HW-part4 - Mathematics 245 Name Functions of a Complex...

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Unformatted text preview: Mathematics 245 Name Functions of a Complex Variable Final Exam (Friday section) M2h6-5/60 d. Show in a diagram the image, in the w-plane, of the Square 0 < ‘ < 1 1n the Z-plane (2 = x + 1 y) under Jan. 22, 1960, 6:00 P.M. to 8:00 P.M. o<y<1 each of the mappings: (1) w = e72 Please write on these sheets. (2) w = i z + 1 I. (8 credits) Express the following numbers in the form a + b1: (3) w = (1 + 1)z ‘ (2 + 51)(3 + i) - 25 f (4 + 31) = (1 + i)4 = elm/2 elm/W [n1 = = , tn(l + 1) 31 = II. (8 credits) Place the appropriate sign (<, >, <, 2, a, or g )* ‘in the following expressions, where a, B,... are complex numbers: Ia + Bl lal + la! Iu- BI lei-Isl Iaal IaI-Ial Im(a + a) Im(a) + Im(5) a; [ale The last of these symbols means merely that there is no relation between the quantities: either one may be larger or they may be equal. <3} |a+a|2 mm 1..- l l {MM Ial +151 la+el la-al / III. (14 credits) Complete the following definitions: 1. The series co + cl +---+ on +--- is said to be absolutely convergent if 2. The series ¢o(z) + ”1(2) +-"+ ¢n(z) +"' is said to be uniformly convergent for z in a set S if 3. A function f(z) is said to be analxtic in an (open) domain D if h. A point set in the complex plane is said to be an open set if 5. A point set in the complex plane is said to be simplx connected if 6. The function f(z) is said to have an isolated singularity at z = 20 if 7. The function f(z) is said to have a simple pole at 2320 11' IV. (12 credits) Give the power-series expansions about 2 a 0, and the radii of convergence, for the following . functions, using the format of the 1-s-2 example: function series radius of conv. ez =ll+z+%z2 +----¢-(—I]§!)zn +--- 00 cos 2 = w 1 1 + Z = 2n(3 + z) = H + N bl 1| V. VI. (10 credits) For each of the following functions u(x,y), give a function v(x,y) (the so-called harmonic conjugate of u(x.y)) such that u(x,y) + i v(x,y) of z - x + i y in the unit circle Izl < 1 ; if no such function exists, write "none" u(X.y) = u(my) = u(X.¥) = u(X.¥) = u(x,y) = (l# credits) x, v(x,y) = x2 - ye. V(x.y) = x2 + ye, V(X:Y) = tnx/(x + 2)2 + ye, v(x,y) = ex cos y, V(XnY) = premises (there may be more than one) by encircling the corresponding letters (a, b, etc.). 1. If the real-valued functions u = u(x,y) and V = V(x:Y) u v v _ _ u 3—); =37 and 3? — 3-3; in a domain D, the function f(x + i y)»= u + i v is necessarily ‘ a. b. c. d. e. 2. If have continuous derivatives satisfying continuous in D. a constant. analytic in D. bounded in D. an entire function. f(z) is an entire function (i.e. analytic for all _ u - (gig) is an analytic function Mark all the correct conclusions from the stated z), and if If(z)l < 1 for |z| > 1, then, a |f(z)| < l for all z b. f(z) is a constant c. f(z) is real-valued d. f(z) = o co 3. If f(z) is defined by E anz" whenever this 0 (n) this series converges, and if lanl g l/n! for all n, then 2 a- |f(2)l s,|e l b. f(z) is an entire function e. 1/T(z) is an entire function d. f(z) is an entire function e. |r(z)| g e|z‘ VII. (8 credits) Complete each of the following statements to indicate the values of the complex variable 2 for which the statement is valid: (Example: lezl < 1 for Regz) < 0) co 1. If the series 2;:(3) anJ converges for z = l + i, it converges absolutely for 2. Ieizl < 2 for 3. sinaz + coszz = 1 for H. Re(£n z) < 0 for Q55) the path of in being in each case the circle described cowise: [2 dz f3 dz 1% dz fsin(z£ sin 2 1"1r‘- Z £35) VIII.(10 credits) values of the following integrals, |z|=1 FINAL EXAM Closed Book Answer 6 questions. 1. Find the radius of convergence of the power series 2 anz , given th at 2 en (a) :5?» an O converges co 5 (33-7175) dz (b) an? X zn+1 ‘ |2|=1 sin 2 (C) E = ‘ 2_—____-éE-TE n IZ'=n (Z - 1000) 2 . Given that sinz z(z2 - 1) + _ g” a Zn n - GD Prof. L. Bers Math 2&5 Sec A January 21;, 1962 00 n O in some domain, find all possible values of a_2. 3. Compute +00 COS X § 2‘“ -G 1 + X 1;. Given that f(z) is holomorphic for Izl s 00, and that for real x, - 1 < x < 1, lf(x)| s e . What can you say about f? -1/x2 £39 Let f(z) be entire. Assume that f(z) # o for z ¢ n (n = 1,2, ...) and that (ii 76 362 U f(z) L, f(z) I2] = n-1/2 Izl = n + 1/2. Is this function transcendental? Find all functions f(z) such that f(z) is holomorphic except perhaps for z = 0,1,2,3,14., 21 'f(z)| < <:’1oo lzl 26 _ |2-1|I2-21|2-3l|2-h| and gfdz=l, Sfdz=-3, |2|=3/2 l2l=5/2 Sfdz= ffdz=0 IZI=7/2 lzl = 10 Let f(z) be holomorphic for Izl<2. Where does the series £3- 1 dnf(z) £3;, ET ‘Ezh“ n=0 converge? Let f(z) be holomorphic for Izlg 1, f(0) = o, r(1) = 1 and If(z)l: 1 for [Z] = 1. Show that lfv(1)l i 1. State and prove the maximum modulus theorem. State and prove Liouville's theorem. (5% Professor Nirenberg FINAL EXAMINATION Complex Variables 1. Find the radius of convergence of each of the following series (1) The power series in (2—1) about i of the function 2 2 e +1032 (ii) in + cos 941)" 2n 0 no (111) 2: 2"1 sin n2 O 2. Find the Laurent series expansion near 0:) in powers of z of the function f(2) = z- z— . Is it regular at on ? 3. (a) State Rouche"s theorem. (b) let f(z) be analytic in the strip |Im 2| < 10 with |f(z)l < 1 . Prove that cos z + f(z) has an infinite number of zeros in the strip. 4. Evaluate the integral f 10512: dx lML): around the half circle Q for R large. By taking its —R O R real part evaluate an j“ leng dx . o ‘t + x 5. (a) The function f(z) is analytic in the punctured disc 0 ‘ IZI ‘ 1 . “and LN “I‘ll continuous in 0 < Izl .E 12 and |f(z)|_<_ 1 on [2! = 1 . Prove that k lf(z)l _<_ 1 everywhere. (b) Assuming furthermore that on Izl s 1 , Re f(2) = y = Im z , determine the function f , proving your statement . ...
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