3 Meromorphic Functions and the Logarithm
Theorem 7.1 The coefficients of the power series expansion of f are given by
n=0
an(z− z0)n
that converges in that disc.
Theorem 7.1 The coefficients of the power series expansion of f are given by
an= 1 2πrn
2π 0
f (z0+ reiθ)e−inθdθ for all n≥ 0 and 0 < r < R. Moreover,
0 = 1 2πrn
2π 0
f (z0+ reiθ)e−inθdθ whenever n < 0.
102 Chapter 3. MEROMORPHIC FUNCTIONS AND THE LOGARITHM orientation. Choosing ζ = z0+ reiθfor the parametrization of this circle, we find that for n≥ 0
Finally, even when n < 0, our calculation shows that we still have the identity and by Cauchy’s theorem the last integral vanishes.
The interpretation of this theorem is as follows. Consider f (z0+ reiθ) as the restriction to the circle of a holomorphic function on the closure of a disc centered at z0 with radius r. Then its Fourier coefficients vanish if n < 0, while those for n≥ 0 are equal (up to a factor of rn) to coefficients of the power series of the holomorphic function f . The property of the vanishing of the Fourier coefficients for n < 0 reveals another special characteristic of holomorphic functions (and in particular their restrictions to any circle).
Next, since a0= f (z0), we obtain the following corollary.
Corollary 7.2 (Mean-value property) If f is holomorphic in a disc DR(z0), then
Taking the real parts of both sides, we obtain the following conse-quence.
8. Exercises 103 Corollary 7.3 If f is holomorphic in a disc DR(z0), and u = Re(f ), then
u(z0) = 1 2π
2π 0
u(z0+ reiθ) dθ, for any 0 < r < R.
Recall that u is harmonic whenever f is holomorphic, and in fact, the above corollary is a property enjoyed by every harmonic function in the disc DR(z0). This follows from Exercise 12 in Chapter 2, which shows that every harmonic function in a disc is the real part of a holomorphic function in that disc.
8 Exercises
1. Using Euler’s formula
sin πz = eiπz− e−iπz
2i ,
show that the complex zeros of sin πz are exactly at the integers, and that they are each of order 1.
Calculate the residue of 1/ sin πz at z = n∈ Z.
2. Evaluate the integral
∞
−∞
dx 1 + x4. Where are the poles of 1/(1 + z4)?
3. Show that
∞
−∞
cos x
x2+ a2dx = πe−a
a , for a > 0.
4. Show that
∞
−∞
x sin x
x2+ a2 dx = πe−a, for all a > 0.
5. Use contour integration to show that
∞
−∞
e−2πixξ
(1 + x2)2dx = π
2(1 + 2π|ξ|)e−2π|ξ|
for all ξ real.
104 Chapter 3. MEROMORPHIC FUNCTIONS AND THE LOGARITHM 6. Show that
∞
−∞
dx
(1 + x2)n+1 = 1· 3 · 5 · · · (2n − 1) 2· 4 · 6 · · · (2n) · π.
7. Prove that
2π
0
dθ
(a + cos θ)2 = 2πa
(a2− 1)3/2, whenever a > 1.
8. Prove that
2π
0
dθ
a + b cos θ= 2π
√a2− b2
if a >|b| and a, b ∈ R.
9. Show that
1
0
log(sin πx) dx =− log 2.
[Hint: Use the contour shown in Figure 9.]
0 1
Figure 9. Contour in Exercise 9
10. Show that if a > 0, then
∞
0
log x
x2+ a2dx = π 2alog a.
[Hint: Use the contour in Figure 10.]
11. Show that if|a| < 1, then
2π
0
log|1 − aeiθ| dθ = 0.
8. Exercises 105
−R − R
ia
Figure 10. Contour in Exercise 10
Then, prove that the above result remains true if we assume only that|a| ≤ 1.
12. Suppose u is not an integer. Prove that
∞ n=−∞
1
(u + n)2 = π2 (sin πu)2 by integrating
f (z) = π cot πz (u + z)2
over the circle|z| = RN= N + 1/2 (N integral, N≥ |u|), adding the residues of f inside the circle, and letting N tend to infinity.
Note. Two other derivations of this identity, using Fourier series, were given in Book I.
13. Suppose f (z) is holomorphic in a punctured disc Dr(z0)− {z0}. Suppose also that
|f(z)| ≤ A|z − z0|−1+
for some > 0, and all z near z0. Show that the singularity of f at z0is removable.
14. Prove that all entire functions that are also injective take the form f (z) = az + b with a, b∈ C, and a = 0.
[Hint: Apply the Casorati-Weierstrass theorem to f (1/z).]
15. Use the Cauchy inequalities or the maximum modulus principle to solve the following problems:
(a) Prove that if f is an entire function that satisfies sup
|z|=R|f(z)| ≤ ARk+ B
106 Chapter 3. MEROMORPHIC FUNCTIONS AND THE LOGARITHM for all R > 0, and for some integer k≥ 0 and some constants A, B > 0, then f is a polynomial of degree≤ k.
(b) Show that if f is holomorphic in the unit disc, is bounded, and converges uniformly to zero in the sector θ < arg z < ϕ as|z| → 1, then f = 0.
(c) Let w1, . . . , wnbe points on the unit circle in the complex plane. Prove that there exists a point z on the unit circle such that the product of the distances from z to the points wj, 1≤ j ≤ n, is at least 1. Conclude that there exists a point w on the unit circle such that the product of the distances from w to the points wj, 1≤ j ≤ n, is exactly equal to 1.
(d) Show that if the real part of an entire function f is bounded, then f is constant.
16. Suppose f and g are holomorphic in a region containing the disc |z| ≤ 1.
Suppose that f has a simple zero at z = 0 and vanishes nowhere else in|z| ≤ 1.
Let
f(z) = f (z) + g(z).
Show that if is sufficiently small, then (a) f(z) has a unique zero in|z| ≤ 1, and
(b) if zis this zero, the mapping → zis continuous.
17. Let f be non-constant and holomorphic in an open set containing the closed unit disc.
(a) Show that if|f(z)| = 1 whenever |z| = 1, then the image of f contains the unit disc. [Hint: One must show that f (z) = w0has a root for every w0∈ D.
To do this, it suffices to show that f (z) = 0 has a root (why?). Use the maximum modulus principle to conclude.]
(b) If |f(z)| ≥ 1 whenever |z| = 1 and there exists a point z0∈ D such that
|f(z0)| < 1, then the image of f contains the unit disc.
18. Give another proof of the Cauchy integral formula f (z) = 1
2πi
C
f (ζ) ζ− zdζ using homotopy of curves.
[Hint: Deform the circle C to a small circle centered at z, and note that the quotient (f (ζ)− f(z))/(ζ − z) is bounded.]
19. Prove the maximum principle for harmonic functions, that is:
8. Exercises 107 (a) If u is a non-constant real-valued harmonic function in a region Ω, then u
cannot attain a maximum (or a minimum) in Ω.
(b) Suppose that Ω is a region with compact closure Ω. If u is harmonic in Ω and continuous in Ω, then
sup
z∈Ω
|u(z)| ≤ sup
z∈Ω−Ω
|u(z)|.
[Hint: To prove the first part, assume that u attains a local maximum at z0. Let f be holomorphic near z0 with u = Re(f ), and show that f is not open. The second part follows directly from the first.]
20. This exercise shows how the mean square convergence dominates the uniform convergence of analytic functions. If U is an open subset ofC we use the notation
fL2(U )=
U
|f(z)|2dxdy
1/2
for the mean square norm, and
fL∞(U )= sup
z∈U|f(z)|
for the sup norm.
(a) If f is holomorphic in a neighborhood of the disc Dr(z0), show that for any 0 < s < r there exists a constant C > 0 (which depends on s and r) such that
fL∞(Ds(z0))≤ CfL2(Dr(z0)).
(b) Prove that if {fn} is a Cauchy sequence of holomorphic functions in the mean square norm · L2(U ), then the sequence {fn} converges uniformly on every compact subset of U to a holomorphic function.
[Hint: Use the mean-value property.]
21. Certain sets have geometric properties that guarantee they are simply con-nected.
(a) An open set Ω⊂ C is convex if for any two points in Ω, the straight line segment between them is contained in Ω. Prove that a convex open set is simply connected.
(b) More generally, an open set Ω⊂ C is star-shaped if there exists a point z0 ∈ Ω such that for any z ∈ Ω, the straight line segment between z and z0
is contained in Ω. Prove that a star-shaped open set is simply connected.
Conclude that the slit planeC − {(−∞, 0]} (and more generally any sector, convex or not) is simply connected.
108 Chapter 3. MEROMORPHIC FUNCTIONS AND THE LOGARITHM (c) What are other examples of open sets that are simply connected?
22. Show that there is no holomorphic function f in the unit discD that extends continuously to ∂D such that f(z) = 1/z for z ∈ ∂D.
9 Problems
1.∗ Consider a holomorphic map on the unit disc f :D → C which satisfies f (0) = 0. By the open mapping theorem, the image f (D) contains a small disc centered at the origin. We then ask: does there exist r > 0 such that for all f :D → C with f(0) = 0, we have Dr(0)⊂ f(D)?
(a) Show that with no further restrictions on f , no such r exists. It suffices to find a sequence of functions{fn} holomorphic in D such that 1/n /∈ f(D).
Compute fn(0), and discuss.
(b) Assume in addition that f also satisfies f(0) = 1. Show that despite this new assumption, there exists no r > 0 satisfying the desired condition.
[Hint: Try f(z) = (ez/− 1).]
The Koebe-Bieberbach theorem states that if in addition to f (0) = 0 and f(0) = 1 we also assume that f is injective, then such an r exists and the best possible value is r = 1/4.
(c) As a first step, show that if h(z) = 1z+ c0+ c1z + c2z2+· · · is analytic and injective for 0 <|z| < 1, then∞
n=1n|cn|2≤ 1.
[Hint: Calculate the area of the complement of h(Dρ(0)− {0}) where 0 < ρ < 1, and let ρ→ 1.]
(d) If f (z) = z + a2z2+· · · satisfies the hypotheses of the theorem, show that there exists another function g satisfying the hypotheses of the theorem such that g2(z) = f (z2).
[Hint: f (z)/z is nowhere vanishing so there exists ψ such that ψ2(z) = f (z)/z and ψ(0) = 1. Check that g(z) = zψ(z2) is injective.]
(e) With the notation of the previous part, show that|a2| ≤ 2, and that equality holds if and only if
f (z) = z
(1− eiθz)2 for some θ∈ R.
[Hint: What is the power series expansion of 1/g(z)? Use part (c).]
(f) If h(z) =1z + c0+ c1z + c2z2+· · · is injective on D and avoids the values z1 and z2, show that|z1− z2| ≤ 4.
[Hint: Look at the second coefficient in the power series expansion of 1/(h(z)− zj).]
9. Problems 109 (g) Complete the proof of the theorem. [Hint: If f avoids w, then 1/f avoids 0
and 1/w.]
2. Let u be a harmonic function in the unit disc that is continuous on its closure.
Deduce Poisson’s integral formula
u(z0) = 1
and we recover the expression for the Poisson kernel derived in the exercises of the previous chapter.
[Hint: Set u0(z) = u(T (z)) where
T (z) = z0− z 1− z0z.
Prove that u0 is harmonic. Then apply the mean value theorem to u0, and make a change of variables in the integral.]
3. If f (z) is holomorphic in the deleted neighborhood{0 < |z − z0| < r} and has a pole of order k at z0, then we can write
f (z) = a−k
(z− z0)k+· · · + a−1
(z− z0)+ g(z)
where g is holomorphic in the disc{|z − z0| < r}. There is a generalization of this expansion that holds even if z0 is an essential singularity. This is a special case of the Laurent series expansion, which is valid in an even more general setting.
Let f be holomorphic in a region containing the annulus{z : r1≤ |z − z0| ≤ r2}
where the series converges absolutely in the interior of the annulus. To prove this, it suffices to write
f (z) = 1
Here Cr1 and Cr2 are the circles bounding the annulus.
110 Chapter 3. MEROMORPHIC FUNCTIONS AND THE LOGARITHM 4.∗Suppose Ω is a bounded region. Let L be a (two-way infinite) line that intersects Ω. Assume that Ω∩ L is an interval I. Choosing an orientation for L, we can define Ωl and Ωr to be the subregions of Ω lying strictly to the left or right of L, with Ω = Ωl∪ I ∪ Ωr a disjoint union. If Ωl and Ωr are simply connected, then Ω is simply connected.
5.∗Let
g(z) = 1 2πi
M
−M
h(x) x− zdx where h is continuous and supported in [−M, M].
(a) Prove that the function g is holomorphic in C − [−M, M], and vanishes at infinity, that is, lim|z|→∞|g(z)| = 0. Moreover, the “jump” of g across [−M, M] is h, that is,
h(x) = lim
→0,>0g(x + i )− g(x − i ).
[Hint: Express the difference g(x + i )− g(x − i ) in terms of a convolution of h with the Poisson kernel.]
(b) If h satisfies a mild smoothness condition, for instance a H¨older condition with exponent α, that is,|h(x) − h(y)| ≤ C|x − y|αfor some C > 0 and all x, y∈ [−M, M], then g(x + i ) and g(x − i ) converge uniformly to functions g+(x) and g−(x) as → 0. Then, g can be characterized as the unique holomorphic function that satisfies:
(i) g is holomorphic outside [−M, M], (ii) g vanishes at infinity,
(iii) g(x + i ) and g(x− i ) converge uniformly as → 0 to functions g+(x) and g−(x) with
g+(x)− g−(x) = h(x).
[Hint: If G is another function satisfying these conditions, g− G is entire.]