Lecture 2: Convergence of Series and Sums
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Motivation: functions given by power series
We will define:
ez = 1 + z +
1
2!
z2 +
1
3!
z3 +
for all z
We will prove:
1
Lecture 19: Convergence of power series
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Radius of convergence
Basic fact about power series
1
X
ak z k , with coefficients ak 2 C,
1
k =0
X
there is a real number
Lecture 21: Power series expansions of
analytic functions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Theorem from last lecture (R = radius convergence)
The series f (z) =
X
ak z k is an analytic function o
Lecture 22: Power series expansions of
analytic functions II
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
If f (z) is analytic on E, then f (z) equals its Taylor expansion
f (z) =
P1
k =0 ak
(z
z0 )k ,
ak =
Math 427A (Autumn Quarter, 2015)
Hart Smith
Instructor: Hart Smith, C-441 Padelford Hall, 685-2902.
Office Hours: 2:404:00 pm Monday and Wednesday.
Text: Complex Variables, by Joseph Taylor (AMS, 2011)
Grading: Your grade for the course will be based on t
Math 427 Midterm Study Topics
Section 1.1: Properties of C: multiplication, inverse, conjugation, modulus and
triangle inequality, Theorem 1.1.7.
Section 1.2: Definition of convergence; simple examples.
Section 1.3: Properties of the function e z , includ
Lecture 17: Properties of the Index Function
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Theorem
If is a closed path in C that does not touch z, then
Z
1
1
ind (z) =
dw
2i w z
is an integer k , called the i
Lecture 15: Cauchys Integral Formula
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Last lecture
Cauchys Theorem for a triangle
If f (z) is analytic on a convex open set E, then for any
Z
f (z) dz = 0
E,
@
An
Lecture 20: Power series and analytic
functions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Theorem
If the power series
X
ak z k has radius of convergence R, then
k =0
the series converges uniformly on cfw_
Lecture 18: Uniform convergence
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Last lecture
Index function for the unit circle
Let trace the unit circle: (t) = e it , t [0, 2].
(
1 , |z| < 1 ,
Then:
ind (z) =
Lecture 12: Contour integrals II
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Contour integral
Definition
If f (z) is a continuous function on E C, and (t) : [a, b] E
is a smooth curve, the contour integral
Lecture 16: The Index Function
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Cauchy integral formula
If E is convex open,
Z
a closed path in E that does not touch z0 ,
f (w)
dw = f (z0 )
w z0
Z
1
w
z0
dw
The
Lecture 14: Cauchys Theorem and
anti-derivatives
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Last lecture
Cauchys Theorem for a triangle
If f (z) is a smooth analytic function on an open set E , then
Z
f (
Lecture 11: Contour integrals
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Curves in the complex plane
Definition
A curve is a continuous map (t) : [a, b] ! C, some [a, b] R.
Example: A straight line curve f
Lecture 24: Isolated singularities
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Definition: Suppose f (z) is defined for 0 < |z z0 | < r .
We say that lim f (z) = if, for all M , there is > 0 so that
zz0
|f
Lecture 26: Essential singularities;
Harmonic functions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Theorem: assume f analytic on Dr (z0 ) \ cfw_z0
If f has an essential singularity at z0 , then for all w
Lecture 3: The exponential function
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Definition
For a complex number z, we define
ez =
X
zk
= 1+z +
k!
1
2!
z2 +
k =0
Some immediate facts:
e0 = 1
e x is a real nu
Lecture 4: Trig and n-th root functions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Definition
For a complex number z, we define
cos z =
sin z =
X
k =0
X
(1)k
z 2k
= 1
(2k )!
(1)k
z 2k +1
= z
(2k + 1)!
k =0
Lecture 5: The log function; topology on C
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Recall: polar form for z 6= 0 :
z = r e i , r = |z|.
is called the argument of z, we write = arg(z).
Notation. If I is
Lecture 6: Topology on C, continuous functions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Open subsets of C
Notation
- C is the set of complex numbers, R the set of real numbers.
- Dr (z0 ) = cfw_w : |w z0
Lecture 9: Complex differentiation
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Differentiability over C
Assume E C is open, and f (z) is a function from E to C.
Definition
We say that f (z) is differentiabl
Lecture 10: The Cauchy-Riemann equations
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Cauchy-Riemann equations
Write z = x + iy , and write
f (x + iy ) = u(x, y ) + iv (x, y )
where u(x, y ) and v (x, y ) ar
Lecture 8: Branches of multi-valued functions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
A multi-valued function f (z) on E C is a function that
assigns a set of complex values to each z E.
Examples:
log z
Lecture 7: Continuous functions;
Differentiable functions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Continuous functions from C to C .
Idea: f (z) continuous if: |f (w) f (z)| is small if |w z| is.
Defini
Lecture 25: Analytic functions with poles
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
The structure of poles
If f (z) =
h(z)
, h(z) analytic on Dr (z0 ) and h(z0 ) 6= 0,
(z z0 )m
we say f (z) has a pole of
Lecture 27: The maximum modulus theorem
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Theorem
Assume that f (t) is a continuous, real valued function on [a, b],
Z b
1
f (t) dt = M,
and f (t) M for all t [a, b
Lecture 23: Zeroes of analytic functions
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Assume f (z) analytic on E C, and f (z0 ) = 0 . If |z z0 | < R :
f (z) =
X
ak (z z0 )k ,
k =1
ak =
f (k ) (z0 )
k!
Two po
Lecture 28: Schwarzs Lemma
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
Assume that: f (z) is analytic on D1 (0) = cfw_z : |z| < 1,
and continuous on D1 (0) = cfw_z : |z| 1.
By the Maximum Modulus Theorem:
Lecture 13: Cauchys Theorem
Hart Smith
Department of Mathematics
University of Washington, Seattle
Math 427, Autumn 2015
More on contour integrals
Useful notation:
If z0 , z1 C , [z0 , z1 ] = straight line path from z0 to z1
[z0 , z1 ] = (1 t)z0 + t z1