COMPLEX ANAYLSIS, Assignments of Music

Download COMPLEX ANAYLSIS and more Assignments Music in PDF only on Docsity! math2069 Complex Analysis Problems 2017 1 The University of New South Wales School of Mathematics and Statistics MATH2069 MATHEMATICS 2A: COMPLEX ANAYLSIS Questions marked [K] are key or core questions. The more difficult questions are marked [H]. Reference is made to four textbooks: Spiegel: M.R. Spiegel, Complex Variables (McGraw-Hill); Church5: R.V. Churchill & J.W. Brown, Complex Variables & Applications, (5th ed.) (McGraw-Hill). Church6: J.W. Brown and R.V. Churchill, Complex Variables & Applications (6th ed.) (McGraw-Hill) Church7: J.W. Brown and R.V. Churchill, Complex Variables & Applications (7th ed.) (McGraw-Hill) The purposes of this set of problems are as follows: 1. To provide common tutorial material; 2. To specify the course in great detail; 3. To serve as a guide to the recommended textbook and its predecessors. 1 The Complex Plane Note: This introduction to complex numbers includes: Basic topology of the complex plane; Functions of a complex variable; Functions as mappings. References: Spiegel chapter 1 also page 200. Church5 chapter 1 also pages 28–31, 207–212. Church6 chapter 1 also pages 28–31, 245–249. Church7 chapter 1 also pages 33–42, 299–305. 1 Evaluate: a. (2 + 3i)3; b. (2 + 7i)/(3− 2i); c. (2 + 3i)3/(1 + 3i)2; d. < ( (3− 7i)4 ) ; e. = ( (3− 7i)4 ) ; f. |3− 7i|4. [K]2 Sketch the regions in the complex plane specified by: a. 2 ≤ |z − i| ≤ 3; b. 2 < |z − i| < 3; c. =(z) > 0; d. |z + 2|+ |z| = 3; e. |z + 2|+ |z| < 3; f. |z + 2|+ |z| > 3. [K]3 Show that if |z| = 10 then 497 ≤ |z3 + 5iz2 − 3| ≤ 1503. 4 a. Express the following complex numbers 1; −1; √ 3 + i; −7 + 7i in the polar form r(cos θ + i sin θ). b. Verify (−1 + i)7 = −8(1 + i) both directly and using the polar form. [K]5 a. Find all solutions of z5 + i = 0. b. Factorize z8 − 15z4 − 16 over C and hence over R. c. Find the real factorization of z4 + 4. math2069 Complex Analysis Problems 2017 2 6 a. Suppose both c and (1 + ic)5 are real (c 6= 0). Show that c = ± √ 5± 2 √ 5. Now use another method to show that either c = ± tan 36◦ or c = ± tan 72◦. b. Suppose α/(α2 + 1) is real, where α = a+ ib, b 6= 0, α 6= ±i. Show that αᾱ = 1. 7 Let z and w be complex numbers with |z| < 1, |w| < 1. a. Show that |1− zw|2 − |z − w|2 = (1− |z|2)(1− |w|2). b. Hence or otherwise prove that |(z − w)/(1− zw)| < 1. [H]8 a. Show that for 0 < θ < 2π: < ( 1− ei(n+1)θ 1− eiθ ) = 1 2 + sin(n+ 12)θ 2 sin θ2 . b. Show that for 0 < θ < 2π: = ( 1− ei(n+1)θ 1− eiθ ) = 1 2 cot θ 2 − cos(n+ 12)θ 2 sin θ2 . c. Use the results above to find C = 1 + cos θ + cos 2θ + · · ·+ cosnθ and S = sin θ + sin 2θ + · · ·+ sinnθ. [K]9 Are the following regions in the plane (1) open (2) connected and (3) domains? a. the real numbers; b. the first quadrant including its boundary; c. the first quadrant excluding its boundary; d. the complement of the unit circle; e. the six regions of question 2; f. C \ Z = {z ∈ C : z /∈ Z}. 10 a. Show that |z1 + z2|2 + |z1 − z2|2 = 2|z1|2 + 2|z2|2 for all z1 and z2 in C. b. Give a geometrical interpretation of this result. [K]11 The point 1 + i is rotated anticlockwise through π 6 about the origin. Find its image. [H]12 Show that the triangle in the complex plane whose vertices are the origin and the points w1 and w2 is equilateral if and only if w 2 1 + w 2 2 = w1w2. [H]13 Four distinct points including 0; 0, z1, z2, z3 lie on a circle. Show that the points 1 z1 , 1 z2 , 1 z3 are collinear, i.e. lie on a straight line. [K]14 Find (i.e. simplify in terms of real and imaginary parts x and y) and sketch the following sets a. {z ∈ C : <(z2) > 0}; b. {z ∈ C : <(z2) > 1}; c. {z ∈ C : < ( z z − 1 ) = 0}; d. {z ∈ C : = ( 1 z ) ≥ 1}. [K]15 Show the set of complex numbers z satisfying ∣∣∣∣ z + 12z + 3 ∣∣∣∣ = 1 is a circle and find its centre and radius. [K]16 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. math2069 Complex Analysis Problems 2017 5 [K]34 a. Explain why the function f defined by f(z) = (z2 + 1)/(z + 1) is analytic except for z = −1; b. Where is the function f defined by f(z) = (z7 + 1)/(z3 − 1) analytic? [H]35 Sketch some of the curves x2 − y2 = constant, 2xy = constant in the plane. In general, show that if the functions u : D → R and v : D → R, with D an open set in R2, satisfy the Cauchy-Riemann equations, then the curves u(x, y) = constant intersect the curves v(x, y) = constant orthogonally. [K]36 For each of the following functions u, defined as shown, determine which are harmonic in C ' R2. Find a harmonic conjugate if it exists: a. u(x, y) = x3 − 3xy2; b. u(x, y) = xex cos y − yex sin y; c. u(x, y) = coshx cos y; d. u(x, y) = x4 − 2x2y2 + y4; e. u(x, y) = 3x− 2xy. [K]37 For the following harmonic functions u : R2 → R, find a harmonic conjugate v : R2 → R for u and express the analytic function f = u+ iv : C→ C as a function of z alone. a. u(x, y) = y3 − 3yx2 + 2xy; b. u(x, y) = x3y − xy3; c. u(x, y) = e−y(x sinx+ y cos(x)); d. u(x, y) = sin(x) sinh(y) + xy. [The 2nd part for c. and d. will require some definitions from section 4.] 38 a. Assume that the function u given by u(x, y) = ax4 + bx3y + cx2y2 + dxy3 + ey4, for a, b, c, d, e real constants, is harmonic on R2. Find the linear constraints this puts on a, b, c, d, e and show that u is a linear combination of the functions (x, y) 7−→ x4 − 6x2y2 + y4 and (x, y) 7−→ x3y − xy3. b. The real constants α, β are chosen so that the function u defined by u(x, y) = x3 + αx2y + βxy2 is harmonic on R2. Find α and β and determine a conjugate harmonic function, v, to u. [K]39 Show that the function u defined by u(x, y) = x+2 cos 2x cosh 2y is harmonic and find a harmonic conjugate v. Hence express the function f defined by f(z) = u(x, y)+ iv(x, y) as a function of z, where z = x+ iy. 4 Exponential, trigonometric and hyperbolic functions Note: This section includes the solution of trigonometric and hyperbolic equations References: Spiegel chapter 3 pages 34–36, 44–48. Church5 chapter 3 sections 22 – 25. Church6 chapter 3 sections 23 – 25. Church7 chapter 3 sections 28, 33, 34. 40 Express exp(3 + πi) in Cartesian form and show that exp(z + πi) = − exp(z) for all z ∈ C. [K]41 Find all solutions of the following equations: a. ez = e2−3i; b. ez = 2 + i √ 3; c. ez = −5i. math2069 Complex Analysis Problems 2017 6 42 Express the following complex numbers in the form a+ ib, for a and b real: a. sin i; b. cos(2− i); c. tan(1− i). [K]43 a. If z = x+ iy,with x and y real, show that | sin z|2 = sin2 x+ sinh2 y. b. Show that | cos z|2 − | sin z|2 = cos 2x. c. Show that | cos z|2 + | sin z|2 ≥ 1 with equality only if z is a real number. 44 For the mapping f(z) = sinh z, find and sketch the image of a. <(z) = c; b. =(z) = d; [H]c. {z ∈ C : <(z) > 0, 0 < =(z) < π/2}; [H]d. {z ∈ C : <(z) < 0, −π/2 ≤ =(z) ≤ π/2}. 45 Find all solutions z ∈ C of the following equations: a. cos z = cos 2; b. sin z = sin 2; c. cosh z = cosh 2; d. cos z = cosh 2; e. sin z = sinh 2; f. cosh z = sin 2. [K]46 Solve the following equations over C: a. cos z = −1; b. cos z = 2; c. cosh z = 2i e. sinh z = 4i; d. sin z = 2i; f. cos z + sin z = i. 5 The Principal Logarithm & Complex Exponents Note: The principal value logarithm is written Log. References: Spiegel chapter 2 pages 36, 46, 48. Church5 chapter 3 sections 26 – 29. Church6 chapter 3 sections 26 – 29. Church7 chapter 3 sections 29, 30 31, 32, 35 [K]47 Find log(z) and Log(z) for each of the following: a. z = 1; b. z = i; c. z = −3 + i. 48 Find a pair z1, z2 of complex numbers for which Log(z1) + Log(z2) = Log(z1z2) and find a pair z1, z2 of complex numbers for which Log(z1) + Log(z2) 6= Log(z1z2). [K]49 Where is the function f defined by f(z) = (z2 + 4)−1 Log(z + 2i) analytic? [K]50 Find exp Log(10i) and Log exp(10i) in Cartesian form. [K]51 Find the Cartesian forms of Log (( exp(34πi) )2) and 2 Log ( exp(34πi) ) . [K]52 Find the Cartesian form of each of the following: a. ii; b. p.v. ii; c. p.v. (1 + i√3 2 )−31−i;d. p.v. [(−1 + i)2]i. 53 Find all the values determined by (−1 − i)i. Give < ( p.v.(−1− i)i ) to 3 significant figures. 54 Determine the Cartesian form of all complex numbers given by iLog i. math2069 Complex Analysis Problems 2017 7 6 Arcs, Contour Integrals and Anti-Derivatives References: Spiegel chapter 4 pages 92–93, 98. Church5 chapter 4 sections 30 – 34. Church6 chapter 4 sections 30 – 35. Church7 chapter 4 sections 36 – 43 [K]55 Calculate I = ∫ C z dz where C is the straight line segment from z = i to z = 1 + 2i. [K]56 Let γ denote the boundary of the triangle with vertices 0, 1 and 1+i taken anticlockwise. Evaluate ∮ γ <(z) dz and ∮ γ z dz. [K]57 Evaluate ∫ γ z dz where a. γ is the straight line from −1 + 2i to 3 + 5i; b. γ is the upper semicircle of unit radius from −1 to 1. [K]58 Let Γ denote the semicircular contour in the upper half plane with centre 1 and radius 2 taken from 3 to −1. Evaluate ∫ Γ z dz and ∫ Γ z dz. [K]59 Let γ denote the anticlockwise contour consisting of the above semicircle taken with its base. Evaluate ∫ γ z dz and ∫ γ z dz. 60 Evaluate ∫ γ f(z) dz where z = x+ iy, with x and y real, and f is the given function: a. f(z) = 2x− 3iy, γ is the ellipse {cos t+ 2i sin t : 0 ≤ t ≤ 2π}; b. f(z) = x2, γ is the parabola y = 2x2 from x = 0 to x = 2; c. f(z) = x2, γ is the parabola y = 2x2 from x = 2 to x = 0; d. f(z) = z̄, γ is the ellipse x = a cos t, y = b sin t, for 0 ≤ t < 2π. [K]61 Show that if =(z) ≥ 0, then |eiz| ≤ 1. Now let R > 1 be a real constant. Deduce that ∣∣∣∣ eizz4 + 1 ∣∣∣∣ ≤ 1R4 − 1 for z on the semi-circle {z ∈ C : |z| = R, =(z) ≥ 0}. This sort of inequality will be needed in Theme 11. [K]62 Let ΓR = be the semicircular contour {Reiθ : 0 ≤ θ ≤ π}, where R > 1. Use the answer to question 61 to show that ∣∣∣∣∫ ΓR eizdz z4 + 1 ∣∣∣∣ ≤ π RR4 − 1 . [K]63 Let γ be any contour from 1− i to 1 + i. Evaluate the following integrals: a. ∫ γ 4z3 dz; b. ∫ γ cos z dz; c. ∫ γ sin 2z dz. [K]64 Let γ be the semi-circle from 2i to −2i that passes through −2 in the positive direction. Find ∫ γ z −1 dz a. from the definition; b. by using a suitable branch of log as an anti-derivative. 65 Let Γ be any contour from exp(2πi/3) to exp(−2πi/3) which lies entirely on the left hand side of the imaginary axis. Evaluate ∫ Γ z −1 dz. math2069 Complex Analysis Problems 2017 10 81 Find the Taylor series for z 7→ (1 + z)1/2 in powers of z which is valid for |z| < 1. 82 Find the first four terms in the Taylor series for ez(1 + z)1/2 valid for |z| < 1. Laurent Series [K]83 Let f(z) = z (z − 1)(z + 4) . a. Find the largest annuli or open discs centred at −1 in which f is analytic. (There are 3 such regions). b. For each of the regions in a. above, find the corresponding Laurent series for f about −1. [K]84 Give two Laurent series in powers of z for the function f given by f(z) = z−3(1− z)−1 and specify the regions in which those series are valid. [K]85 a. Let f be the function given by f(z) = 2z − 4 z2 − 4z + 3 . Find the Laurent series for f that is valid for |z − 1| > 2; b. Assuming that the function f given by f(z) = (z− 2i)−1− (z+ i)−1 has a Laurent series of the form ∑∞ n=−∞ an(z−1)n which converges at z = 3, find the coefficients an. c. Find the Laurent series up to terms in z3 about z = 0 valid at z = 12 for the function given by f(z) = cosec z and state the radius of convergence. 86 Let f be the function given by f(z) = 1 z2 − 1 . a. Find the Laurent series for f in powers of z+ 2 that converges where z = 0. What is its annulus of convergence? b. Find the Laurent series for f in powers of z+ 2 that converges where z = 3. What is its annulus of convergence? c. Find the Laurent series for f in powers of z+2i that converges where z = 3. What is its annulus of convergence? 87 Show that the function f(z) = cosec(1/z) does not have a Laurent series about 0 converging to f in 0 < |z| < r for any r > 0. [H]88 Let f(z) = z ez − 1 . a. Show f has a removable singularity at z = 0. Find the largest value of r such that f is analytic in the annulus 0 < |z| < r. b. Let ∞∑ n=0 anz n be the Laurent series for f in 0 < |z| < r for r in a. above. Find a0. Show that f(z) + 12z is an even function in z, and hence a1 = − 1 2 , an = 0 for n ≥ 3 and n odd. c. The Bernoulli numbers Bn are defined by z ez − 1 = ∞∑ n=0 Bnz n n! . Find a recurrence relation for Bn in terms of B0, B1, . . . Bn−1, by cross-multiplying by the Maclaurin series of ez − 1. Hence calculate Bn for n = 0 to 6. math2069 Complex Analysis Problems 2017 11 9 Singularities and the Method of Residues References: Spiegel chapter 7 pages 172–173, 176–184. Church5 chapter 6 sections 53 – 57. Church6 chapter 6 sections 53 – 57. Church7 chapter 6 sections 62 – 69. [K]89 In each of the six cases below write down the principal part of the function f at each of its singular points. Determine if each singular point is a pole, an essential singularity or a removable singularity of the given function. a. f(z) = z−1 sin z; b. f(z) = z−1 cos z; c. f(z) = z2/(1 + z); d. f(z) = ez/(z2 − 1); e. f(z) = (3− z)−3; f. f(z) = z exp(1/z). [K]90 Write down the residue of each function of question 89 at each of its singularities. [K]91 Let f be defined by f(z) = exp ( z−1 ) . Find the Laurent series for f in powers of z and show that f has an essential singularity at z = 0. [K]92 a. Find the order of the zero of −1 + cos2 z at z = nπ, n ∈ Z; b. Find the order of the pole at z = 0 for the function z 7→ (−1 + cos2 z)−4; c. Find the order of the zero of sin3 z at z = nπ, n ∈ Z; d. Find the order of all the poles and zeros of (−1 + cos2 z)/ sin3 z; e. Find the order of the pole at z = 0 for the function f given by f(z) = (e2z − 1− z)−21 · (ez − 1− z)−10 · sin7(z3) · (z − cos z)13 · (1− cosh z)9. [K]93 Find the residues of each of the following functions at each of their singularities: a. ez z2 + 1 ; b. sinh z (z − i)3 . c. z2 − 3z + 1 (z2 − 1)(z − 1) ; d. ez cosh z ; e. tan z. [K]94 Use the residue theorem to evaluate the following integrals. All contours are taken once anticlockwise. a. ∮ |z|=2 ez z2(z + 1) dz; b. ∮ |z−i|=3 ez 2 − 1 z3 − iz2 dz; c. ∮ |z|=3 sin z z2 − z dz; d. ∮ |z|=9 z3 tan z dz. [K]95 Evaluate ∫ γ z−1(z2 + 1)−1 tanh z dz for the following closed contours γ, where in each case θ ∈ [−π, π]: a. γ = {2 + eiθ}; b. γ = {2eiθ}; c. γ = {5i+ eiθ}. 96 Evaluate ∮ Γ (iz + 1)−1(z2 + 1)−1 cosh(πz) dz where Γ is the contour |z| = 2 taken anti- clockwise. 97 a. Evaluate ∮ γ zne(1/z) dz where γ is the unit circle taken anticlockwise and n is an integer. b. Evaluate ∮ γ zn sin 1 z dz, where γ, n are as in part a. math2069 Complex Analysis Problems 2017 12 10 The Z-transform 98 If Z{fn} = ∑∞ n=0 fnz −n = F (z), prove that Z{nfn} = −zF ′(z). Using Z{un} = z/(z − 1) and this result, or otherwise, find Z{n}. [K]99 Evaluate the following Z-transforms: a. Z{n2}; b. Z{n3}; f. Z{(n+ 1)2}; d. Z{nan}, a ∈ R; e. Z{n sinnθ}; f. Z{an cosnθ}, a ∈ R. [K]100 Evaluate the following inverse Z-transforms by any suitable method. a. Z−1 { 7z z2 − 5z − 6 } ; b. Z−1 { z z2 + z + 1 } ; c. Z−1 { z2 + 5z z2 − 3z − 4 } ; [K]101 Solve the following difference equations for the given initial conditions using the Z- transform: a. yn+2 + 2yn+1 − 3yn = 0, y0 = 1, y1 = 0; b. yn+2 − yn = 2, y0 = 4, y1 = 1; c. 2yn+2 − yn+1 − yn = 15n2n, y0 = −2, y1 = 1. 11 Real Improper Integrals References: Spiegel chapter 7 pages 179–183. Church5 chapter 6 sections 58 – 59. Church6 chapter 7 sections 60 – 61. Church7 chapter 7 sections 71 – 74. In the following problems, a > 0 is a real constant and the recommended contour is the boundary of the half disc |z| ≤ R, =(z) ≥ 0 taken once anticlockwise. You may assume that the integrals converge. [K]102 Evaluate the following integrals: a. ∫ ∞ −∞ dx (x2 + 1)2 ; b. ∫ ∞ 0 x2 dx (x2 + 1)(x2 + 9) ; c. ∫ ∞ −∞ x2 dx x4 + 1 ; d. ∫ ∞ −∞ x2 dx (x2 + 1)(x2 + x+ 1) . 103 Evaluate ∫ ∞ −∞ eix dx x2 + a2 and hence ∫ ∞ −∞ cosx dx x2 + a2 . [K]104 Evaluate the following integrals: a. ∫ ∞ −∞ cos 3x dx x2 + 2x+ 2 ; b. ∫ ∞ −∞ sinx dx x2 − 4x+ 5 . 105 Evaluate ∫ ∞ −∞ cosx dx (x2 + 1)(x2 + 2x+ 5) . math2069 Complex Analysis Problems 2017 15 41 a. z = 2− (3 + 2kπ)i; b. 12 ln(7) + i[tan −1 √ 3 2 + 2kπ]; c. ln(5)− i(2k + 1 2)π. 42 a. isinh 1; b. cos 2 cosh 1 + i sin 2 sinh 1; c. tan 1sech21− i tanh 1 sec2 1 1 + tanh2 1 tan2 1 . 44 a. ( u sinh c )2 + ( v cosh c )2 = 1 if c 6= 0, {iv : −1 ≤ v ≤ 1} if c = 0. b. ( v sin d )2 − ( u cos d )2 = 1, v sin d ≥ 1 if d 6= kπ/2 for k ∈ Z, v = 0 if d = nπ for n ∈ Z, {iv : v ≥ 1} for d = (2n + 1)π/2 and n ∈ Z even, {iv : v ≤ −1} for d = (2n + 1)π/2 and n ∈ Z odd. c. {w ∈ C : <(w) > 0 and =(w) > 0} d. {w ∈ C : <(w) < 0} ∪ {w ∈ C : <(w) = 0, |=(w)| > 1}. 45 a. ±(2 + 2kπ); b. (2 + 2kπ), −2 + (2k + 1)π; c. ±(2 + 2kπi); d. ±(2i + 2kπ); e. (2k+12)π±icosh −1(sinh2) = (2k+12)π±i ln(sinh 2+ √ sinh2 2− 1); f. ±i(cos−1(sin 2)+2kπ). 46 a. 2kπ+π; b. 2kπ±i ln(2+ √ 3); c. 2kπi±(ln(2+ √ 5)+12πi); d. ± ln(4+ √ 15)+(2kπ+12π)i; e. kπ + (−1)ni ln(2 + √ 5); f. z = (nπ − 14π) + (−1) ni sinh−1 1√ 2 . 47 a. log 1 = 2kπi, Log 1 = 0; b. log i = πi(2k + 12), Log i = 1 2πi; c. log(−3 + i) = 1 2 ln(10)− i tan −1 1 3 + iπ(2k + 1), Log(−3 + i) = 1 2 ln(10)− i tan −1 1 3 + iπ. 49 Everywhere except at z = ±2i and on the half line x ≤ 0, y = −2. 50 10i, (10− 4π)i. 51 −iπ/2 and 3iπ/2 respectively. 52 a. exp(−12π + 2kπ); b. exp(−π/2); c. −e π d. eπ/2 (cos ln 2 + i sin ln 2). 53 exp(i ln √ 2 + 34π + 2kπ); 9.92. 54 exp(−π 2(k + 14)). 55 2− i. 56 12 i 57 a. 29 2 − 11i; b. −iπ. 58 −4; −4(1− iπ). 60 a. 10πi; b. 83 + 16i; c. − 8 3 − 16i; d. 2πiab. 63 a. 0 64 πi 65 2πi/3 66 a. The largest domain in which Log(z2 − 1) is analytic is C \H where H = {z ∈ R : |z| ≤ 1} ∪ {iy : y ∈ R}. b. Part i) is correct – correct partial fractions and anti-derivatives Log(z− 1) and Log(z+ 1) are analytic in the domain C \ {z ∈ R : z ≤ 1} which contains γ. Part ii) is not a correct use of the Anti-derivative Theorem (and so may be incorrect) as γ goes outside the maximal domain of analyticity of Log(z2 − 1) i.e. γ crosses the imaginary axis which is part of H above. c. i(2 tan−1(1/2) − π) ≈ −2.214i. (The answer from ii) is 2i(π − tan−1(2)) ≈ 4.069i (incorrect)). 67 f is analytic in C \H where H = {−s± i : s ≥ 0} 68 a. 2πie; b. −49π exp( 2 3 i); c. − 1 2πi. 69 0. 70 −πi/16. 71 0 (n 6= −1), 2πi (n = −1). 72 0 for n ≥ 0, 2πi/m! if n = −m− 1. 73 0, ±e, ±e2, ±(e2 − e). 74 a. e−1 ∑∞ n=0(−1)n(z − 1)n/n!, radius = ∞; b. radius = ∞; c. −1 6 ∑∞ n=0 ( z − 2 2 )n − 1 3 ∑∞ n=0(−1)n(z − 2)n, radius = 1. math2069 Complex Analysis Problems 2017 16 75 a. i) |z − 1| < 2. ii) −1/6 − (z − 1)/36 − 7(z − 1)2/216 − · · · . iii) an = −(1/2n+1 + (−1)n/3n+1)/5. b. i) |z| < 1/3. ii) 1 + 4z + 25z2/2 + · · · . iii) an = n∑ k‘=0 3n−k k! . c. i) |z| < π/2. ii)1+3z2/2+29z4/24+· · · . iii) f(z) = ∞∑ n=0 bnz 2n where k∑ n=0 bn (−1)k−n/(2k− 2n)! = 1/k! for k ≥ 0. d. i) |z−1| < 4. ii) e2/4+7e2(z−1)/16+25e2(z−1)2/64+· · · , iii) an = e2 4n+1 n∑ k=0 8k(−1)n−k k! . e. i) |z| < 1. ii) 1 − 3z2/2 + 37z4/24 + · · · . iii) a2n = (−1)n n∑ k=0 1/(2k)! for all n ≥ 0 and an = 0 for all n odd. f. i) |z| < π/4. ii) 1 + 3z2 + 19z4/3 + · · · . iii) f(z) = ∞∑ n=0 bnz 2n where k∑ n=0 bn (−4)n−k/(2n− 2k)! = 1 for k ≥ 0. g. i) |z| < π/2. ii) 1 + z2/2 + 5z4/24 + · · · . iii) f(z) = sec z = ∞∑ n=0 bnz 2n where k∑ n=0 bn (−1)n−k/(2n− 2k)! = { 1 if k = 0 0 if k > 0 . 76 a. 1 (1− z)2 ; b. z (1− z)2 ; c. 1 + z (1− z)3 = d dz ( z (1− z)2 ) ; d. z + z2 (1− z)3 ; e. −Log(1−z). 77 √ 2. 79 ∑∞ n=0(z − i)n/(1− i)n+1, radius = √ 2. 80 cosh z = − ∑∞ n=0 1 (2n)!(z − iπ) 2n; z5 = ∑5 n=0 an(z − iπ)n where a0 = iπ5, a1 = 5π4, a2 = −10π3i, a3 = −10π2, a4 = 5πi, a5 = 1. 81 1 + z2 − z2 8 + z3 16 − · · · . 82 1 + 3z 2 + 7z2 8 + 17z3 48 . 83 a. The 3 regions are I) |z + 1| < 2, II) 2 < |z + 1| < 3, III) |z + 1| > 3. b. For I), f(z) = ∞∑ n=0 [ − 1 10 · 1 2n + 4 15 · ( −1 3 )n] (z + 1)n For II), f(z) = 1 5 ∞∑ n=0 2n (z + 1)−n−1 + 4 15 ∞∑ n=0 ( −1 3 )n (z + 1)n = ∞∑ k=−∞ ak(z + 1) k where ak = { 2−k−1/5 if k < 0 4(−1)k/(15 · 3k) if k ≥ 0 For III), f(z) = ∞∑ n=0 [ 2n 5 + 4(−3)n 5 ] (z+1)−n−1. 84 ∑∞ n=0 z n−3, 0 < |z| < 1, − ∑∞ n=0 z −(n+4), |z| > 1. 85 a. 1 z − 1 + ∑∞ n=0 2n (z − 1)n+1 ; b. (−1)n (1− 2i)n+1 for n ≥ 0, a−n = (−1)n(1 + i)n−1 for n ≥ 1; c. z−1 + 1 6 z + 7 360 z3 + . . . . 88 a. 2π; b. a0 = 1; c. p−1∑ n=0 ( p n ) Bn = 0 for p ≥ 2, B0 = 1, B1 = −1/2, B2 = 1/6, B4 = −1/30, B6 = 1/42, B3 = B5 = 0. math2069 Complex Analysis Problems 2017 17 89 a. z = 0, removable; b. z = 0, pole, z−1; c. z = −1, pole, (z + 1)−1; d. poles at z = ±1, e2(z−1) , −1 2e(z+1) ; e. z = 3, pole, −(z − 3) −3; f. z = 0, essential, ∑ m>1 z 1−m/m! 90 a. 0; b. 1; c. 1; d. e2 ,− 1 2e ; e. 0 f. 1 2 . 92 a. 2; b. 8; c. 3; d. simple pole at z = nπ, n ∈ Z; e. pole of order 2. 93 a. z = i, ei/(2i); z = −i, e−i−2i ; b. z = i, 1 2(i sin 1). c. z = −1; 5 4 ; z = 1;− 1 4 ; d. z = iπ2 + kπi;1; 94 a. 2πi/e; b. 2πi(1− e−1); c. 2πi sin 1. 95 a. 0; b. 0; c. 16/(12− 27π2). 96 0. 97 a. 0 if n < −1, 2πi/(n + 1)! if n ≥ 0, 2πi if n = −1; b. 0 if n is odd, 0 if n < 0, (−1)n/22πi/(n+ 1)! if n ≥ 0 is even. 98 Z{n} = z/(z − 1)2 99 a. z(z+ 1)/(z− 1)3; b. z(z2 + 4z+ 1)/(z− 1)4; c. z2(z+ 1)/(z− 1)3; d. az/(z−a)2; e. z(z2 − 1) sin θ/(z2 − 2z cos θ + 1)2; f. z(z − a cos θ)/(z2 − 2az cos θ + a2) 100 a. {6n − (−1)n}; b. {(2/ √ 3) sin(2nπ/3)}; c. {95 × (4) n − 15(−1) n}. 101 a. yn = 1 4(3 + (−3) n); b. yn = 2(−1)n + n+ 2; c. yn = (3n− 425 )2 n − 185 ( −12 )n + 10. 102 a. π/2; b. π/8; c. π/ √ 2; d. √ 3π/3. 103 πe−a/a. 104 a. πe−3 cos 3; b. πe−1 sin 2. 105 π40{8e −1 + e−2(4 sin 1− 2 cos 1)}. 107 I1 = −π/96; I2 = 0. 109 πi/4. 111 I2 = 49π Last revision : February 2017. 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