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Several complex variables and complex manifolds;. 9. Real analysis and PDE (harmonic functions, elliptic equations and distributions). This course covers some ...
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1 Basic complex analysis...................... 2 2 The simply-connected Riemann surfaces............ 39 3 Entire and meromorphic functions............... 59 4 Conformal mapping....................... 81 5 Elliptic functions and elliptic curves.............. 109
Complex analysis is a nexus for many mathematical fields, including:
This course covers some basic material on both the geometric and analytic aspects of complex analysis in one variable.
Prerequisites: Background in real analysis and basic differential topology (such as covering spaces and differential forms), and a first course in complex analysis.
(These exercises are review.)
(a) Find a smooth 1-form β on R^3 such that α = dβ. (b) Define consistent orientations for T and ∂T. (c) Using your choices in (ii), compute
T α^ and^
∂T β^ directly, and check that they agree. (Why should they agree?)
The complex numbers are formally defined as the field C = R[i], where i^2 = −1. They are represented in the Euclidean plane by z = (x, y) = x+iy. There are two square-roots of −1 in C; the number i is the one with positive imaginary part. An important role is played by the Galois involution z 7 → z. We define |z|^2 = N (z) = zz = x^2 + y^2. (Compare the case of a real quadratic field, where N (a + b
d) = a^2 − db^2 gives an indefinite form.) Compatibility of |z| with the Euclidean metric justifies the identification of C and R^2. We also see that z is a field: 1/z = z/|z|. It is also convenient to describe complex numbers by polar coordinates z = [r, θ] = r(cos θ + i sin θ).
Here r = |z| and θ = arg z ∈ R/ 2 πZ. (The multivaluedness of arg z requires care but is also the ultimate source of powerful results such as Cauchy’s integral formula.) We then have
[r 1 , θ 1 ][r 2 , θ 2 ] = [r 1 r 2 , θ 1 + θ 2 ].
In particular, the linear maps f (z) = az + b, a 6 = 0, of C to itself, preserve angles and orientations. This formula should be proved geometrically: in fact, it is a consequence of the formula |ab| = |a||b| and properties of similar triangles. It can then be used to derive the addition formulas for sine and cosine (in Ahflors the reverse logic is applied).
Algebraic closure. A critical feature of the complex numbers is that they are algebraically closed; every polynomial has a root. (A proof will be reviewed below). Classically, the complex numbers were introducing in the course of solv- ing real cubic equations. Staring with x^3 + ax + b = 0 one can make a Tschirnhaus transformation so a = 0. This is done by introducing a new variable y = cx^2 + d such that
yi =
y^2 i = 0; even when a and b are real, it may be necessary to choose c complex (the discriminant of the equation for c is 27b^2 + 4a^3 .) It is negative when the cubic has only one real root; this can be checked by looking at the product of the values of the cubic at its max and min. Analytic functions. Let U be an open set in C and f : U → C a function. We say f is analytic if
f ′(z) = lim t→ 0
f (z + t) − f (z) t
exists for all z ∈ U. It is crucial here that t approaches zero through arbitrary values in C. Remarkably, this condition implies that f is a smooth (C∞) function. For example, polynomials are analytic, as are rational functions away from their poles. Note that any real linear function φ : C → C has the form φ(v) = av+bv. The condition of analytic says that Dfz (v) = f ′(z)v; in other words, the v part is absent. To make this point systematically, for a general C^1 function F : U → C we define
dF dz
dF dx
i
dF dy
and
dF dz
dF dx
i
dF dy
We then have
DFz (v) =
dF dz v +
dz v.
We can also write complex-valued 1-form dF as
dF = ∂F + ∂F =
dF dz dz +
dF dz dz
Thus F is analytic iff ∂F = 0; these are the Cauchy-Riemann equations. We note that (d/dz)zn^ = nzn−^1 ; a polynomial p(z, z) behaves as if these variables are independent.
Sources of analytic functions.
∑n 0 anz n (^) : C → C. The ring of polynomials C[z] is an integral domain and a unique factorization domain, since C is a field. Indeed, since C is algebraically closed, fact every polynomial factors into linear terms. It is useful to add the allowed value ∞ to obtain the Riemann sphere Ĉ = C ∪ {∞}. Then rational functions (ratios f (z) = p(z)/q(z) of rel- atively prime polynomials, with the denominator not identically zero) determine rational maps f : C → C. The rational functions C(z) are the same as the field of fractions for the domain C[z]. We set f (z) = ∞ if q(z) = 0; these points are called the poles of f.
0 an(z)f^ (z) n (^) = 0 for some rational functions an(z); these arise, at least formally, when
f (z + λ) = f (z) for all λ ∈ Λ, a lattice in C; or f (g(z)) = f (z) for all g ∈ Γ ⊂ Aut(H). The elliptic modular functions f : H → C with have the property that f (z) = f (z + 1) = f (− 1 /z), and hence f ((az + b)/(cz + d)) = f (z) for all
( (^) a b c d
ez^ = lim(1 + z/n)n.
The triangle with vertices 0, 1 and 1 + iθ/n has a hypotenuse of length 1 + O(1/n^2 ) and an angle at 0 of θ + O(1/n^2 ). Thus one finds geo- metrically that zn = (1 + iθ/n)n^ satisfies |zn| → 1 and arg zn → θ; in other words, eiθ^ = cos θ + i sin θ. In particular, eπi^ = −1 (Euler).
We now return to the general theory of analytic functions. Let U be a compact, connected, smoothly bounded region in C, and let f : U → C be a continuous function such that f : U → C is analytic. We then have:
Theorem 1.1 (Cauchy)∫ For any analytic function f : U → C, we have
∂U f^ (z)^ dz^ = 0.
Remark. It is critical to know the definition of such a path integral. (For example, f (z) = 1 is analytic, its average over the circle is 1, and yet
S^1 1 dz^ = 0; why is this?) If γ : [a, b] → C parameterizes an arc, then we define ∫
γ
f (z) dz =
∫ (^) b
a
f (γ(t)) γ′(t) dt.
Alternatively, we choose a sequence of points z 1 ,... , zn close together along γ, and then define
∫
γ
f (z) = lim
n∑− 1
1
f (zi)(zi+1 − zi).
(If the loop is closed, we choose zn = z 1 ). This should not be confused with the integral with respect to arclength: ∫
γ
f (z) |dz| = lim
f (zi)|zi+1 − zi|.
Note that the former depends on a choice of orientation of γ, while the latter does not.
Proof of Cauchy’s formula: (i) observe that d(f dz) = (∂f )dz dz = 0 and apply Stokes’ theorem. (ii — Goursat). Cut the region U into small squares, observe that on these squares∫ f (z) ≈ az + b, and use the fact that
∂U (az^ +^ b)^ dz^ = 0.
Aside: distributions. The first proof implicitly assumes f is C^1 , while the second does not. (To see where C^1 is used, suppose α = u dx + v dy and dα = 0 on a square S. In the proof that
∂S α^ = 0, we integrate^ vdy over the vertical sides of S and observe that this is the same as integrating dv/dx dx dy over the square. But if α is not C^1 , we don’t know that dv/dx is integrable.) More generally, we say a distribution (e.g. an L^1 function f ) is (weakly) analytic if
f ∂φ = 0 for every φ ∈ C c∞ (U ). By convolution with a smooth function (a mollifier), any weakly analytic function is a limit of C∞^ analytic functions. We will see below that uniform limits of C∞^ analytic functions are C∞, so even weakly analytic functions are actually smooth. An important and standard computation in differential forms shows that, for f ∈ C 0 ∞ (C), we have ∫
C
(∂f ) ∧ dz z
C−∆(epsilon)
d(f (z) dz/z) = −
S^1 ()
f (z) z dz ≈ − 2 πif (0),
and hence ∂(dz/z) = 2πiδ 0 , as a current. More on the Cauchy–Riemann equations and with minimal smoothness assumptions can be found in [GM].
Cauchy’s integral formula: Differentiability and power series. Because of Cauchy’s theorem, only one integral has to be explicitly evaluated in
Corollary 1.4 An analytic function has at least one singularity on its circle of convergence.
That is, if f can be extended analytically from B(p, R) to B(p, R′), then the radius of convergence is at least R′. So there must be some obstruction to making such an extension, if 1/R = lim sup |ak|^1 /k.
Example: Fibonacci numbers. Let f (z) =
anzn, where an is the nth Fibonacci number. We have (a 0 , a 1 , a 2 , a 3 ,.. .) = (1, 1 , 2 , 3 , 5 , 8 ,.. .). Since an = an− 1 + an− 2 , except for n = 0, we get
f (z) = (z + z^2 )f (z) + 1
and so f (z) = 1/(1 − z − z^2 ). This has a singularity at z = 1/γ and thus lim sup |an|^1 /n^ = γ, where γ = (1 +
5)/2 = 1. 618... is the golden ratio (slightly more than the number of kilometers in a mile). Note that
(an − αγn)zn^ = f (z) − α/(1 − γz) has radius of convergence |γ| > 1, if we choose the constant α to cancel the pole of f (z) at z = 1/γ. Thus |an − αγn| → 0. (In fact α = γ/
Theorem 1.5 A power series represents a rational function iff its coeffi- cients satisfy a recurrence relation.
Aside: Pisot numbers. The golden ratio is an example of a Pisot number; as we have just seen, it has the property that d(γn, Z) → 0 as n → ∞. It is an unsolved problem to show that if α > 1 satisfies d(αn, Z) → 0, then α is an algebraic number. Kronecker’s theorem asserts that
aizi^ is a rational function iff deter- minants of the matrices ai,i+j , 0 ≤ i, j ≤ n are zero for all n sufficiently large [Sa, §I.3] Question: why are 10:09 and 8:18 such pleasant times? [Mon].
How to compute π. The power series for the arctangent is easy to evaluate by relating it to
dx/(1 + x^2 ). Thus we get:
f (z) = tan−^1 (z) = z − z^3 /3 + z^5 /5 + z^7 /7 + · · ·
which suggest correctly that
π 4
However the convergence is very slow, since the error after n terms is on the order of 1/n.
In general, to accelerate the convergence of the sums sn = sn(0) =
∑n 0 ai, we can set sn(1) = (sn + sn+1)/2 and take its limit instead. This will be especially efficacious for an alternating series. But why not repeat the process again and again? To this end we define inductively:
sn(k + 1) = sn(k) + sn+1(k) 2
If we have the terms (a 0 ,... , an) at our disposal, we can then compute
en = s 0 (n) = 2−n
∑n
0
n k
sn.
For f (z) =
akzk^ as above, we find that
0
ak = 3. 121... but 4
0
k
ak = 3. 141592653589793273... ,
which agrees with π to 16 decimal places (!) The reason this works is that if we write F (y) = f (y/(1 − y)) =
bkyk, then en =
∑n 0 bk(1/2)
k. Assuming f is analytic in the unit disk, F is analytic
in the halfplane Re(y) ≤ 1 /2. But if f (z) is analytic in a neighborhood of z = 1, then F (y) is analytic in B(0, r) for some r > 1 /2, and hence its power series converges geometrically fast at y = 1/2. In the case at hand, f (z) has singularities at z = ±i (and nowhere else). Then F (z) has singularities at ±i/(1 + i), so its radius of convergence is 1 /
2 which is bigger than 1/2. In fact we expect the error to be roughly of size 2−n/^2 after taking n terms, and 2−^50 = 9 × 10 −^16 , consistent with the results above. To justify this explanation, we must show that
en =
∑^ n
0
bk 2 −k. (1.2)
To this end, first note that:
1 (1 − y)n+^
0
n + k k
yk.
This can be seen by either differentiating both sides, starting with n = 0, or by observing that the coefficient of yk^ is the number of ways of writing
In this section we develop the maximum princple and related to ideas that lead to compactness of spaces of analytic functions.
Mean value and maximum principle. On the circle |z| = r, with z = reiθ, we have dz/z = i dθ. Thus Cauchy’s formula gives
f (0) =
2 πi
S^1 (r)
f (z)
dz z
2 π
S^1 (r)
f (z) dθ.
In other words, analytic functions satisfy:
Theorem 1.9 (The mean-value formula) The value of f (p) is the av- erage of f (z) over S^1 (p, r).
Corollary 1.10 (The Maximum Principle) A nonconstant analytic func- tion does not achieve its maximum in U.
Proof. Suppose f (z) achieves its maximum at p ∈ U. Then f (p) is the average of f (z) over a small circle S^1 (p, r). Moreover, |f (z)| ≤ |f (p)| on this circle. The only way the average can agree is if f (z) = f (p) on S^1 (p, r). But then f is constant on an arc, so it is constant in U.
Corollary 1.11 If U is compact, then supU |f | = sup∂U |f |.
Cauchy’s bound and algebraic completeness of C. Suppose f is analytic on B(p, R) and let M (R) = sup|z−p|=R |f (z)|; then Cauchy’s bound (1.1) becomes: |f (n)(p)| n!
Rn^
On the other hand, if U = C — so f is an entire function — then the bound above forces M (R) to grow unless some derivative vanishes identically. Thus we find:
Theorem 1.12 A bounded entire function is a constant. More generally, if M (R) = O(Rn), then f is a polynomial of degree at most n.
Corollary 1.13 Any polynomial f ∈ C[z] of degree 1 or more has a zero in C.
Otherwise 1/f (z) would be a nonconstant, bounded entire function. Al- ternatively, 1/f (z) → ∞ as |z| → ∞, so we obtain a violation of the maxi- mum principle.
Corollary 1.14 There is no conformal homeomorphism between C and ∆.
(An analytic map with no critical points is said to be conformal, because it preserves angles.) The Picard theorem says an entire function can omit at most one value in C. Here is an easy form (cf. Weierstrass–Casorati, to be proved later):
Theorem 1.15 Let f : C → C be an entire function. Then the closure of its image is either a single point, or C.
Proof. If p 6 ∈ f (C), then 1/(f (z) − p) is a bounded entire function, hence constant.
Aside: quasiconformal maps. A diffeomorphism f : U → V between domains in C is quasiconformal if supU |∂f /∂f | < ∞. Many qualitative the- orems for conformal maps also hold for quasiconformal maps. For example, there is no quasiconformal homeomorphism between C and ∆.
Parseval’s theorem. The power series of an analytic function on the ball B(0, R) also contains information about its L^2 -norm on the circle |z| = R: namely if f (z) =
anzn, then we have: ∑ |an|^2 R^2 n^ =
2 π
|z|=R
|f (z)|^2 dθ.
This comes from the fact that the functions zn^ are orthogonal in L^2 (S^1 ). It also gives another important perspective on holomorphic functions: they are the elements in L^2 (S^1 ) with positive Fourier coefficients, and hence give a half–dimensional subspace of this infinite–dimensional space. Cauchy’s bound on a disk also implies that if f is small, then f ′^ is also small, at least if we are not too near the edge of U.
Theorem 1.16 Let f (z) be analytic on U and bounded by M. Then |f ′(z)| ≤ M/d(z, ∂U ).
Corollary 1.17 If fn are analytic functions and fn → f uniformly, then f ′^ exists and f (^) n′ → f ′^ locally uniformly.
f (z) = (z − p)ng(z),
where g(p) 6 = 0 and g(z) is analytic near p (so O(g, p) = 0).
f (z) = a−n (z − p)n^
a− 1 z − p
n=
anzn
near p. The germs of functions at p with finite Laurent tails form a local field, with ord(f, p) as its discrete valuation. (Compare Qp, where vp(pna/b) = n.)
Bounded functions and essential singularities.
Theorem 1.22 Every isolated singularity of a bounded analytic function is removable.
Proof. Suppose the singularity is at z = 0, and write f (z) as a Laurent series
anzn. Then for any r > 0 we have
Res(f, 0) = a− 1 =
2 πi
S^1 (r)
f (z) dz.
But if |f | ≤ M then this integral tends to zero as r → 0, and hence a− 1 = 0. Similarly a−(n+1) = Res(znf (z), 0) = 0 for all n ≥ 0. Thus the Laurent power series gives an extension of f to an analytic function at z = 0.
Corollary 1.23 (Weierstrass-Casorati) If f (z) has an essential singu- larity at p, then there exist zn → p such that f (zn) is dense in C.
Proof. Otherwise there is a neighborhood U of p such that f (U −{p}) omits some ball B(q, r) in C. But then g(z) = 1/(f (z) − q) is bounded on U , and hence analytic at p, with a zero of finite order. Then f (z) = q + 1/g(z) has at worst a pole at p, not an essential singularity.
Aside: several complex variables. A function f (z 1 ,... , zn) is analytic if it is C∞^ and df /dzi = 0 for i = 1,... , n. (There are several other equivalent definitions). Using the same type of argument and some basic facts from several complex variables, it is easy to show that if V ⊂ U ⊂ Cn^ is an analytic hypersurfaces, and f : U − V → C is a bounded analytic function, then f extends to all of U. More remarkably, every analytic function extends across V if codim(V ) ≥
Theorem 1.24 (Hartogs) Let f (z, t) be analytic on ∆^2 − ∆(r)^2 , with r <
Proof. Define F (z, t) = (2πi)−^1
S^1 f^ (ζ, t)/(z^ −^ ζ)^ dζ. Since the integrand is holomorphic on as a function of (z, t), so is F. But F (z, t) = f (z, t) for |t| > r, so it provides the desired extension.
In this section we will discuss complex integrals for analytic functions f (z) with isolated singularities: these are controlled by the residues of f. We will use the residue theorem to show that (nonconstant) analytic functions are open maps, and to evaluate definite integrals.
The residue. A critical role is played by the residue of f (z) at p, defined by Res(f, p) = a− 1. It satisfies ∫
γ
f (z) dz = 2πi Res(f, p)
for any small loop encircling the point p in U. Thus the residue is intrin- sically an invariant of the 1-form f (z) dz, not the function f (z). (If we regard f (z) dz as a 1-form, then its residue is invariant under change of coordinates.)
Theorem 1.25 (The residue theorem) Let f : U → C be a function which is analytic apart from a finite set of isolated singularities. We then have: (^) ∫
∂U
f (z) dz = 2πi
p∈U
Res(f, p).
Corollary 1.28 Let p(z) = zd^ + a 1 zd−^1 +... + ad, and suppose |ai| < Ri/d. Then p(z) has d zeros inside the disk |z| < R.
Proof. Write p(z) = f (z) + g(z) with f (z) = zd, and let U = B(0, R); then on ∂U , we have |f | = Rd^ and |g| < Rd; now apply Rouch´e’s Theorem.
Geometric picture. Suppose for simplicity that U is a disk, and p 6 ∈ f (∂U ). The the number of solutions to f (z) = p in U , counted with multi- plicity, is:
N (f (z) − p, U ) =
2 πi
∂U
d log(f (z) − p) =
2 π
∂U
d arg(f (z) − p)·
This is nothing more than the winding number of f (∂U ) around p. Observe that N (f (z) − p, U ) is a locally constant function of p on C − f (∂U ), zero on the noncompact component. This shows:
Theorem 1.29 Suppose p ∈ U and f (p) 6 ∈ f (∂U ). Then f (U ) contains the component of C − f (∂U ) in which p lies.
Corollary 1.30 A nonconstant analytic function is an open mapping.
Proof. Suppose f (p) = q. Then p is an isolated zero of f (z) − q. Choose a small ball U such that f (z) 6 = q on ∂U. Then f (U ) contains the component of C − f (∂U ) in which q lies.
The open mapping theorem gives an alternate proof of the maximum principle and its strict version:
Corollary 1.31 If |f | achieves its maximum in U , then f is a constant.
Moreover, the geometric picture of the argument principle yields:
Theorem 1.32 For any p ∈ C−f (∂U ), the number of solutions to f (z) = p in U is the same as the winding number of f (∂U ) around p.
Using isolation of zeros, we have:
Corollary 1.33 If f ′(p) 6 = 0, then f is a local homeomorphism at p.
In fact for r sufficiently small and q close to p, we have
f −^1 (q) =
2 π
B(p,r)
zf ′(z) dz f (z) − q
This shows:
Corollary 1.34 The local inverse of f is analytic wherever it exists.
The Weierstrass preparation theorem. The same idea can be used to analyze functions of two (or more) complex variables. First a definition: we say f (z, t) is analytic on C^2 if it is analytic in each variable separately. This implies that f is locally given by a convergent power series
aij (z − z 0 )i(w − w 0 )j^. A function of one complex variable looks like a polynomial: f (z) = zng(z). In particular, its zero set agrees with the zero set of a polynomial. Similar statements hold in C^2 , but we must allow polynomials with analytic coefficients. To be precise, suppose ft(z) = f (z, t) is analytic near (0, 0) on C^2 , f 0 (0) = 0, but f 0 (z) is not identically zero.
Theorem 1.35 There exists analytic functions ai(t) on the unit disk, and a nowhere vanishing analytic function ht(z), such that
ft(z) = (zn^ + a 1 (t)zn−^1 + · · · + an(t))ht(z)
for (z, t) near (0, 0), and ai(0) = 0.
For the proof, use the fact that
S^1 ()(z
kf ′ t (z)/ft(z))^ dz^ is analytic in^ t and gives the sums of the powers of the zeros of ft near the origin.
Algebraicity. It can also be shown that if f (z, t) = 0 has an isolated singularity at (0, 0), then one can make an analytic change of coordinates so the f (Z, T ) becomes a polynomial in (Z, T ). See [Ad] and references therein.
Aside: the smooth case. These results also hold for smooth mappings f once one finds a way to count the number of solutions to f (z) = p correctly. (Some may count negatively, and the zeros are only isolated for generic values of p.)
Aside: Linking numbers and intersection multiplicities of curves in C^2. Counting the number of zeros of y = f (x) at x = 0 is the same as counting the multiplicity of intersection between the curves y = 0 and y = f (x) in C^2 , at (0, 0).
