Exam (elaborations) TEST BANK FOR Functions of One Complex Variable I By Andreas Kleefeld

Exam (elaborations) TEST BANK FOR Functions of One Complex Variable I By Andreas Kleefeld Solutions Manual for Functions of One Complex Variable I, Second Edition1 © Copyright by Andreas Kleefeld, 2009 All Rights Reserved2 1by John B. Conway 2Last updated on January, 7th 2013 Contents 1 The Complex Number System 1 1.1 The real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The field of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 The complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Polar representation and roots of complex numbers . . . . . . . . . . . . . . . . . . . . . 5 1.5 Lines and half planes in the complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 The extended plane and its spherical representation . . . . . . . . . . . . . . . . . . . . . 7 2 Metric Spaces and the Topology of C 9 2.1 Definitions and examples of metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Sequences and completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Elementary Properties and Examples of Analytic Functions 21 3.1 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Analytic functions as mappings. Möbius transformations . . . . . . . . . . . . . . . . . . 31 4 Complex Integration 42 4.1 Riemann-Stieltjes integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Power series representation of analytic functions . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Zeros of an analytic function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 The index of a closed curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5 Cauchy’s Theorem and Integral Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.6 The homotopic version of Cauchy’s Theorem and simple connectivity . . . . . . . . . . . 63 4.7 Counting zeros; the Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . 66 4.8 Goursat’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 Singularities 68 5.1 Classification of singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3 The Argument Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3 6 The Maximum Modulus Theorem 84 6.1 The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2 Schwarz’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.3 Convex functions and Hadamard’s Three Circles Theorem . . . . . . . . . . . . . . . . . 88 6.4 The Phragmén-Lindelöf Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7 Compactness and Convergence in the Space of Analytic Functions 93 7.1 The space of continuous functions C(G, ) . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2 Spaces of analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.3 Spaces of meromorphic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.4 The Riemann Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.5 The Weierstrass Factorization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.6 Factorization of the sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.7 The gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.8 The Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8 Runge’s Theorem 123 8.1 Runge’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.2 Simple connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.3 Mittag-Leffler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 9 Analytic Continuation and Riemann Surfaces 130 9.1 Schwarz Reflection Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 9.2 Analytic Continuation Along a Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.3 Monodromy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9.4 Topological Spaces and Neighborhood Systems . . . . . . . . . . . . . . . . . . . . . . . 132 9.5 The Sheaf of Germs of Analytic Functions on an Open Set . . . . . . . . . . . . . . . . . 133 9.6 Analytic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 9.7 Covering spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10 Harmonic Functions 137 10.1 Basic properties of harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 10.2 Harmonic functions on a disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 10.3 Subharmonic and superharmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . 144 10.4 The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 10.5 Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 11 Entire Functions 151 11.1 Jensen’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 11.2 The genus and order of an entire function . . . . . . . . . . . . . . . . . . . . . . . . . . 153 11.3 Hadamard Factorization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 12 The Range of an Analytic Function 161 12.1 Bloch’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.2 The Little Picard Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 12.3 Schottky’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 12.4 The Great Picard Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4 Chapter 1 The Complex Number System 1.1 The real numbers No exercises are assigned in this section. 1.2 The field of complex numbers Exercise 1. Find the real and imaginary parts of the following: 1 z ; z − a z + a (a 2 R); z3; 3 + 5i 7i + 1 ; 0BBBB@ −1 + ip3 2 1CCCCA 3 ; 0BBBB@ −1 − ip3 2 1CCCCA 6 ; in; 1 + i p2 !n for 2  n  8. Solution. Let z = x + iy. Then a) Re 1 z ! = x x2 + y2 = Re(z) |z|2 Im 1 z ! = − y x2 + y2 = − Im(z) |z|2 b) Re  z − a z + a  = x2 + y2 − a2 x2 + y2 + 2ax + a2 = |z|2 − a2 |z|2 + 2aRe(z) + a2 Im  z − a z + a  = 2ya x2 + y2 + 2xa + a2 = 2Im(z)a |z|2 + 2aRe(z) + a2 c) Re  z3  = x3 − 3xy2 = Re(z)3 − 3Re(z)Im(z)2 Im  z3  = 3x2y − y3 = 3Re(z)2Im(z) − Im(z)3 1 d) Re 3 + 5i 7i + 1 ! = 19 25 Im 3 + 5i 7i + 1 ! = − 8 25 e) Re 0BBBBBB@ 0BBBB@ −1 + ip3 2 1CCCCA 31CCCCCCA = 1 Im 0BBBBBB@ 0BBBB@ −1 + ip3 2 1CCCCA 31CCCCCCA = 0 f) Re 0BBBBBB@ 0BBBB@ −1 − ip3 2 1CCCCA 61CCCCCCA = 1 Im 0BBBBBB@ 0BBBB@ −1 − ip3 2 1CCCCA 61CCCCCCA = 0 g) Re (in) = 8>>>>><>>>>>: 0, n is odd 1, n 2 {4k : k 2 Z} −1, n 2 {2 + 4k : k 2 Z} Im(in) = 8>>>>><>>>>>: 0, n is even 1, n 2 {1 + 4k : k 2 Z} −1, n 2 {3 + 4k : k 2 Z} 2 h) Re 1 + i p2 !n! = 8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>: 0, n = 2 − p2 2 , n = 3 −1, n = 4 − p2 2 , n = 5 0, n = 6 p2 2 , n = 7 1, n = 8 Im 1 + i p2 !n! = 8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>: 1, n = 2 p2 2 , n = 3 0, n = 4 − p2 2 , n = 5 −1, n = 6 − p2 2 , n = 7 0, n = 8 Exercise 2. Find the absolute value and conjugate of each of the following: −2 + i; −3; (2 + i)(4 + 3i); 3 − i p2 + 3i ; i i + 3 ; (1 + i)6; i17. Solution. It is easy to calculate: a) z = −2 + i, |z| = p5, ¯z = −2 − i b) z = −3, |z| = 3, ¯z = −3 c) z = (2 + i)(4 + 3i) = 5 + 10i, |z| = 5p5, ¯z = 5 − 10i d z = 3 − i p2 + 3i , |z| = 1 11 p110, ¯z = 3 + i p2 − 3i e) z = i i + 3 = 1 10 + 3 10 i, |z| = 1 10 p10, ¯z = 1 10 − 3 10 i f) z = (1 + i)6 = −8i, |z| = 8, ¯z = 8i g) i17 = i, |z| = 1, ¯z = −i Exercise 3. Show that z is a real number if and only if z = ¯z. 3 Solution. Let z = x + iy. ): If z is a real number, then z = x (y = 0). This implies ¯z = x and therefore z = ¯z. ,: If z = ¯z, then we must have x + iy = x − iy for all x, y 2 R. This implies y = −y which is true if y = 0 and therefore z = x. This means that z is a real number. Exercise 4. If z and w are complex numbers, prove the following equations: |z + w|2 = |z|2 + 2Re(zw¯ ) + |w|2. |z − w|2 = |z|2 − 2Re(zw¯ ) + |w|2. |z + w|2 + |z − w|2 = 2  |z|2 + |w|2  . Solution. We can easily verify that ¯¯z = z. Thus |z + w|2 = (z + w)(z + w) = (z + w)(z¯ + w¯ ) = zz¯ + zw¯ + wz¯ + ww¯ = |z|2 + |w|2 + zw¯ + z¯w = |z|2 + |w|2 + z ¯ w + ¯z¯¯w = |z|2 + |w|2 + zw¯ + zw¯ = |z|2 + |w|2 + 2 zw¯ + zw¯ 2 = |z|2 + |w|2 + 2Re(zw¯ ) = |z|2 + 2Re(zw¯ ) + |w|2. |z − w|2 = (z − w)(z − w) = (z − w)(z¯ − w¯ ) = zz¯ − zw¯ − wz¯ + ww¯ = |z|2 + |w|2 − zw¯ − z¯w = |z|2 + |w|2 − z ¯ w − ¯z¯¯w = |z|2 + |w|2 − zw¯ − zw¯ = |z|2 + |w|2 − 2 zw¯ + zw¯ 2 = |z|2 + |w|2 − 2Re(zw¯ ) = |z|2 − 2Re(zw¯ ) + |w|2. |z + w|2 + |z − w|2 = |z|2 + Re(zw¯ ) + |w|2 + |z|2 − Re(zw¯ ) + |w|2 = |z|2 + |w|2 + |z|2 + |w|2 = 2|z|2 + 2|w|2 = 2  |z|2 + |w|2  . Exercise 5. Use induction to prove that for z = z1 + . . . + zn; w = w1w2 . . .wn: |w| = |w1| . . . |wn|; z¯ = z¯1 + . . . + z¯n;w¯ = w¯ 1 . . . w¯ n. Solution. Not available. Exercise 6. Let R(z) be a rational function of z. Show that R(z) = R(¯z) if all the coefficients in R(z) are real. Solution. Let R(z) be a rational function of z, that is R(z) = anzn + an−1zn−1 + . . . a0 bmzm + bm−1zm−1 + . . . b0 where n,m are nonnegative integers. Let all coefficients of R(z) be real, that is a0, a1, . . . , an, b0, b1, . . . , bm 2 R. Then R(z) = anzn + an−1zn−1 + . . . a0 bmzm + bm−1zm−1 + . . . b0 = anzn + an−1zn−1 + . . . a0 bmzm + bm−1zm−1 + . . . b0 = anzn + an−1zn−1 + . . . a0 bmzm + bm−1zm−1 + . . . b0 = an ¯zn + an−1 ¯zn−1 + . . . a0 bm¯zm + bm−1 ¯zm−1 + . . . b0 = R(¯z). 4 1.3 The complex plane Exercise 1. Prove (3.4) and give necessary and sufficient conditions for equality. Solution. Let z and w be complex numbers. Then ||z| − |w|| = ||z − w + w| − |w||  ||z − w| + |w| − |w|| = ||z − w|| = |z − w| Notice that |z| and |w| is the distance from z and w, respectively, to the origin while |z − w| is the distance between z and w. Considering the construction of the implied triangle, in order to guarantee equality, it is necessary and sufficient that ||z| − |w|| = |z − w| () (|z| − |w|)2 = |z − w|2 () (|z| − |w|)2 = |z|2 − 2Re(zw¯ ) + |w|2 () |z|2 − 2|z||w| + |w|2 = |z|2 − 2Re(zw¯ ) + |w|2 () |z||w| = Re(zw¯ ) () |zw¯ | = Re(zw¯ ) Equivalently, this is zw¯  0. Multiplying this by w w , we get zw¯ · w w = |w|2 · zw  0 if w , 0. If t = zw =  1 |w|2  · |w|2 · zw . Then t  0 and z = tw. Exercise 2. Show that equality occurs in (3.3) if and only if zk/zl  0 for any integers k and l, 1  k, l  n, for which zl , 0. Solution. Not available. Exercise 3. Let a 2 R and c > 0 be fixed. Describe the set of points z satisfying |z − a| − |z + a| = 2c for every possible choice of a and c. Now let a be any complex number and, using a rotation of the plane, describe the locus of points satisfying the above equation. Solution. Not available. 1.4 Polar representation and roots of complex numbers Exercise 1. Find the sixth roots of unity. Solution. Start with z6 = 1 and z = rcis(), therefore r6cis(6) = 1. Hence r = 1 and  = 2k 6 with k 2 {−3, −2, −1, 0, 1, 2}. The following table gives a list of principle values of arguments and the corresponding value of the root of the equation z6 = 1. 0 = 0 z0 = 1 1 =  3 z1 = cis( 3 ) 2 = 2 3 z2 = cis( 2 3 ) 3 =  z3 = cis() = −1 4 = −2 3 z4 = cis( −2 3 ) 5 = − 3 z5 = cis( − 3 ) 5 Exercise 2. Calculate the following: a) the square roots of i b) the cube roots of i c) the square roots of p3 + 3i Solution. c) The square roots of p3 + 3i. Let z = p3 + 3i. Then r = |z| = qp3 2 + 32 = p12 and  = tan−1  3 p3  =  3 . So, the 2 distinct roots of z are given by p2 r  cos +2k n + i sin +2k n  where k = 0, 1. Specifically, pz = p4 12 cos 3 + 2k 2 + i sin 3 + 2k 2 ! . Therefore, the square roots of z, zk, are given by z0 = p4 12  cos  6 + i sin  6  = p4 12  p3 2 + 1 2 i  z1 = p4 12  cos 7 6 + i sin 7 6  = p4 12  − p3 2 − 1 2 i  . So, in rectangular form, the second roots of z are given by  4p108 2 , 4p12 2  and  − 4p108 2 , − 4p12 2  . Exercise 3. A primitive nth root of unity is a complex number a such that 1, a, a2, ..., an−1 are distinct nth roots of unity. Show that if a and b are primitive nth and mth roots of unity, respectively, then ab is a kth root of unity for some integer k. What is the smallest value of k? What can be said if a and b are nonprimitive roots of unity? Solution. Not available. Exercise 4. Use the binomial equation (a + b)n = Xn k=0 n k ! an−kbk, where n k ! = n! k!(n − k)! , and compare the real and imaginary parts of each side of de Moivre’s formula to obtain the formulas: cos n = cosn  − n 2 ! cosn−2  sin2  + n 4 ! cosn−4  sin4  − . . . sin n = n 1 ! cosn−1  sin  − n 3 ! cosn−3  sin3  + . . . Solution. Not available. Exercise 5. Let z = cis 2 n for an integer n  2. Show that 1 + z + . . . + zn−1 = 0. 6 Solution. The summation of the finite geometric sequence 1, z, z2, . . . , zn−1 can be calculated as Pn j=1 z j−1 = zn−1 z−1 . We want to show that zn is an nth root of unity. So, using de Moivre’s formula, zn =  cis  2 n n = cis  n · 2 n  = cis(2) = 1. It follows that 1 + z + z2 + ... + zn−1 = zn−1 z−1 = 1−1 z−1 = 0 as required. Exercise 6. Show that '(t) = cis t is a group homomorphism of the additive group R onto the multiplicative group T = {z : |z| = 1}. Solution. Not available. Exercise 7. If z 2 C and Re(zn)  0 for every positive integer n, show that z is a non-negative real number. Solution. Let n be an arbitrary but fixed positive integer and let z 2 C and Re(zn)  0. Since zn = rn(cos(n) + i sin(n)), we have Re(zn) = rn cos(n)  0. If z = 0, then we are done, since r = 0 and Re(zn) = 0. Therefore, assume z , 0, then r > 0. Thus Re(zn) = rn cos(n)  0 implies cos(n)  0 for all n. This implies  = 0 as we will show next. Clearly,  < [/2, 3/2]. If  2 (0, /2), then there exists a k 2 {2, 3, . . .} such that  k+1   <  k . If we choose n = k + 1, we have   n < (k + 1) k which is impossible since cos(n)  0. Similarly, we can derive a contradiction if we assume  2 (3/2, 2). Then 2 − /k   < 2 − /(k + 1) for some k 2 {2, 3, . . .}. 1.5 Lines and half planes in the complex plane Exercise 1. Let C be the circle {z : |z − c| = r}, r > 0; let a = c + rcis  and put L =  z : Im  z − a b  = 0  where b = cis. Find necessary and sufficient conditions in terms of  that L be tangent to C at a. Solution. Not available. 1.6 The extended plane and its spherical representation Exercise 1. Give the details in the derivation of (6.7) and (6.8). Solution. Not available. Exercise 2. For each of the following points in C, give the corresponding point of S : 0, 1 + i, 3 + 2i