1. 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.

2. 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.

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

4. w=f (z) is not easy to graph as real functions are.
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

5. 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

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

7. a parabola with vertex aty 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’

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

9. means: the limit of as z approaches z0 is w0
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

10. 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

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

12. 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.

13. 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.

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

15. since
and
“ ”
But

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

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

18. 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.

19. 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

20. Ex 2.

21. Ex 3.

22. 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

23. A composition of continuous function is continuous.
A polynomial is continuous in the entire plane because of (11), section 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.

24. 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.

25. 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.

26. 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.

27. 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.

28. 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.

29. Differentiation Formulas
n a positive integer.

30. (F continuous at z)

31. 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

32. 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

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

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

35. 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.

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

37. 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

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

39. 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

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

41. 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,

42. from (2) (5),
Thm. p53…

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

45. 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 .

46. 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

47. 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.

48. 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

49. 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.

50. 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.
“ ”

51. 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.

52. 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.

53. 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

54. 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.

55. 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.

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