Stein_Complex_Analysis复分析答案

SOLUTIONS/HINTS TO THE EXERCISES FROM COMPLEX ANALYSIS BY STEIN AND SHAKARCHI ROBERT C. RHOADES Abstract. This contains the solutions or hints to many of the exercises from the Complex Analysis book by Elias Stein and Rami Shakarchi. I worked these problems during the Spring of 2006 while I was taking a Complex Analysis course taught by Andreas Seeger at the University of Wisconsin - Madison. I am grateful to him for his wonderful lectures and helpful conversations about some of the problems discussed below. Contents 1. Chapter 1. Preliminaries to Complex Analysis 2 2. Chapter 2. Cauchy’s Theorem and Its Applications 8 3. Chapter 3. Meromorphic Functions and the Logarithm 9 4. Chapter 4. The Fourier Transform 10 5. Chapter 5: Entire Functions 11 6. Chapter 6. The Gamma and Zeta Functions 13 7. Chapter 7: The Zeta Function and Prime Number Theorem 17 8. Chapter 8: Conformal Mappings 20 9. Chapter 9: An Introduction to Elliptic Functions 23 10. Chapter 10: Applications of Theta Functions 25 Date: September 5, 2006. The author is thankful for an NSF graduate research fellowship and a National Physical Science Consortium graduate fellowship supported by the NSA. 1 He 高亮 2 ROBERT C. RHOADES 1. Chapter 1. Preliminaries to Complex Analysis Exercise 1. Describe geometrically the sets of points z in the complex plane defined by the following relations: (1) |z − z1| = |z − z2| where z1, z2 ∈ C. (2) 1/z = z. (3) Re(z) = 3. (4) Re(z) > c, (resp., ≥ c) where c ∈ R. (5) Re(az + b) > 0 where a, b ∈ C. (6) |z| = Re(z) + 1. (7) Im(z) = c with c ∈ R. Solution 1. (1) It is the line in the complex plane consisting of all points that are an equal distance from both z1 and z2. Equivalently the perpendicular bisector of the segment between z1 and z2 in the complex plane. (2) It is the unit circle. (3) It is the line where all the numbers on the line have real part equal to 3. (4) In the first case it is the open half plane with all numbers with real part greater than c. In the second case it is the closed half plane with the same condition. (5) (6) Calculate |z|2 = x2 + y2 = (x+ 1)2 = x2 + 2x + 1. So we are left with y2 = 2x + 1. Thus the complex numbers defined by this relation is a parabola opening to the “right”. (7) This is a line. Exercise 2. Let 〈·, ·〉 denote the usual inner product in R2. In other words, if Z = (z1, y1) and W = (x2, y2), then 〈Z,W 〉 = x1x2 + y1y2. Similarly, we may define a Hermitian inner product (·, ·) in C by (z, w) = zw. The term Hermitian is used to describe the fact that (·, ·) is not symmetric, but rather satisfies the relation (z, w) = (w, z) for all z, w ∈ C. Show that 〈z, w〉 = 1 2 [(z, w) + (w, z)] = Re(z, w), where we use the usual identification z = x+ iy ∈ C with (x, y) ∈ R2. Solution 2. This is a straightforward calculation 1 2 [(z, w) + (w, z)] = 1 2 (zw + wz) = Re(z, w)Re(zw) = 1 2 ((z1 + z2i)(w1 + iw2) + (w1 + iw2)(z1 − iz2)) = z1w1 + z2w2. Exercise 3. With ω = seiφ, where s ≥ 0 and φ ∈ R, solve the equation zn = ω in C where n is a natural number. How many solutions are there? SOLUTIONS/HINTS TO THE EXERCISES FROM COMPLEX ANALYSIS BY STEIN AND SHAKARCHI 3 Solution 3.zn = seiφ implies that z = s 1 n ei( φ n+ 2piik n ), where k = 0, 1, · · · , n − 1 and s 1n is the real nth root of the positive number s. There are n solutions as there should be since we are finding the roots of a degree n polynomial in the algebraically closed field C. Exercise 4. Show that it is impossible to define a total ordering on C. In other words, one cannot find a relation � between complex numbers so that: (1) For any two complex numbers z, w, one and only one of the following is true: z � w, w � z or z = w. (2) For all z1, z2, z3 ∈ C the relation z1 � z2 implies z1 + z3 � z2 + z3. (3) Moreover, for all z1, z2, z3 ∈ C with z3 � 0, then z1 � z2 implies z1z3 � z2z3. Solution 4. Suppose, for a contradiction, that i � 0, then −1 = i · i � 0 · i = 0. Now we get −i � −1 · i � 0. Therefore i − i � i + 0 = i. But this contradicts our assumption. We obtain a similar situation in the case 0 � i. So we must have i = 0. But then for all z ∈ C we have z · i = z · 0 = 0 Repeating we have z = 0 for all z ∈ C. So this relation would give a trivial total ordering. Exercise 5. A set Ω is said to be pathwise connected if any two points in Ω can be joined by a (piecewise-smooth) curve entirely contained in Ω. The purpose of this exercise is to prove that an open set Ω is pathwise connected if and only if Ω is connected. (1) Suppose first that Ω is open and pathwise connected, and that it can be written as Ω = Ω1 ∪ Ω2 where Ω1 and Ω2 are disjoint non-empty open sets. Choose two points ω1 ∈ Ω1 and ω2 ∈ Ω2 and let γ denote a curve in Ω joining ω1 to ω2. Consider a parametrization z : [0, 1]→ Ω of this curve with z(0) = ω1 and z(1) = ω2, and let t∗ = sup 0≤t≤1 {t : z(s) ∈ Ω1 for all 0 ≤ s < t}. Arrive at a contradiction by considering the point z(t∗). (2) Conversely, suppose that Ω is open and connected. Fix a point w ∈ Ω and let Ω1 ⊂ Ω denote the set of all points that can be joined to w by a curve contained in Ω. Also, let Ω2 ⊂ Ω denote the set of all points that cannot be joined to w by a curve in Ω. Prove that both Ω1 and Ω2 are open, disjoint and their union is Ω. Finally, since Ω1 is non-empty (why?) conclude that Ω = Ω1 as desired. The proof actually shows that the regularity and type of curves we used to define pathwise connectedness can be relaxed without changing the equivalence between the two definitions when Ω is open. For instance, we may take all curves to be continuous, or simply polygonal lines. Solution 5. Following the first part, assume for a contradiction that z(t∗) ∈ Ω1. Since Ω1 is open there exists a ball B(z(t∗), δ) ⊂ Ω1. Now by assumption z(t∗ + �) ∈ Ω2. Thus |z(t∗ + �)− z(t∗)| > δ for all � > 0. But this is a contradiction since z is smooth. Define Ω1 and Ω2 as in the problem. First to see that Ω1 is open let z ∈ Ω1. Then since Ω is open and z ∈ Ω we know that there exists a ball B(z, δ ⊂ Ω. We claim that this ball is actually inside of Ω1. If we prove this claim then we have established that Ω1 is open. Let s ∈ B(z, δ) and consider f : [0, 1] → C given by f(t) = st + z(1 − t). Then |f(t) − z| = t|s − z| < δ. So the image of f is contained in B(z, δ) ⊂ Ω. By concatenating the paths from w to z and z to s we see that s ∈ Ω1. Finally we will prove that Ω2 is also open. Suppose that Ω2 is not open. Then for there is some z ∈ Ω2 such that every ball around z contains a point of Ω1. So that B(z, δ) ⊂ Ω is one such ball, with s ∈ Ω1∩B(z, δ). Then as in the previous paragraph we can use the straight line path to connect 4 ROBERT C. RHOADES z to s and the path has imagine inside B(z, δ) ⊂ Ω. Therefore, w is path connected to s which is path connected to z. Therefore, by concatenating paths, we see that z ∈ Ω1 which contradicts the definition of Ω2. So Ω2 must be open. Now Ω1 is non-empty since w ∈ Ω1. Therefore, by connectedness, Ω2 = ∅. Remark. This argument works in any metric space. Exercise 6. Let Ω be an open set in C and z ∈ Ω. The connected component (or simply the component) of Ω containing z is the set Cz of all points w in Ω that can be joined to z by a curve entirely contained in Ω. (1) Check first that Cz is open and connected. Then, show that w ∈ Cz defines an equivalence relation, that is (i) z ∈ Cz, (ii) w ∈ Cz implies z ∈ Cw, and (iii) if w ∈ Cz and z ∈ Cζ , then w ∈ Cζ . Thus Ω is the union of all its connected components, and two components are either disjoint or coincide. (2) Show that Ω can have only countably many distinct connected components. (3) Prove that if Ω is the complement of a compact set, then Ω has only one unbounded com- ponent. Solution 6. (1) (i) the trivial path works. (ii) Running the path in reverse works. (iii) We have a path from ζ to z and from z to w. Concatenating the paths gets the job done. (2) The set of all elements of the form q + iq′ where q, q′ ∈ Q is countable. Each component contains a point of the form q+ iq′, since each Cz is open, we can be seen from the previous exercise. (3) IfK is compact then it is closed and bounded. So it is contained in an open disc with bounded radius and center the origin. So then the complement of that open disc is contained in Ω. Then if Ω is not connected it must have a component contained in the large disc. But thus it is bounded. So we see that Ω can have at most one unbounded component. Exercise 7. The family of mappings introduced here plays an important role in complex analysis. These mappings, sometimes called Blaschke factors, will reappear in various applications in later chapters. (1) Let z, w be two complex numbers such that zw 6= 1. Prove that∣∣∣∣ w − z1− wz ∣∣∣∣ < 1 if |z| < 1 and |w| < 1, and also that ∣∣∣∣ w − z1− wz ∣∣∣∣ = 1 if |z| = 1 or |w| = 1. (2) Prove that for a fixed w in the unit disc D, the mapping F : z 7→ w − z 1− wz satisfies the following conditions (a) F maps the unit disc to itself (that is, F : D→ D), and is holomorphic. (b) F interchanges 0 and w, namely F (0) = w and F (w) = 0. (c) |F (z)| = 1 if |z| = 1. SOLUTIONS/HINTS TO THE EXERCISES FROM COMPLEX ANALYSIS BY STEIN AND SHAKARCHI 5 (d) F : D→ D is bijective. Solution 7. (a) Suppose that |w| < 1 and |z| = 1, then we have | w − z 1− wz | = | w − z z − w | = 1, since |frac1z| = 1. Since |w| < 1 we see that the function f(z) := w−z1−wz is holomorphic in D. Thus by the maximum modulus principle it satisfies |f(z)| < 1 in D because it is non-constant. A straightforward calculation can also give the result. (b) We already showed that F (D) ⊂ D. Clearly, F (0) = w and F (w) = 0. Also from (a) we had F (∂D) ⊆ ∂D. Bijective can be shown by computing F−1. Exercise 8. Suppose U and V are open sets in the complex plane. Prove that if f : U → V and g : V → C are two functions that are differentiable (in the real sense, that is, as functions of the two real variables x and y), and h = g ◦ f , then ∂h ∂z = ∂g ∂z ∂f ∂z + ∂g ∂z ∂f ∂z and ∂h ∂z = ∂g ∂z ∂f ∂z + ∂g ∂z ∂f ∂z . This is the complex version of the chain rule. Solution 8. We see have ∂11 ∂111 gz ∂11 ∂111 fz = Exercise 9. Show that in polar coordinates, the Cauchy-Riemann equations take the form ∂u ∂r = 1 r ∂v ∂θ and 1 r ∂u ∂θ = −∂v ∂r . Use these equations to show that the logarithm function defined by log(z) = log(r) + iθ where z = reiθ with − pi < θ < pi is holomorphic in the region r > 0 and −pi < θ < pi. Here the second logarithm is the standard real valued one. Solution 9. Exercise 10. Show that 4 ∂ ∂z ∂ ∂z = 4 ∂ ∂z ∂ ∂z = ∆, where ∆ is the Laplacian ∆ = ∂2 ∂x2 + ∂2 ∂y2 . Solution 10. Exercise 11. Use exercise 10 to prove that if f is holomorphic in the open set Ω, then the real and imaginary parts of f are harmonic; that is, their Laplacian is zero. 6 ROBERT C. RHOADES Solution 11. Exercise 12. Consider the function defined by f(x+ iy) = √ |x||y|, where x, y ∈ R. Show that f satisfies the Cauchy-Riemann equations at the origin, yet f is not holomorphic at 0. Exercise 13. Suppose that f is holomorphic in an open set Ω. Prove that in any one of the following cases: (1) Re(f) is constant (2) Im(f) is constant; (3) |f | is constant; one can conclude that f is constant. Exercise 14. Suppose {an}Nn=1 and {bn}Nn=1 are two finite sequences of complex numbers. Let Bk = ∑k n=1 bn denote the partial sums of the series ∑ bn with the convention B0 = 0. Prove the summation by parts formula N∑ n=M anbn = aNBN − aMBM−1 − N−1∑ n=M (an+1 − an)Bn. Exercise 15. Abel’s theorem. Suppose ∑∞ n=1 an converges. Prove that lim r→1,r<1 ∞∑ n=1 rnan = ∞∑ n=1 an. Exercise 16. Determine the radius of convergence of the series ∑∞ n=1 anz n when: (1) an = (log n)2 (2) an = n! (3) an = n 2 4n+3n (4) an = (n!)3/(3n)! (5) Find the radius of convergence of the hypergeometric series F (α, β, γ; z) = 1 + ∞∑ n=1 α(α+ 1) · · · (α+ n− 1)β(β + 1) · · · (β + n− 1) n!γ(γ + 1) · · · (γ + n− 1) z n. Here α, β ∈ C and γ 6= 0,−1,−2, · · · (6) Find the radius of convergence of the Bessel function of order r: Jr = (z 2 )r ∞∑ n=0 (−1)n n!(n+ r)! (z 2 )2n , where r is a positive integer. Exercise 17. Show that if {an}∞n=0 is a sequence of non-zero complex numbers such that lim n→∞ |an+1| |an| = L, SOLUTIONS/HINTS TO THE EXERCISES FROM COMPLEX ANALYSIS BY STEIN AND SHAKARCHI 7 then lim n→∞ |an| 1 n . In particular, this exercise shows that when applicable, the ratio test can be used to calculate the radius of convergence of a power series. Exercise 18. Let f be a power series centered at the origin. Prove that f has a power series expansion around any point in its disc of convergence. Exercise 19. Prove the following: (1) The power series ∑ nzn does not converge on any point of the unit circle. (2) The power series ∑ zn/n2 converges at every point of the unit circle (3) The power series ∑ zn/n converges at every point of the unit circle except z = 1. 8 ROBERT C. RHOADES 2. Chapter 2. Cauchy’s Theorem and Its Applications SOLUTIONS/HINTS TO THE EXERCISES FROM COMPLEX ANALYSIS BY STEIN AND SHAKARCHI 9 3. Chapter 3. Meromorphic Functions and the Logarithm 10 ROBERT C. RHOADES 4. Chapter 4. The Fourier Transform Solution 1. SOLUTIONS/HINTS TO THE EXERCISES FROM COMPLEX ANALYSIS BY STEIN AND SHAKARCHI 11 5. Chapter 5: Entire Functions Solution 1. We follow the proof of Jensen’s formula that is given in the book. We keep step 1 exactly the same. Following step 2, we have set g = f/ψ1 · ψN , then g is holomorphic and bounded near each zj . So it suffices to prove the theorem for Blaschke factors and for bounded functions that vanish nowhere. Functions that vanish nowhere are treated in Step 3. It remains to show the result for Blaschke factors. We have log |ψα(0)| = log |α| = log |α|+ 12pi ∫ 2pi 0 log |ψα(eiθ)|dθ, since |ψα(z)| = 1 for z ∈ ∂D. Solution 2. (a) Noting that |zn| = O(ez�) for all � > 0 we see that the order of growth of p is 0. (b) Clearly the order of growth is n. (c) The order of growth is ee z is infinite. Since |z|k = O(ez) we know that there is no k and constants A and B, such that ee |z| ≤ AeBzk . Solution 3. Solution 4. For (a) see the hint. For (b): let r = min 1, t and R = max 1, t. So there are eight (m,n) such that r ≤ |nit+m| ≤ R, namely (0,±1), (±1, 0), (±1,±1). There are 16 (m,n) such that 2r ≤ |nit+m| ≤ 2R. In general there are 8k tuples with kr ≤ |nit+m| ≤ kR. Thus with S(k) := ∑ (m,n) with max(m,n)≤k 1 |int+m|α , we have 8 Rα k∑ j=1 1 jα−1 ≤ S(k) ≤ 8 rα−1 k∑ j=1 1 jα−1 . Solution 5.To show entire let CR(z0) be a circle of radius R centered at z0. Then we have∫ C1(z0) Fα(z)dz = ∫ 2pi 0 ∫ ∞ −∞ e−|t| α+2piiz0t+e iθ2piitdtieiθdθ = ∫ ∞ −∞ e−|t| α+2piiz0t ∫ 2pi 0 e2piie iθtieiθdθdt = 0. We can switch the order of integration by Fubini’s theorem since the integrand is L1. The final equality is established since ∫ C(z0) e2piiztdz = 0. This shows that the function is holormophic in any disc by Morera’s theorem. Thus it is entire. Solution 6. By the product formula for sine we have sin(pi/2) = pi 2 ∞∏ n=1 ( 1− 1 4n2 ) = pi 2 ∞∏ n=1 ( (2n+ 1)(2n− 1) (2n)(2n) ) . Solution 7. Solution 8. 12 ROBERT C. RHOADES Solution 9. Solution 10.(a) ez − 1 is order 1 and has zeros precisely when z = 2piin for n ∈ Z. Thus we have ez − 1 = eAz+Bz ∏ n 6=0 ( 1− z 2piin ) ez/2piin = eAz+Bz ∏ n≥1 ( 1 + z2 4pi2n2 ) . Multiplying this equation by e−z/2 we are left with two expressions both of which must be odd. Thus we see that we must have A = 1/2. Also considering limz→0(ez − 1)/z = 1 we see that we must have B = 0. (b) cos(piz) is also order 1 and has zeros at 2n+12 for all n ∈ Z. Thus we are left with cos(piz) = eAz+B ∏ n∈Z ( 1− 2z (2n+ 1) ) e2z/(2n+1) = eAz+B ∏ n≥1 ( 1− 4z 2 (2n+ 1)2 ) . Since cosine is even we see that A = 0. Also letting z = 0 we see that we must have B = 0. Solution 11. By Hadamard’s theorem if it omits a and b, then we have f(z) − a = ep(z) and f(z) − b = eq(z) for polynomials p and q. Then ep(z) − eq(z) = C for some constant C. Letting z tend to infinity we see that the leading terms of the polynomials p and q must be the same. Say it is anzn. Then, considering the limit as z tends to infinity of ep(z)−anz n − eq(z)−anzn = Ce−anzn , we see that the next leading terms are also equal. Proceeding by induction shows that p = q, but then this is a contradiction since it would imply that b = a. But we assume they are distinct. Solution 12. If f has finite order and never vanishes then we have f(z) = Cep(z) for some polynomial, this follows from Hadamard’s theorem. So then f ′(z) = Cp′(z)ep(z). Since we assume that the derivatives are never 0, we see that p′ is a constant function. Solution 13. ez − z is entire of order 1. So ez − z = eAz+B∏n (1− zan) ez/an . If there are only finitely many zeros then we have ez − z = eA˜z+B˜Q(z) for some polynomial Q. But then Q(z) = ez − z eA˜z+B˜ = O ( e(1−A˜)z ) . This is only possible if Q is a constant and 1 − A˜ = 0. In that case we have z = Cez for some constant C = 1− eB˜ , but this is clearly not possible. Solution 14. Let k < ρ < k + 1, and say we have only finitely many zeros, then as in the previous problem, F (z) = ep(z)Q(z) for some polynomial p of degree at most k and Q a polynomial. Now F has order of growth less than or equal to k, since Q has order of growth 0. This contradicts the assumption that F has order of growth ρ. Solution 15. Every meromorphic function, f , is holomorphic with poles at some sequence p1, p2, . . .. Then there is an entire function with zeros precisely at the pj with desired multiplicity. Call this function g. Then fg has no poles and is holomorphic except possibly at the pj , but since there are no poles at the pj , we see that we have an entire function, say h. Then f = g/h. For the second part of the problem construct two entire functions and take their quotient. Solution 16. SOLUTIONS/HINTS TO THE EXERCISES FROM COMPLEX ANALYSIS BY STEIN AND SHAKARCHI 13 6. Chapter 6. The Gamma and Zeta Functions Solution 1. We have 1 Γ(s) = eγss ∏ n≥1 ( 1 + s n ) e−s/n = lim N→∞ es( PN n=1 1 n−logN)s N∏ n=1 n+ s n e−s/n = lim N→∞ elogN −s N ! s(s+1) · · · (s+N). Solution 2. The easier part is to deduce the identity from the product expansion of sine and the desired identity from this problem. To do so first divide by (1 + a + b), we want to make the substation a = s− 1 and b = −s. But to do so we must substitute a = s− δ and b = −s and then let δ tend to 1 and use the fact that Γ(x)/x→ 1 as x→ 0. We are then left with 1 (1− s)s ∏ n>1 n(n− 1) (n− s)(n− 1− s) = 1 (s− 1)s ∏ n>1 1 (1− s/n) (1 + s/(n− 1)) . Thus we may deduce that (Γ(s)Γ(1− s))−1 = s ∏ n≥1 ( 1− s n ) ∏ n>1 ( 1 + s n+ 1 ) = s ∞∏ n=1 ( 1− s 2 n2 ) = sin(pis) pi . This is the desired result. To prove the identity use Lemma 1.2 on page 161 and induction to establish that (6.1) Γ(a+ 1)Γ(b+ 1) Γ(a+ b+ 1) = ( N ! N∏ n=1 (a+ b+ n) (a+ n)(b+ n) ) Γ(a+N + 1)Γ(b+N + 1) N !Γ(a+ b+N + 1) . Now let a′, b′ be integers such that a′ ≤ a ≤ a′ + 1, b′ ≤ b ≤ b′ + 1, a′ + b′ ≤ a + b < a′ + b′ + 1. Then we have the inequalities (a′ +N + 1)!(b′ +N + 1)! N ! (a′ + b′ +N + 2)! ≤ Γ(a+N + 1)Γ(b+N + 1) N !Γ(a+ b+N + 1) ≤ (a ′ +N + 2)!(b′ +N + 2)! N ! (a′ + b′ +N + 1)! . We claim that for any A and B non-negative integers, lim N→∞ (A+N)!(B +N)! N !(A+B +N)! = 1. With this in hand or a slight modification of this we can easily see that letting N go to infinity in (6.1), gives the result. To prove the necessary limit use Stirling’s approximation to the factorial and standard analysis. One useful limit to know is( 1 + α N2 )N → 1, as N →∞. It occured to me later that exercise 1 can be used prove this result fairly easily so long as you can prove existence of the infinity product. Solution 3. Notice that ∏m j=1(2j + 1)