SFU , . ' -
17. CauchgGoursat Theorems
1 Review
E] A contour {2(t) : X(t) + £90) : o g t g b} is said to be :imom'f'r if x and g are differentiable.
Cl A contour is surge if it does not cross itself.
C] A contour is ctc: d and Sir:._.")t<'. if 2(a) = 2(b
MATH 322 Complex Analysis
24. Examples of Laurent Series
1
Recall
J A sequence (zn ) converges to z if limn zn = z.
n=0
J (xn + iyn ) x + iy
J The series
n=0 zn
(xn ) x
(yn ) y
and
converges if the sequence of partial sums (
n
j=0 zj )n=0
converges.
J A n
MATH 322 Complex Analysis
10. Logarithm identities and powers
1
Review
J ez ex eiy
(z = x + iy),
J ez+2i = ez periodic
J ez can be negative
J log z ln r + i ( + 2n)
nZ
<
J elog z = z
J Log z ln |z| + i Arg(z)
J log ez = z + 2ni (n = 0, 1, 2, . . . )
J
2
MATH 322 Complex Analysis
22. Introduction to series
1
Recall real-valued Taylor Series
Recall Taylor series from real variable calculus:
ex = 1 + x 1 /1! + x 2 /2! + x 3 /3! + =
n=0
xn
n!
1. Allow one to evaluate an approximation of a function up to some
MATH 322 Complex Analysis
14. Contour integration
1
Review
1. If w : R C and w(t) = u(t) + iv(t) dene
b
a w(t)dt
b
a u(t)dt
+i
b
a v(t)dt.
2. Many standard results hold by applying them to the component integrals and recombining. Eg. The
fundamental theor
MATH 322 Complex Analysis
15. Antiderivatives
1
Review
J Fundamental Theorem of Calculus implies Suppose that on the interval a t b, W (t) = w(t)
and w(t) is continuous. Then
b
a
w(t)dt = W (b) W (a) = W (t)|b .
a
J A contour cfw_z(t) = x(t) + iy(t) : a t
MATH 322 Complex Analysis
27. Residues and Poles
1
Residues and Poles
Def. A point z0 is an isolated singularity of f (z) if
1. f (z) is not analytic at z0 ;
2. f (z) is analytic in the punctured disc about z0 : 0 < |z z0 | < R1 .
z0
2
Examples
1. f (z) =
MATH 322 Complex Analysis
25. Absolute and Uniform Convergence
1
Recall
J A sucient condition for convergence: Remainder(N) =
J A series
zn is absolutely convergent if the series
N zn
0.
|zn | is convergent.
J Theorem. Suppose that a function f is analyt
MATH 322 Complex Analysis
7. Cauchy Riemann Equations, part two
1
Review
J f (z0 ) = limzz0
f (z)f (z0 )
zz0
J f (z) = limz0
f (z+z)f (z)
z
with
z = z z0
J Know the outline of this proof:
Theorem. Suppose that f (z) = u(x, y)+iv(x, y) and that f (z) exist
MATH 322 Complex Analysis
9. Introduction to exp and log
1
The exponential function
We dene the exponential function ez by writing
ez ex eiy
(z = x + iy),
where by Eulers formula
eiy = cos y + i sin y. (The value of y is in radians.)
Recall the basic prop
MATH 322 Complex Analysis
6. Cauchy Riemann Equations
1
Review
J f (z0 ) = limzz0
f (z)f (z0 )
zz0
J
f (z + z) f (z)
z0
z
f (z) = lim
with
z = z z0
(1)
J f (z) = |z|2 only has a derivative at 0. We determine this by rst computing the limit in Equation (1)
MATH 322 Complex Analysis
2. Introduction to complex functions
1
Review
So far we have seen the basic denition of complex numbers, and how to add, multiply and divide them.
We have seen how to represent them in the Argand plane.
Key skills:
1. Know the de
MATH 322 Complex Analysis
13. Introduction to Integration
1
Outline of this section
1. Dene and compute denite integrals of complex functions of a real variable w(t).
2. Decription of contours: (t) = x(t) + iy(t), t = a.b.
3. Countour integrals for functi
MATH 322 Complex Analysis
30. Unexpected (real) applications of residue calculus
To end, we return to the theme
Entre deux vrits du domaine rel, le chemin le plus facile et le plus court passe bien
souvent par le domaine complexe.
-PAUL PAINLEVE
It has be
MATH 322 C A
1
18. Cauchy Integral Formula
Review
o Anti-derivative Theorem. Suppose () is continuous on domain D. Then the following statements
are all true or all false:
1. () has an anti-derivative on D;
2. Fix 1 , 2 D. All integrals C dened by all con
~."».u\ 3'3th '-~:'.1»s;'.'-.
. M;'t:[t~t£.tt l' ' 1., .
Integration strategies
1. Contour parameterization and direct calculation no restriction on analgticitg
2. Contour deformation I path indepedence use antiderivative or an "easier" contour
3. Holes
NI\1H\II(.NIR l\l\lif"ll\
MATHEMATICS i.t- :H w . \
20. A compiex analgsis harvest
1 Review
1:] Theorem (CIF) Let f be analgtic evergwhere inside and on a simple closed contour C taken in the
positive sense (CCW). If 20 is ang point interior to C then
f
The principal square root f(z)=z1/2
Images generated by Dr. David Muraki, 2006
Simon Fraser University MATH 322-3 Complex Analysis
The sine function f(z)=sin(z)
Images generated by Dr. David Muraki, 2006
Simon Fraser University MATH 322-3 Complex Analysis
MATH 322 Complex Analysis
23. Laurent series
1
Recall
J A sequence (zn ) converges to z if limn zn = z.
n=0
J (xn + iyn ) x + iy
n=0 zn
J The series
(xn ) x
and
(yn ) y
converges if the sequence of partial sums (
n
j=0 zj )n=0
converges.
J A neccessary co
MATH 322 Complex Analysis
11. Trigonometric functions
What is sin i?
1
By Eulers formula,
eix = cos x + i sin x
We can deduce
and eix = cos x i sin x
eix eix = 2i sin x
x R.
and eix + eix = 2 cos x.
We solve for sin x, cos x and deduce
sin x =
We conseque
MATH 322 Complex Analysis
18. Cauchy Integral Formula
1
Review
J Anti-derivative Theorem. Suppose f (z) is continuous on domain D. Then the following statements
are all true or all false:
1. f (z) has an anti-derivative on D;
2. Fix z1 , z2 D. All integra
MATH 322 Complex Analysis
8. Analytic and Harmonic Functions
1
Review
J f (z0 ) = limzz0
f (z)f (z0 )
zz0
J f (z) = limz0
f (z+z)f (z)
z
with
z = z z0
J Know the outline of this proof:
Theorem. Suppose that f (z) = u(x, y) + iv(x, y) and that f (z) exists
MATH 322 Complex Analysis
21. Maximum modulus principle
1
Review
J Theorem (pg. 54/sec 18) If a function f is continuous throughout a region R that is both closed
and bounded, then there exists a nonnegative real number M such that
|f (z)| M
for all point
MATH 322 Complex Analysis
29. The order of a zero
1
Recall
Def. residue Resz=z0 f (z) The coecient b1 is called the residue of f (z) at z0 :
b1 =
1
2i
C
f (z) dz = Resz=z0 f (z).
Theorem. (Cauchy Residue Theorem) If f (z) has a nite number of isolated pol
MATH 322 Complex Analysis
16. Review
The exam will cover material up until the end of class Friday October 11, 2005.
1
Topics
1. Complex numbers
(a) expressed in rectangular and exponential form
(b) conjugates
(c) moduli
(d) How to compute the n-th root o
MATH 322 Complex Analysis
28. The order of a pole
1
Recall
Def. residue Resz=z0 f (z) The coecient b1 is called the residue of f (z) at z0 :
b1 =
1
2i
C
f (z) dz = Resz=z0 f (z).
Theorem. (Cauchy Residue Theorem) If f (z) has a nite number of isolated pol
MATH 322 Complex Analysis
5. The complex derivative
1
Review
J |z| = |x + iy| =
1.1
x 2 + y2 = |rei | = r.
Limits
J Denition. Limit:
lim f (z) = w0
zz0
By denition this means: for any > 0, there is a > 0 such that
|f (z) w0 | <
whenever
0 < |z z0 | < .
J
MATH 322 Complex Analysis
3. Limits
1
Review
|z| = |x + iy| =
x 2 + y2 = |rei | = r.
We did not directly cover it, but you should review the following useful facts about modulus, all of
which can be derived from the distance function properties in R2
T
MATH 322
Assignment #10
Series, Residues & Real-Valued Integrals
submit your write-up for problems A-B by 12:01pm, Friday 25 November.
submit your write-up for problems C-D by 12:01pm, Friday 02 December.
please attach the cover sheet to the back of yo
Introduction to Graph Theory
Allen Dickson
October 2006
1
The K
onigsberg Bridge Problem
The city of Konigsberg was located on the Pregel river in Prussia. The river divided the city into four separate landmasses, including the island of Kneiphopf.
These