exists

71. 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 .

72. 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

73. 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.

74. 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

75. 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.

76. 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.
“ ”

77. 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.

78. 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 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.

79. 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

80. 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.

81. 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.

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

83. 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

84. 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.

85. can be used to verify the additive property

86. Ex : There are values of z such that

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

88. Ex:

89. when y is real.
in Exercise 7.
unbounded

90. 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)

91. 25. Hyperbolic Functions
(3)
(4)

92. Frequently used identities

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

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

95. 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),

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

97. 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,

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

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

102. 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

103. 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.

104. 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.

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

106. Example:
But
since
similarly,

108. 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

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

110. 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

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

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

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

114. 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

115. 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

116. tch-prob

117. 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

118. 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

119. ( 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

120. 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

121. 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

122. 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

123. 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

124. 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

125. 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

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

127. Ex2.
tch-prob

128. 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

129. 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

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

131. 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

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

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

134. 35. Examples
tch-prob

135. 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

136. 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)
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137. 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)
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138. 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).
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139. Since an antiderivative of F1(z) is
We can write
(cf. p. 100, Ex4)
Replace the integrand by the branch
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140. tch-prob

141. 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.
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142. 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
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143. 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.
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144. 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
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145. 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.
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146. Example:
C is any positively oriented simple closed contour surrounding the origin.
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147. 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)
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148. 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,
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149. Non-negative constant arbitrary
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150. 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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151. tch-prob

152. 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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153. tch-prob

154. 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
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155. 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
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156. 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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157. tch-prob

158. 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
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159. 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.
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160. From (3) and (5)
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161. 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.
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162. tch-prob

163. 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.
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164. 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.
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165. 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
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166. 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.
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167. An infinite series
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168. 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.
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169. 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)
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170. 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.
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171. The Cauchy integral formula applies:
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172. tch-prob

173. 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
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174. The analyticity of g(z) in the disk ensures
the existence of a Maclaurin series representation:
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175. 45 Examples Ex1. Since is entire
It has a Maclaurin series representation which is valid for all z.
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176. Ex2. Find Maclaurin series representation of
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177. Ex4.
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178. Ex5.
為Laurent series 預告
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179. 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
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180. where
Pf: see textbook.
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181. 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
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182. Ex2.
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183. 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
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184. (b) f(z) in D2
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185. (c) f(z) in D3
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186. 48. Absolute and uniform convergence of power series
Thm1.
(1)
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187. 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.
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188. 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.
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189. then that series is uniformly convergent in the closed disk
Corollary.
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190. 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,
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191. 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.
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192. 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.
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193. 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.
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194. 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
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195. 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
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196. 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.
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197. 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
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198. 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.
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199. Ex4.
湊出z-2在分母
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200. Ex5.
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201. 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
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202. Ex1.
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203. 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:
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204. Ex2.
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205. 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
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206. (i) Type 1.
Ex1.
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207. 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.
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208. * 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.
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209. 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.
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210. 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.
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211. an infinite number of these points clearly lie in any given neighborhood of the origin.
tch-prob

212. 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
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213. Pf: “<=“
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214. “=>”
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215. Ex1.
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216. Need to write out the Laurent series for f(z) as in Ex 2. Sec. 55.
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217. Ex4.
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218. 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.
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219. 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.
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220. 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,
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221. 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
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222. Ex4
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223. 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
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224. 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.
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225. 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)
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226. 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.
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227. 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 z 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
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228. 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
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229. 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.
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230. tch-prob

231. 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.

232. 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

233. 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)

234. (1)
(3)
if exist

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

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

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

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

241. Take real part

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

243. Suppose f is analytic at all points

244. Example 2.
Sol:

246. But from Jordan’s Lemma

247. 62. Definite Integrals Involving Sines and Cosines

250. 63. Indented Paths

252. Ex1.
Consider a simple closed contour

253. Jordan’s Lemma

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

256. Then

258. 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.

260. Positive:
Negative:

261. 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

262. Pf.

266. Rouche’s theorem
Thm 2.
Pf.

269. 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

271. Jordan’s inequality

273. 67. Example
Exercise 12
When t is real

274. Ex1.

277. Chap 8 Mapping by Elementary Functions
68. Linear Transformations

279. 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

283. Ex1.

284. Ex2.
Ex3.

285. 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

287. Denominator=0

288. 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.

289. A linear fractional transformation

290. 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.

291. Ex2:

292. 71. An Implicit Form
The equation

293. Ex1.

295. Ex2.

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

299. Ex1.
Ex2.

300. 73. Exponential and Logarithmic Transformations

301. Ex1

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

303. Ex3.

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

308. Ex2.
Ex3.
Ex4.

309. 75. Mapping by Branches of

310. Ex1
Ex2

312. 76. Square roots of polynomials
Ex1.

313. Ex.2

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

318. Ex1.

319. Ex2. Consider two smooth arcs

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

321. Ex4.

322. 80. Further Properties
We know
length

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

326. Ex2.

327. 81. Harmonic Conjugates
Recall

329. P Q

330. Ex.

331. 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.

332. 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:

333. 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:

334. Ex2.

335. Ex3.
arctan
arctan

336. 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.

337. Thm: suppose that a transformation

338. Pf:

342. Ex.