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
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 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
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 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
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
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
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
~."».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
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
Assignment #9
Due: 5PM, Friday November 29, 2013
Assignments are due on the date indicated by 5:00 pm in the course dropbox. Absolutely no late
assignments will be graded. A mark of zero will be recorded in this case. Include a s
MATH 322 Complex Analysis
Assignment #8
Due: 5PM, Wednesday November 20, 2013
Assignments are due on the date indicated by 5:00 pm in the course dropbox. Absolutely no late
assignments will be graded. A mark of zero will be recorded in this case. Include
MATH 322 Complex Analysis
Assignment #7
Due: 5PM, Wednesday November 6, 2013
Assignments are due on the date indicated by 5:00 pm in the course dropbox. Absolutely no late
assignments will be graded. A mark of zero will be recorded in this case. Include a
MATH 322 Complex Analysis
Assignment #6
Due: 5PM, Wednesday October 30, 2013
Assignments are due on the date indicated by 5:00 pm in the course dropbox. Absolutely no late
assignments will be graded. A mark of zero will be recorded in this case. Include a
MATH 322 Complex Analysis
Assignment #5
Due: 5PM, Wednesday October 9, 2013
Assignments are due on the date indicated by 5:00 pm in the course dropbox. Absolutely no late
assignments will be graded. A mark of zero will be recorded in this case. Include a
MATH 322 Complex Analysis
Assignment #4
Due: 5PM, Wednesday October 2, 2013
Assignments are due on the date indicated by 5:00 pm in the course dropbox. Absolutely no late
assignments will be graded. A mark of zero will be recorded in this case. Include a
MATH 322 Complex Analysis
Assignment #3
Due: 5PM, Wednesday September 25, 2013
Assignments are due on the date indicated by 5:00 pm in the course dropbox. Absolutely no late
assignments will be graded. A mark of zero will be recorded in this case. Include
MATH 322 Complex Analysis
Assignment #2
Due: 5PM, Wednesday September 18, 2013
Assignments are due on the date indicated by 5:00 pm in the course dropbox. Absolutely no late
assignments will be graded. A mark of zero will be recorded in this case. Include