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MATH 311: COMPLEX ANALYSIS MAPPINGS LECTURE
1. The complex exponential
The exponential function
exp : C C cfw_0
is dened to be
exp(x + iy ) = ex eiy
where eiy = cos y + i sin y.
It is natural to think of the inputs to the exponential map in cartesian coor
A FAR-REACHING LITTLE INTEGRAL
Let
r be any positive real number, and r be the circle of radius r centered at
the origin, traversed once counterclockwise,
n be any integer, and fn (z ) = z n . This function is undened at z = 0 if n
is negative.
The natu
GEOMETRY OF THE CAUCHYRIEMANN EQUATIONS
The usual picture-explanations given to interpret the divergence and the curl are
not entirely satisfying. Working with the polar coordinate system further quanties
the ideas and makes them more coherent by applying
COMPLEX DIFFERENTIABILITY AND VECTOR
DIFFERENTIABILITY
1. The Little-oh Notation
Denition 1.1. Let U be an open superset of 0 in Rn . A function
: U Rm
is called an o(h)-function if for every > 0 there exists some > 0 such that
hU
|h|
= |(h)| |h|.
The g
STEREOGRAPHIC PROJECTION AND THE MERCATOR MAP
1. Stereographic Projection
Let
S 2 = cfw_(x, y, z ) R3 : x2 + y 2 + z 2 = 1
be the unit sphere, and let n denote the north pole (0, 0, 1). Identify the complex
plane C with the (x, y )-plane in R3 .
The stere
COMPACTNESS AND UNIFORMITY
1. Compactness and Uniform Continuity
Let K be a compact subset of C, and let f : K C be pointwise continuous.
Then f is uniformly continuous.
The topological proof of this fact proceeds as follows. Pointwise continuity says
tha
SUFFICIENT CONDITIONS FOR f AT A POINT
Let be a region in C, and let f : C be a function. Write f = u + iv , and
write z = x + iy for points z of . For the derivative f (z ) to exist a point z of
it is necessary but not sucient that the CauchyRiemann equ
FORMS OF THE CAUCHYRIEMANN EQUATIONS
Let be a region in C, and let f : C be a function. View points
of the domain either as vectors z = (x, y ) or as complex numbers z = x + iy .
View f either as a vector-valued function f = (u, v ) or as a complex-valued
BASIC SYMBOL-PATTERNS FOR SETS AND MAPS
Let X and Y be sets, assuming no further structure whatsoever. Let
f : X Y
be a function, again subject to no assumptions. Let I be any index set. Let
S,
cfw_Si : i I
be arbitrary subsets of X , and let
T,
cfw_Ti :
MATH 311: COMPLEX ANALYSIS PREVIEW LECTURE
1. Introduction
Complex analysis: not very complicated, not much analysis.
The analysis is whats called soft analysis some integrals and derivatives,
but very few epsilons and grungy estimates once Cauchys Theore
MATH 311: COMPLEX ANALYSIS TOPOLOGY LECTURE
1. Topology
Engage with Marsden section 1.4 to taste. If things seem unduly complicated,
ask me whether they can be simplied (cf. connectedness).
Structures of increasing generality:
(R, dR ),
n
(C, dC )
(R , dR
STEREOGRAPHIC PROJECTION IS CONFORMAL
Let
S 2 = cfw_(x, y, z ) R3 : x2 + y 2 + z 2 = 1
be the unit sphere, and let n denote the north pole (0, 0, 1). Identify the complex
plane C with the (x, y )-plane in R3 . The stereographic projection map,
: S 2 n C,
MATH 311: COMPLEX ANALYSIS COMPLEX NUMBERS
LECTURE
1. Complex Numbers
The set of complex numbers is
C = cfw_x + iy : x, y R
where
i2 = 1
and ix = xi for all x R.
The real and imaginary parts of a complex number z = x + iy are
Re(z ) = x,
Im(z ) = y
(so th
MATH 311: COMPLEX ANALYSIS INTEGRATION LECTURE
1. First definition of the complex integral
The data are
a region C,
a continuous function f : C,
a C 1 -path : [a, b] .
In general, for a complex-valued function over a real interval,
: [a, b] C,
(t) = U