Fall 2003

If you find a mistake, omission, etc., please let me know by e-mail.

The orange balls mark our current location in the course, and the current problem set.

Initial handout (DVI, PDF)

Concerning Ahlfors I.3, on ``linear transformations'' (usually called ``fractional linear transformations'' nowadays):

The Riemann sphere is the special case n=1, k=**C**
of n-dimensional *projective space* over a field k,
usually denoted by **P**^{n}(k).
This consists of the lines (1-dimensional subspaces) of a
k-vector space V of dimension n+1; in other words,
**P**^{n}(k)=(V-{0}) / k^{*}.
The general point of **P**^{n}(k) is
(x_{0}:x_{1}:x_{2}:...:x_{n})
where the coordinates x_{i} are elements of k,
not all zero, separated by colons rather than commas
to emphasize that proportional (n+1)-tuples
yield the same point in n-dimensional projective space.
We may also write this space as **P**(V)
when we do not want to commit ourselves to a choice of coordinates.
A *d-dimensional subspace* of **P**(V)
is the subset obtained from a (d+1)-dimensional subspace of V.
It is thus itself a projective space, of dimension d.
In particular, when d=n-1 we call the subspace a ``hyperplane''.
The complement of a hyperplane is an ordinary affine n-dimensional
space over k. [An ``affine space'' is a vector space
without a choice of origin; for instance the Euclidean plane
is an affine 2-dimensional space over **R**.]

The linear group GL(V)=GL_{n+1}(k)
acts on **P**(V)=**P**^{n}(k).
The action is not faithful (unless k is the field of 2 elements):
the kernel is the group of scalar matrices.
We thus obtain an action on **P**(V)
by the quotient group GL(V) / k^{*},
usually known as the *projective linear group*
PGL(V)=PGL_{n+1}(k).
It is a standard fact from linear algebra that
GL(V) acts simply transitively on the ordered bases for V.
Likewise, the affine linear group (the semidirect product
of GL(V) with V, acting on the affine space associated to V)
acts simply transitively on the ordered (n+1)-tuples of points of V
not contained in an affine hyperplane.
Finally, PGL(V) acts simply transitively
on the ordered (n+2)-tuples of points of **P**(V)
no n+1 of which are contained in a hyperplane.
When n=1, this condition reduces to the requirement
that the 1+2=3 points be distinct, and we recover
the fact that PGL_{2}(k) acts simply 3-transitively
on the projective line over k.

We shall soon see that the fact that fractional linear transformations preserve angles between circles is a very special case of the fundamental result that analytic functions with nonzero derivative are conformal (angle-preserving).

Warning: while inversion takes circles
to circles, it does not in general take the center of a circle
to the center of the inverted circle. In fact the stabilizer
in PGL_{2}(**C**) of a circle
acts transitively on the circle's complement
in **P**^{1}(**C**),
so all points not on the circle are equivalent to the center:
a sequence of inversions can take a circle to itself
but take its center to any desired point on the Riemann sphere
as long as that point isn't on the circle!

We did not require that a function of a complex variable
be *continuously* differentiable to qualify as analytic.
This was not an oversight: as Goursat showed,
Cauchy's integral theorem and all its wonderful consequences
can be proved without the hypothesis of continuous differentiability
(which is then seen to be redundant). See Ahlfors, p. 88-89.

One explanation for why Cauchy's integral formula is such a powerful tool is that it represents an arbitrary analytic function as a linear combination of functions 1/(z-a) from which it inherits the desirable properties of smoothness, Taylor series, etc. (The ``linear combination'' is actually an integral over z, but it has much the same effect.)

An easy generalization of Liouville's Theorem:
if f is an entire function
such that f(z)=o(|z|^{M}) as |z| goes to infinity, then f
is a polynomial of degree less than M. By the Fundamental Theorem
of Algebra, it follows that f can be written as a constant multiple
of the product of fewer than M factors z-z_{j}
(the z_{j} being the zeros of f, with multiplicity).
Later in the term we shall see that if f satisfies
the much weaker growth condition f(z)<<exp(|z|^{M})
then f still has a product expansion ranging over zeros of f,
no more than R^{M+o(1)} of which
can fall in a large circle |z|<R
(though the total number of zeros may be infinite in this setting).

We saw that the Fourier expansion for an *analytic* function
on the circle can be recovered from the formula for Laurent series.
Likewise several other manifestation of Fourier analysis
(notably, some cases of evaluation and inversion of one-dimensional
Fourier transforms, and Poisson summation) can be obtained from
contour integration when the functions in question extend
to nice enough analytic functions of a complex variable.

The argument principle (Ahlfors, Thm.20 on p.123, see also
Thm.10 on p.103) has the following interpretation:
if *f* has neither pole nor zero on gamma,
then the winding number of *f*(gamma) around zero
equals the difference between number of zeros and poles of *f*
in the interior of gamma, weighted according to multiplicity
and winding number. In particular, if there are no zeros or poles,
the winding number of *f*(gamma) around the origin vanishes.
The Fundamental Theorem of Algebra can now be recovered from the fact
that if *f* is a polynomial of degree n,
and gamma is a big enough circle,
then *f*(gamma) winds n times around the origin.

**Principal values** of integrals. Recall that
the ``improper'' integral of a locally integrable function
over **R** is defined as the limit of its integral
over [-M,M'] as M and M' approach infinity *independently*.
It may happen that the limit does not exist, but that the integrals
over intervals [-M,M] do approach some finite limit; that limit
is then called the *principal value* of the integral.
For example, the integral over **R**
of a locally integrable odd function always exists and equals zero.
Likewise, if a function is locally integrable on an interval (a,b),
except for a singularity at an internal point x, then its integral
over (a,b) is defined to be the limit as e,e' approach zero
of the integral over (a,b)-(x-e,x+e'); the principal value
is the limit as e->0 of the integral over (a,b)-(x-e,x+e),
which may exist even if the integral does not converge.
Contour integration naturally yields evaluations
of both kinds of principal-value integrals;
notably, the integral over **R** of sin(x)/x
can be evaluated (as Pi) in this way.

The theorem that a limit of analytic functions that's uniform
on compacta is itself analytic (Ahlfors, p.138) works --
with essentially the same proof --
for functions from and to arbitrary Riemann surfaces.
(Uniformity makes sense, because the image of a compact set
is itself compact, and so has a finite cover by neighborhoods
identified with subsets of **C**, from which
we inherit a definition of uniform convergence.)
Weierstrass's formulation via uniformly convergent infinite sums
(p.139), and termwise differentiation of such sums,
require that the functions be complex-valued,
though they can still be defined on an arbitrary Riemann surface.
[But see problem 2 on the fifth problem set.]

We noted that by taking z=1/2 in both sides of the partial-fraction
expansion of (Pi/sin(Pi z))^{2} we can recover
Euler's evaluation of zeta(2). More generally, for any k=2,4,6,8,...
we can evaluate zeta(k) as a rational multiple of Pi^{k}
by comparing Laurent expansions about z=0 (or Taylor expansions
about z=1/2) of (Pi/sin(Pi z))^{2} and the sum over n of
(1/(z-n))^{2}. What do we get by expanding about z=1/4 instead?
For a different take on these sums, see
this
recent article from the *American Math. Monthly*.

Last year I taught a course on analytic number theory, for which I wrote some notes on the Hadamard product formula for an entire function of finite order. Here they are, in DVI and PDF. Likewise for my notes from the same course on the Gamma function (DVI, PDF).

The proof in Ahlfors of the Riemann Mapping Theorem
(IV.4.2, Thm.10, p.172ff.) is less mysterious than it may at first seem.
Suppose the result is true. Then f'(z_{0}) must be the largest
value of |g'(z_{0})| over all functions g, univalent or not,
taking Omega to the unit disc. Proof: consider the function
on the unit disc taking z to f^{-1}(g(z)).

For that matter, Ahlfors' definition of a ``normal family'',
and the statement of Theorem 8 might not be what one
would expect from the Arzela-Ascoli theorem. What Ahlfors
is actually defining is the special case
S=**P**^{1}(**C**)
of a normal family of analytic functions from Omega
to a Riemann surface S. In general, a family of continuous functions
on a metric space with a countable covering by compact subspaces
is normal iff it is uniformly bounded and equicontinuous on compacta.
When we're dealing with analytic functions, equicontinuity
is implied by uniform boundedness of the derivative.
When S=**C**,
uniform boundedness of f implies uniform boundedness of f'
by the Cauchy integral formula for f'.
When S is a compact Riemann surface such as
**P**^{1}(**C**),
uniform boundedness of f is automatic,
and uniform boundedness of f'
yields the f'/(1+|f|^{2}) criterion.
Note that Ahlfors might as well have allowed meromorphic functions
in his definition, which would make the situation a bit clearer.

In addition to the tricks we've already seen for constructing
conformal maps to the unit disc, here's how to conformally map
the unit disc to a triangle: use the indefinite integral of
(1-z/a)^{A-1}(1-z/b)^{B-1}(1-z/c)^{C-1}dz,
where a,b,c are on the unit circle
and A,B,C are positive numbers with A+B+C=1.
This takes a,b,c to the vertices of a triangle
with angles Pi*a, Pi*b, Pi*c.
What happens if one of A,B,C is taken to equal zero?
[By following this map with a fractional linear transformation
we may obtain any ``triangle'' bounded by arcs of three
concurrent circles. ``Triangles'' bounded by arcs
of three arbitrary circles will be treated
when we discuss hypergeometric functions.]

Since the space of harmonic functions on a region is invariant
under conformal transformation, we can (and shall) consider
harmonic functions also on Riemann surfaces that might not be equivalent
to subsets of **C**.

The notation *du is a special case of the "Hodge star" acting on differential 1-forms on a region in a real inner-product space of dimension 2: *(a dx + b dy) = (a dy - b dx). In general, the Hodge star takes k-forms on n-space to (n-k)-forms.

As noted already, the fact that a function on a simply-connected region
is harmonic if and only if it is the real part of an analytic function
implies the remarkable fact that a harmonic function is automatically
smooth (and in fact real-analytic). This is an example of a remarkable
property of ``elliptic differential equations'', which holds also
for harmonic functions in higher dimension. Note that already
in dimension 2 a function satisfying the
``hyperbolic differential equation''
u_{xx}-u_{yy}=0,
or equivalently u_{rs}=0 where (r,s)=(x+y,x-y),
need not have any additional differentiability.

Formula (12) on p.181, giving a harmonic function P_{U}(z)
on the interior of the unit disc in terms of its boundary values U(z),
has the following interpretation: let
Sum_{n}a_{n}z^{n}
be the Fourier series for U(z) on |z|=1
(in which a_{n},a_{-n} must be complex conjugates);
then P_{U}(z) is a_{0} plus twice the real part of
Sum_{n}a_{n}z^{n} over positive n.

Much of our development of (sub)harmonic functions
works just as well for regions in **R**^{n}
for any dimension n.
See for instance the last section of the seventh problem set.

Ahlfors' construction of a barrier function on the complement
of a line segment (p.199) doesn't seem to quite work as stated,
since the square root of
(z-*zeta*_{0})/(z-*zeta*_{1})
maps the complement of the line segment
[*zeta*_{0}, *zeta*_{1}]
to a half-plane. However, using any positive exponent less than 1/2
instead of the square root is good enough.

The analysis leading up to Theorem 13 (p.202) is an example of a theme you'll likely see again in differential or algebraic geometry: it can be very useful to represent deRham cohomology (closed modulo exact differential forms on a manifold) by harmonic differentials.

The formula 3n-6 for the number of real parameters of a canonical region
(Ahlfors, p.204) must be modified for n=2 and, of course, for n=1.
We must add 1 and 3 respectively, due to the dimension of the group
of automorphisms of the canonical region, which is respectively
the ``infinite dihedral group'' of symmetries of a circle,
and our old friend PSL_{2}(**R**).

Another characterization of the solution u of the Dirichlet problem
on some bounded region is that u is the continuously differentiable
function on the region that minimizes I(u) the integral of
(du/dx)^{2}+(du/dy)^{2} among all functions
with the same boundary values. (The physical interpretation
of this integral is the energy stored in an electric field
with potential function u.) Here it's easy to show that
for each M the family of all continuously differentiable u
with the same boundary values and I(u) bounded by M
is equicontinuous, and nonempty if M is large enough.
The hard part is showing that the limit of a sequence of functions u
whose I(u) approach the infimum is again continuously differentiable,
and then to show that it is in fact harmonic. The advantage
of this method is that it applies readily to other regions
with compact closure: integrate (du)^(*du) instead of
((du/dx)^{2}+(du/dy)^{2}) dx dy.

At least in the edition that I've been working from, Ahlfors uses the word ``homotop'' (starting at VI.1.4). This is a rather old-fashioned [or Germanic] form of the word we know as ``homotopic''.

The Weierstrass preparation theorem may be stated as follows.
Suppose f(z,w) is an analytic function of two variables
on a neighborhood of (0,0) such that f(0,w)
does not vanish identically but has a zero of order m>0 at w=0.
Then there exists an integer k<=m
and a punctured neighborhood of the origin in the z-plane
on which the equation f(z,w)=0 has k solutions for each z,
each analytic in z^{1/r} for some positive integer r<=k.

The Wronskian of analytic functions
f_{1},f_{2},...,f_{n}
is the zero function if and only if the f_{k}
are linearly dependent over **C**.
Hence any n-dimensional space of analytic functions
is the space of solutions of a unique regular
linear differential equation of order n, namely
Wr_{n+1}(f_{1},f_{2},...,f_{n},w)/Wr_{n}(f_{1},f_{2},...,f_{n})=0.

On a general Riemann surface S,
a linear first-order differential equation looks like w'+pw=0
where p is a differential on S. A linear second-order diff.eq. is
w''+pw'+qw=0, where p is *almost* a differential
(it transforms a bit differently under coordinate changes)
and q is a ``quadratic differential'', given in local coordinates by
f(z)(dz)^{2} for some meromorphic function f.
Likewise for linear differential equations of higher order.

Handout on (lattices and) doubly periodic functions
(DVI, PDF)
**corrected** (typo in second display on p.11)

NB: The TeX command for the Weierstrass P-function is \wp .

The argument for Prop.3, part (ii), shows in general that for any
compact Riemann surface S and any point p_{0} on S,
the field F of rational functions on S is the fraction field
of its subring A consisting of the functions with no poles
except possibly p_{0}, provided that A contains
at least one nonconstant function. Then the determination of A
suffices to reconstruct F.

Zeroth problem set: Review of complex arithmetic (DVI, PDF)

First problem set: inversive geometry and algebra
(DVI, PDF)

**corrected**: parts (iii-iv) of #6
are about unitary matrices *of unit determinant*

Concerning Problem 1(ii):
the group in question is the orthogonal group O(n+1,1),
acting on the vector space spanned by
1,x_{1},x_{2},...,x_{n},|x|^{2}
whose elements
a+b_{1}x_{1}+...+b_{n}x_{n}+c|x|^{2}
of positive discriminant |b|^{2}-4ac yield spheres in
**R**^{n}.
This ought to be much better known than it is.

Second problem set: analytic functions, cont'd; Cauchy applications I (DVI, PDF)

Third problem set: Differentials, residues, Laurent series
(DVI, PDF)

Concerning Problem 3:
There are also elementary but ad-hoc derivations of these power series;
see **abel**, **catalan**, **catalan2**
under ``one-page papers''.

Fourth problem set: Contour integration II (DVI, PDF)

Fifth problem set:
Series of analytic functions; partial fractions and product formula
(DVI, PDF)

**corrected**:
in #4iii, a term L(f) was missing;
in #5i, the sum should extend only over *large* real zeros x_{k}

Sixth problem set: The Gamma function; univalent functions and normal families (DVI, PDF)

Seventh problem set:
Harmonic functions and the Dirichlet problem
(DVI, PDF)

My **apologies** for the bad Problem 5. :-(

Eighth problem set:
Harmonic functions and their uses, cont'd
(DVI, PDF)

**corrected**: see problem 3 (incorrect conclusion)
and problem 4 (missing hypothesis)

Ninth problem set: Hypergeometric functions (DVI, PDF)

Tenth and final problem set:
Doubly-periodic (a.k.a. elliptic) functions
(DVI, PDF)

**corrected**: see problem 4i.