m2pm3l14(11).tex
Lecture 14. 8.2.2011.
3. Connectedness.
The theory in Ch. II is about nice functions (dierentiable ones: Section
2) on nice sets (non-empty, open, connected ones). Connectedness is a topological property, not specic to C. We meet it now r
m2pm3l2.tex
Lecture 2. 11.1.2011.
1. Complex Numbers.
Recall N := cfw_1, 2, 3, ., the set of natural numbers. Also, N0 := cfw_0, 1, 2, . =
N cfw_0.
We can take these for granted, or proceed as follows:
0
1
2
3
cfw_
cfw_0, 1
cfw_0, 1, 2
etc. (John von NEUM
m2pm3l3.tex
Lecture 3. 13.1.2011.
Cartesians v. Polars.
For addition and subtraction, cartesians are convenient: Re and Im add
and subtract nicely. For multiplication and division, polars are convenient:
z1 z2 = (r1 ei1 )(r2 ei2 ) = (r1 r2 )ei(1 +2 ) ,
ri
m2pm3l5(11).tex
Lecture 5. 18.1.2011.
De Moivres Theorem.
in
cos n + i sin n = e
i n
n
= (e ) = (cos + i sin ) =
n
k=0
()
n
cosk ink sinnk .
k
Take real parts: writing c, s for cos , sin ,
()
()
()
()
n n2 2 n n4 4
n n2
n n4
cos n = c
c s+
c s +. . . = c
m2pm3l6(11).tex
Lecture 6. 20.1.2011.
Defn. 1. A neighbourhood (nhd) of a point x with radius r is N (x, r) := cfw_y :
| y x | < r .
2. A set S (in Rd , C,.) is open if each point x S has a neighourhood in
S : x S r > 0, s.t. N (x, r) S .
3. A point x of
m2pm3l8.tex
Lecture 7. 24.1.2011.
6. The Theorems of Bolzano &Weierstrass and Cantor. We quote (proofs
not examinable):
Theorem (Bolzano-Weierstrass). If S is an innite bounded set in Rd
(or C) then S has at least one limit point.
Theorem (Cantor; Nested
m2pm3l8(11).tex
Lecture 8. 25.1.2011.
3. The extended complex plane C is compact.
Proof. By stereographic projection, C , closed and bounded, so compact by Heine-Borel.
Theorem (Heine). If f is a continuous function on a compact set S , f is
uniformly con
m2pm3l9.tex
Lecture 9. 27.1.2011.
Defn. A power series in z C is a series of the form an z n , (an C).
n=0
The series may converge for:
all z (e.g. the exponential series ez = z n /n!);
0
some but not all z (e.g. the geometric series 1/(1 z );
only for z
m2pm3l11.tex
Lecture 10. 31.1.2011.
Chapter II. Holomorphic (Analytic) Functions: Theory
1. Special Complex Functions.
1. Polynomials. f (z ) = a0 + a1 z + . + an z n (ai C, an = 0). This
is a (complex) polynomial of degree n. We shall prove (II.6, Fundam
m2pm3l11.tex
Lecture 11. 1.2.2011.
(iii) Riemann surfaces
Think of log, not going from C to C, but from C to R, where R is a doubly
innite stack of copies Ck of C, one for each k Z, spliced together along
their positive real axes so that + 2 takes one fro
m2pm3l12(11).tex
Lecture 12. 3.2.2011.
So if f is dierentiable at z0 with derivativ e f (z0 ),
> 0, > 0 s.t. z with |z z0 | < ,
f (z ) f (z0 )
f (z0 ) <
z z0
(arg (z z0 ) can be anything!). Write z z0 = h = k + il (k , l real), f = u + iv
(u, v real):
m2pm3l1.tex
Lecture 1. 10.1.2011.
Chapter I. PRELIMINARIES
0. Why complex Analysis?
Complex Analysis appeared in 1545: Ars Magna, Girolano CARDANO
(1501-1576) (all dates are in Dramatis Personae: Who Did What When,
on the course website). Calculus was int