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THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH2621

Higher Complex Analysis Problem Sheet

S2, 2018

Good solutions to these problems include reasons. A few problems may require extensions of ideas or definitions in the course; these may be explained in tutorials. Revision exercises. 1. Show that |ez | = eRe(z) for all z ∈ C.

2. Show that |eiθ − 1| = 2 |sin(θ/2)| for all θ ∈ R (a) using geometry; (b) using the algebraic formula eiθ − 1 = eiθ/2 (eiθ/2 − e−iθ/2 ). Deduce that |eiθ − 1| ≤ |θ| for all θ ∈ R. 3. Give a geometric explanation for the inequality |w| ≤ |w + z| + |z |.

4. Suppose that α = cos t + i sin t where 0 < t < 2π. Show that t 1+α = i cot . 1−α 2

5. Suppose that α 6= 0. Show that α + 1/α is real if and only if α is real or |α| = 1.

6. Expand (4 + i)(5 + 3i) and hence show that

1 3 π = arctan + arctan . 4 4 5

√ 7. By expanding √ (1 + √ 3i)(1 + i), find√cos(7π/ √ 12) and sin(7π/12) in surd form. Answer: (1 − 3)/(2 2) and (1 + 3)/(2 2). 8. (a) Show that |z1 + z2 |2 + |z1 − z2 |2 = 2|z1 |2 + 2|z2 |2 for all z1 , z2 ∈ C. (b) Give a geometrical interpretation of this result.

* 9. Show that the triangle in the complex plane whose vertices are z1 , z2 , z3 is equilateral if and only if z12 + z22 + z 23 = z1 z2 + z2 z3 + z3 z1 . 10. Explain both geometrically and algebraically why |z + w| = |z| + |w| if arg z = arg w. Does the result extend to arbitrary finite sums of complex numbers? 11. Show that any root z of z 4 + z + 3 = 0 satisfies |z| > 1 and that any root z of 4z 4 + z + 1 = 0 satisfies |z| ≤ 1. 12. Suppose that w is an nth root of unity and w 6= 1. Show that 1 + w + w2 + · · · + wn−1 = 0;

hence or otherwise, factorise 1 + z + z 2 + z 3 + z 4 + z 5 over C and over R. 13. (a) Factorise z 8 − 15z 4 − 16 over C and over R. (b) Find the real factorisation of z 4 + 4. Answer: (b) (z 2 + 2z + 2)(z 2 − 2z + 2)

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14. (a) Show that for 0 < θ < 2π :   1 1 − ei(n+1)θ 1 sin(n + 2 )θ + . Re = 2 1 − eiθ 2 sin θ2 (b) Show that for 0 < θ < 2π :   cos(n + 21)θ θ 1 1 − ei(n+1)θ . = cot − Im 2 2 1 − eiθ 2 sin θ2 (c) Find 1 + cos θ + cos 2θ + · · · + cos nθ and sin θ + sin 2θ + · · · + sin nθ. Sets. 15. Are the following sets open, closed, bounded, compact, connected, simply connected, regions or domains? (a) The set S1 = {z ∈ C : |z| < 1} (b) The set S2 = {z ∈ C : |z| ≤ 1} (c) The set S3 = {z = x + iy ∈ C : xy = 1} (d) The set S4 = {z = x + iy ∈ C : xy > 1} (e) The set S5 = {z = x + iy ∈ C : x ≥ 0, y > 0} (f) The set S6 = {z ∈ C : |z + i| + |z − i| < 4} (g) The set S7 = {z ∈ C : |z + i| + |z − i| = 4} (h) The set S8 = C (i) The set S9 = ∅ (the empty set).

16. If a set S consists of a single point {p}, then is p an interior point, an exterior point, or a boundary? Is S open, closed, connected, compact, or a region? †

17. If an open set Ω is simply connected, is its complement connected? If the complement of a bounded open set Ω is connected, is the set simply connected? What if Ω is not bounded? Answer: Not necessarily. Yes. Perhaps.

Functions. 18. Show that the composition of two polynomials is a polynomial, and the composition of two rational functions is a rational function. * 19. Suppose that f is a nonconstant rational function. What can you say about the range of f ? Hint: Write f as p/q and try to solve p(z )/q(z) = λ; this is possible if and only if λ is in the range of f . 20. The point 1 + i is rotated anticlockwise about the origin through π/6. Find its image. √ √ Answer: 12 ( 3 − 1 + i( 3 + 1)) 21. Find the image of (a) the line y = 2x + 5 and (b) the circle |z − 1| = 1 under the transformation w = (1 + i)z − 2. √ Answer: (a) v = −3u − 16, (b) |w + 1 − i| = 2. 22. Find the image of the region {z ∈ C : 0 < Re(z) < π/2, Im(z) > 0} under the transformation w = iz + 2. Express your answer using set notation. Answer: {(w ∈ C : Re w < 2, 0 < Im w < π/2}.

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23. Find the image of the following regions under the mapping w = z −1 : (a) x + y = 4, (b) |z − 1| = 1, (c) |z − 1| ≤ 1, z 6= 0. 24. Find the image of (a) the region |z − 1| ≤ 1 under the mapping w =

z , z+2 1 . (b) the line x + 2y = 2 under the mapping w = z+i 25. Let f (z) = 1/(z − i). Find the image of {z ∈ C : |z| < 1} under f . Answer: The set {w ∈ C : Im w > 1/2}. Fractional linear transformations. * 26. Suppose that a, b, c, d ∈ C and ad − bc = 1. Let T be the fractional linear az + b . transformation z 7→ cz + d (a) Show that if a, b, c, d ∈ R, then T maps the upper half plane onto the upper half plane. Is the converse true? (b) Find conditions on a, b, c, d that ensure that T maps the unit disc {z ∈ C : |z| < 1} onto itself. (c) Show that if a = i, b = −i, c = 1 and d = i, then the corresponding fractional linear transformation maps the upper half plane onto the unit disc. Where does the first quadrant map to? (d) Use your answer to (c) to connect your answers to (a) and (b). Estimating the size of functions. 27. Show that if |z| = R > 1, then    z  R   z 3 + 1  ≤ R3 − 1 . Show also that if |z| = r < 1, then    z  r    z3 + 1  ≤ 1 − r3 .

28. Show that if |z| = R > 1, then    1  1   z 4 + 1  ≤ R4 − 1 . Show also that if |z| = r < 1, then    1  1    z4 + 1  ≤ 1 − r4 .

29. Show that if z is real, then  iz   e  1    z 2 + 1  ≤ |z|2 + 1 .

Is this still true when z is not real? Give two reasons for your answer.

30. Show that if |z| = R > 2 and Im(z) ≥ 0, then  eiz  1 ≤ .  R z    

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Real and imaginary parts of functions. 31. Suppose that f (z) = z 3 −z +1. Write f (x) in the form u(x, y)+iv(x, y), where z = x+iy. Answer: u(x, y) = x3 −3xy∗2−x+1 and v(x, y ) = 3x2 y − y ∗3−y . 32. Suppose that f (z) = ze−z . Write f (x) in the form u(x, y) + iv(x, y), where z = x + iy, and in the form u(r, θ ) + iv(r, θ), where z = reiθ . 33. Suppose that u(x, y) = x2 + y 2 and v(x, y ) = x − y. Can you write u(x, y) + iv(x, y) as a simple function of z? Or of z and z¯? Graphical representation of complex functions. * 34. Suppose that f (z) = az 2 + bz + c, where a, b, c ∈ C and a 6= 0. (a) Show that the image of a straight line ℓ in C under f either lies in a straight line or is a parabola. (b) Show that the image of a straight line ℓ in C under f lies in a straight line if and only if ℓ passes through −b/2a. 35. Suppose that f (z) = ez . Show that the image of a straight line ℓ in C under f is a straight line if and only if ℓ is horizontal.

Limits and continuity. 36. Find the following limits, if they exist, or explain why they do not exist. (a) (c) †

(e)

ez − 1 z→0 z 1/z lim e

(b)

lim

(d)

z→0

xy (x,y )→(0,0) x2 + y 2 lim

†

(f)

z2 − z + 1 − i z→1+i z 2 − 2z + 2 4 lim e1/z lim

z→0

xy2 . (x,y )→(0,0) x2 + y 4 lim

Answer: (a) 1; (b) 1 − i/2; (c) the limit does not exist, because we get different limits as we approach (0, 0) along the x axis from the left and from the right; (d) the limit does not exist, because we get different limits as we approach (0, 0) along different rays (x, y) = (at, bt) as t → 0; (e) as for (d); (f) the limit does not exist, because we get different limits as we approach (0, 0) along different curves (x, y) = (at2 , bt) as t → 0; 37. Use the definition of a limit in each of the following. (a) Show that limz→∞ f (z) = limz→0 f (1/z ). (b) Suppose that limz→z0 g(z) exists and is finite. Show that limz→z0 f (z ) exists if and only if limz→z0 f (z) + g(z) exists. (c) Show that limz→z0 f (az + b) = limw→az0 +b f (w). 38. Prove from the definition that |z| and Re(z ) are continuous.

39. For each of the following functions, state where the function is continuous, and justify your answer: (a) f1 (z) =

z2 + i ; z2 + 1

(b) f2 (z) =

zk ; z¯

(c) f3 (z) = (z − i) Log(z 2 + 1).

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(in (b), k is a positive integer.) Is it possible to extend their definition to a larger domain, maintaining continuity. Answer: (a) z 6= ±i; (b) z 6= 0; if k ≥ 2, then f2 can be extended continuously by setting f2 (0) = 0; (c) z 6= iy, where y ≤ −1 or y ≥ 1; f3 can be extended continuously by setting f3 (i) = 0. Differentiation. 40. Show from the definition, and then using the Cauchy Riemann equations, that f1 (z) = Im(z) and f2 (x + iy ) = 3x + 4iy are nowhere differentiable. 41. Where are the following functions differentiable? Where are they analytic? (a) f1 (z) = z |z |2 ; (b) f2 (x + iy) = x2 + iy2 ; (c) f3 (x + iy) = (xy 2 + 5) + i(y − x2 y − 3); (d) f4 (x + iy) = (3xy 2 + 6x2 − 4x + 3) + i(3y 2 − 3x2 y + 2y + 15); (e) f5 (z) = ez (z − z¯)2 ; (f) f6 (x + iy) = |x| + i|y|. Answer: (a) f1 is diffentiable at 0; (b) f2 is differentiable on the line x = y ; (c) f3 is differentiable on the circle x2 + y 2 = 1; (d) f4 is differentiable on the circle (x + 2)2 + (y − 1)2 = 1; (e) f5 is differentiable on the x axis, that is, y = 0; (f) f6 is differentiable when xy > 0; f1 to f5 are differentiable on sets that are not open, so are not analytic, while f6 is analytic when xy > 0. * 42. Suppose that

( −4 e−z f (z) = 0

when z 6= 0 when z = 0.

Show that the Cauchy–Riemann equations hold in C. Is f entire? Hint: You do not need to compute the partial derivatives, except at 0. Answer: f is analytic in C \ {0}, as there it is made up of differentiable functions, and so the Cauchy–Riemann equations hold in C \ {0}. At 0, ∂u/∂x = ∂u/∂y = ∂v/∂x = ∂v/∂y = 0; hence the Cauchy–Riemann equations also hold at 0. However, f is not continuous at 0, and hence not differentiable at 0. 43. Show that real-valued entire functions are constant. 44. Show that if f and g = f¯ are both differentiable in a domain, then f is constant on that domain. 45. Suppose that f is analytic in Ω, and f ′ is continuous; let g(z) = f (¯ z ). Is g analytic? Where? Answer: f satisfies the Cauchy–Riemann equations in Ω, and so g satisfies ¯ The partial derivatives are continuous, the Cauchy–Riemann equations in Ω. ¯ ¯ and Ω is open, and so g is analytic in Ω 46. Suppose that a ∈ R. Use the polar form of the Cauchy–Riemann equations to find where the function f (z) = |z|a eia Arg(z) is differentiable.

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Connections with multivariable calculus† . 47. Suppose that a linear transformation L of R2 preserves angles between lines passing through the origin. Show that L is the composition of a scalar multiplication and a rotation, and possibly a reflection. 48. A subset S of C is said to be arcwise connected if, given any two points p and q of S, there exists a continuous curve γ : [0, 1] → S such that γ(0) = p and γ(1) = q. Show that if an open set is arcwise connected, then the continuous curve γ may be taken to be polygonal. 49. In the area of mathematics called General Topology, a set S is said to be connected if we cannot find two open sets Ω1 and Ω2 such that Ω1 ∩ Ω2 = ∅, Ω1 6= ∅, Ω2 6= ∅ and (Ω1 ∩ S) ∪ (Ω2 ∩ S) = S . Show that an open set Ω is connected in this sense if and only if it is arcwise connected. What can you say about regions? Harmonic functions. 50. Show that the function u(x, y) = x4 − 6x2 y 2 + y 4 is harmonic and find a harmonic conjugate v. Express f = u + iv as a function of z . Answer: v(x, y) = 4x3 y − 4xy3 + C, where C is a constant; f (z) = z 4 + iC .

51. Find all real a, b, c such that u(x, y) = x3 + ax2 y + bxy2 + cy 3 is harmonic on R2 . For all such a, b, c determine v(x, y) such that f = u + iv is entire and write f as a function of z only. Answer: a = −3c, b = −3, v(x, y) = cx3 + 3x2 y −3cxy 2 −y 3 , f (z) = (1+ ic)z 3 .

52. Suppose u, v are harmonic and satisfy the Cauchy-Riemann equations in R2 . Show that f = u + iv satisfies f ′ (x) = ux (x, 0) − iuy (x, 0) for real x. 53. Using the result of the previous question to guess f , or otherwise, show that the following are harmonic and find analytic functions f of which they are the real parts: (a) x − x3 + 3xy 2 , (b) cos x cosh y , x , where (x, y) 6= (0, 0)), (c) 2 x + y2 x(x2 + y 2 + 1) ⋆ (d) 2 , where (x, y) = 6 ±(0, 1), (x + y 2 )2 + 2(x2 − y 2 ) + 1 (e) y . z ; (e) iz. Answer: (a) z − z 3 ; (b) cos(z); (c) 1/z; (d) 2 z +1 54. Suppose that v is a harmonic conjugate of u. Show that −u is a harmonic conjugate of v .

* 55. Suppose that u is a harmonic function in an open set Ω, and define the function g on Ω by g(z) = ux (x, y) − iuy (x, y), where z = x + iy. Show that g is holomorphic in Ω. If f ∈ H(Ω) and f ′ = g, what can you say about f and u?

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Power series. 56. Find the centre and radius of convergence of the following power series; if possible, identify the sum: ∞ X

(a)

(z − 2)n

(b)

X1 (z − 2)n n!

(d)

n=0 ∞

(c)

n=0 ∞

n=0 ∞

X

(e) (g)

X n=0

n(z − 2)n

(f)

X1 (z − 2)n n

(h)

n=0 ∞

∞ X 1 (z − 2)n 3n

∞ X n=0 ∞

n=1

n2 (z − 2)n

X n=1

1 (z − 2)n n!

3n

1 (z − 2)n . n(n + 1)

Exponential and related functions. 57. (a) Show that |sin z|2 = sin2 x + sinh2 y and |cos z|2 − |sin z|2 = cos 2x. (b) Show that |cos z|2 + |sin z|2 ≥ 1. When does equality hold? 58. Find all z ∈ C such that (a) ez = −2, (b) cos z√= 2, (c) cosh z = −5, (d) exp(z 2 ) = 1. Answer: (c) ± ln(5 + 2 6) + (2k + 1)πi, where k ∈ Z.

59. Find the image of {z ∈ C : 0 < Re(z) < π/2, Im z > 0} under the mappings (a) w = cos z, (b) w = cos2 z, (c) w = cos1/2 z . (Use the principal branch of the square root.) 60. Find the image of the region {z ∈ C : 0 ≤ Re(z) ≤ 1/2, −π ≤ Im(z) ≤ π} under the mapping w = ez . 2

61. Find the maximum √ value of |e−z | for |z| ≥ 5 and 0 ≤ arg z ≤ π/8. Answer: exp(−25/ 2) 62. Find all solutions z ∈ C of (a) cosh z = −1

(b) sin z = 3

(c) sinh z = 2i.

63. Solve (a) sinh z − cosh z = 2i, (b) cos z + sin z = i. Answer: (a) − ln(2) + (2k + 21 )πi.

64. Where is the function f (z) = (z 2 + 4) −1 Log(z + 2i) analytic? Answer: f is analytic on the complement of the set {2i} ∪ {x − 2i : x ≤ 0}.

65. Find all values of (a) ii , and (b) sin−1 are the principal values? √ √10. Which 1 1 Answer: (b) (2k + 2 )π ± i log(10 + 3 11); 2 π − i log(10 + 3 11). √   1 + 3i −3 1−i . 66. Find the principal value of 2 2

67. Evaluate lim (cos z)1/z . (Use the principal branch of the power.) z→0

Answer: e−1/2 , by l’Hˆopital’s rule. 68. Write z = reiθ . Find Re(z i ), Im(z i ), |z i | in terms of r and θ. (Use the principal value of z i .)

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69. Define f (z) = z Log(z) when z 6= 0, and f (0) = 0. Is f continuous or differentiable at 0? Answer: It is continuous but not differentiable. 70. Define functions f and g by f (z) = Log(iz) and g(z) = z −1 Log(z + 1). (a) Where are f and g analytic? (b) Find a branch of log(iz) which is analytic in the region {z : Im(z) > 0}. Answer: (a) f is analytic except when z = iy, where y ≥ 0; while g is analytic except when z = 0 or z = x, where x ≤ −1. (b) iπ/2 + Log(z).

* 71. In this question, √ all square roots are the principal branch. (a) Where is √ z + 1 analytic? √ √ (b) Show that √ z + 1 z − 1 = − z 2 − 1 when Re z < −1. (c) Show that √z 2 − 1√is analytic when Re z < −1. (d) Show that z + 1 z − 1 is analytic on the complement of [−1, 1]. * 72. Where is Log(z + z −1 − 2) analytic? Answer: On the complement of the set {z : |z| = 1} ∪ (−∞, 0]. Contour integrals. 73. (a) Evaluate

Z

z Im(z 2 ) dz, γ

where γ is the unit circle traversed once, anticlockwise. (b) Evaluate Z 2 e|z | Re(z) dz, γ

where γ is the line segment from 0 to 1 + i. Answer: (a) −π (b) (1 + i)(e2 − 1)/4. Z z¯ dz, where γ is 74. Evaluate γ

(a) the straight line from −1 + 2i to 3 + 5i (b) the upper semicircle of unit radius from −1 to 1. Answer: (a) 292 − 11i (b) −iπ . Z 75. (a) Find (ez + z) dz, where γ (t) = 1 + iπt for 0 ≤ t ≤ 1. γ Z (b) Show that Log z dz = −2i, where γ(t) = eiπt for − 21 ≤ t ≤ 12 . γ Z sec2 z dz = 2i tanh 1, for every contour γ in the domain of sec 76. Show that γ

that starts at −i and ends at i.

77. Let γ be any contour from 1 − i to 1 + i. Evaluate the following integrals: Z Z Z 3 (a) 4z dz; (b) cos z dz; (c) sin 2z dz. γ

γ

γ

78. Let γ be the semi-circle from 2i to −2i that passes through −2. Find Z z −1 dz γ

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(a) from the definition; (b) by using a suitable branch of log as a primitive. Answer: πi. Cauchy’s integral formula. 79. Evaluate the following integrals using the Cauchy Integral Formula: Z Z Z (z 2 + 1) dz e2iz dz ez dz . ; (c) ; (b) (a) 2 2 |z |=2 (z − 3)(z − 1) |z |=4 (3z − 1) |z |=2 z − 1 (All contours are traversed once anti-clockwise.) Answer: (a) 2πei (b) −4πe2i/3 /9 (c) −πi/2. 80. Find Z sin πz dz, n γ (z − 1)

where n ∈ Z and γ is a simple closed anticlockwise contour around 1. Answer: 0, unless n = 2, 4, 6, 8, . . . ; in this case, (−1)n/2 π n−1 /(n − 1)!. 81. Let γ be the unit circle {eiθ : − π ≤ θ ≤ π}. Use the Cauchy integral formula to find Z ez z n dz,

γ

where n ∈ Z. Hence evaluate the corresponding real integrals.

82. Define the semicircular arc γR by γR (t) = Reit , where 0 ≤ t ≤ π and R > 1 is a real constant. Let γ be the join of γR and the line segment from −R to R. (a) Show that, if z ∈ Range(γR ), then  iz   e  1   z 2 + 1  ≤ R2 − 1 . (b) Deduce that

Z   

(c) Evaluate

 πR eiz dz  ≤ 2 .  2 R −1 γR (z + 1) Z

(d) Hence find

Z

γ

eiz dz . (z 2 + 1)

∞

cos x dx . (x2 + 1) 0 83. Let f (z) be an entire function that satisfies the inequality |f (z )| ≤ 1 + |z| for all z ∈ C. Show that f (z) = az + b for fixed complex numbers a and b. Hint: Apply the generalized Cauchy integral formula on an arbitrarily large circle to show that f (n) (0) = 0 when n ≥ 2. 2

* 84. By integrating f (z) = e−z round the boundary of the rectangle with corners at a, a + ib, −a + ib and −a, where b > 0, show that Z ∞ Z ∞ 2 −x2 −b2 e−x dx. e cos 2bx dx = e −∞

−b2

The right hand side is e

√

−∞

π.

10 −1

* 85. By integrating f (z) = (1 + z 2 ) around the perimeter of the rectangle with corners at 0, a, a + ib and ib and letting a → ∞, where 0 < b < 1, show that Z ∞ π 1 − b2 + x2 dx = , 2 2 2 2 2 (1 − b + x ) + 4b x 2 0 and

Z

∞ 0

(1 −

b2

1 1+b x . dx = Log 2 2 2 2 1−b + x ) + 4b x 4b 2

* 86. Apply Cauchy’s theorem to the function f (z) = e−z and the sector of the circle |z| = R bounded by a section of the real axis and the linear segment making an angle π/4 with the real axis. Show that the integral over the circular boundary tends to zero as R → ∞ and prove that r Z ∞ Z ∞ 1 π 2 2 . sin(x ) dx = cos(x ) dx = 2 2 0 0 Note: These so-called Fresnel Integrals are used in diffraction problems.

Taylor series, Laurent series, and residues. 87. Expand each of the following functions in a Taylor series about the given point z0 and determine the radius of convergence: (a) f (z) = e−z , z0 = 1; 1 , z0 = 2. (b) f (z) = (z + 2)e3z , z0 = 0; (c) f (z) = 2 z − 5z + 4 ∞ X  3n  1 = 88. Show that z − z 3n+1 in the disc |z| < 1. 2 1+z+z n=0

89. Without finding the series, state the radius of convergence of the Taylor series cos z for the function f given by f (z) = about the point z = 1 + i. z(z + 1)

90. Find the Taylor series for f (z) = (1 + z)1/2 in powers of z in the disc |z| < 1. Use the branch of the square root that is equal to +1 when z = 0...