HW-part9 - HATHEIATICS 213 Final Examination Professor...

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Unformatted text preview: HATHE’IATICS 213 Final Examination Professor Hockey 1. 2. 4. 5. 7. 8. Find all of the fifth roots of 1 + 431 and all values of (4)1“. Assuming the least upper bound property of the real numbers, prove that any Cauchy sequence of complex numbers converges to some complex number. You may not Essume the Cauchy sequence prOperty of the real numbers. Prove it if you need it. For subsets of the plane define compactness, boundedness, closednese, Openedness and connectedness. Illustrate each definition with an example and a counter— example . Define analytic fimction and give as many properties as you can think of which are equivalent to anelyticity. Let so + a z + a2 22 + ... be a power series which converges to 1 z3—l (22 + z + 1)(z2 + 32 - 4) for all z with I z\ sufficiently small. Compute the largest possible 5 such that the series in question converges for I 2 l < 6 . What can you say about the seriesiflz|7€ . Starting from Cauchy's integral formula for circles (which you may assume to be true) prove that every non-constant polynomial has at least one root. Define isolated singularity and describe in two ways the three way classification into movable singularities, poles, and essential singularities. Define residue and find the residue of cos z/(z2 + l) at z' = 1. Outline the main points in the reading period assignment up to and including the forum- lation of the general form of the Cauchy integral theorem. W 64 FINAL EXAMINATION o 133:2: ggédan Duo: Wed. Doc. 16, 2:00 P.M. Directions: 1. Anower so many problems so you can - but at loeot 6 problems. More are needed for an A 2. Bo neat and accurate. . 3. Delete any straightforward computationl. 4. You may 2;; work Jointly. w 5. You may uoo any notoo or roforoncoo. If a rooult is done in some reference, do not rocspy it, but meroyy give the page and name of the reference. 6. Paporo handed in after 2:00 P.M. will not be accepted. ' 7. Graded paper: may be pickod up in my office between 1-2 P.M. on Thuro., Die. 17. If you prefer, you may give me a oolf-eddreoeod envelope. 8. PLEASE DO THE PROBLEMS IN THE 0 ' ON THIS SHEET. RDER LISTED l. (a) Prove that the root: of the polynomial n n-l p(z) = z + an_lz + ... + a O depend continuously on the ceeEicionto, (egg. (b) If f 1: analytic in a neighborhood of 20 and if f has a zero of order n) 1 at 2., prove that g(z) = {(z) - a has n simple zorooo in a neighborhood of z. for all oufficiontly small i complex number: a. A PAUSE FOR SOME NOTATION: K = open unit diok, conterod at O. Kr = opon diok of radiuo r< l, centered at Go U, V = simply connocted opon note in D . Math. 609, pago 2 2. Lot 1' ho analytic inficv. If 11m \r(z)|= HDU 8’! constant , prove that Hz) l—EJMz) for oomo h analytic oome conotanto oJ . c . 3. Lot r bo analytic in |r(z)|5u in x and £03) 3! O. Provo that (numbor of zoln Ig,)$ l 1.3(f(0)l . g kg r M (Hint: considor g(z)) A. Find the following com I: U—o K, whoro U in (a) intorooction of {I - 1[ < 1}, (b) rogion outoido of 12 4 1): 1E , (c) Kn {x>t}. 5. Suppooo tho linoar tron w = 8(2) map: tho unit circlo onto tho rtth 3(0) = 1. (a) On what circlo muLio? (b) Provo that ovory nio circle io a pooaiblo valuo of SQ). 6. Lot f bo an ioonorphi‘rtoiwu and lot Ur=f(Kr). (a) If h io on autor‘ U loaving No) fixod, provo that Ur . (Hint: apply tho Schwarz Lomma:f ). (b) If U io canvox,; 0,. 1o aloo convox. (Hint: lot [zllsand conoidor amt) = '32) + tf(z) ). (c) Moro gonorally, ii opon disk in TE, provo that HE) is con‘IHint: Provo thoro io an Moth. 609; D85. 3 automorphilm q of K ouch that q(E) = Kr, and conoidor foq'l) 7. Lot 1’ and 1“ ho analytic in K with F univalont thoro, and m U = r(x), v = F(K). If Mo) = No) and lid V, provo that thoro io a function g analytic in K ouch that r s Fag , with (guns lzL Moroovor, ohow that f(K,.)C Fair), 8. Look up tho dofinition of "natural boundary" for on analytic function. Prcvo that/:3; connoctod opon oot D in tho complox plano, than a function f analytic in D having D on a natural boundary. (Romarkz if you can not do tho gonoral caoo, then do what you can). 9. Lot f bo an ontiro analytic function with tho praportloo: (o) f(x 4- 21’) = f(x) for any roal x, (b) (112.1 5 exp ah) , for all z and oamo a) O. Provo~ that f has tho form n f(z) =2 akcikz , whoro 115 a. =-n $1 final lunatic: l. 2. 3. 4. 5. 6. 7. 8. was 221 (@ January 22, 1960 9:15 am. Define normal {will of functions. Show that a fanin of functions analytic and Winsregionisnonalthsre. - ' State and prove the Rimnn lapping theorem for a simply connected region. and schli let the functions 23(1) analytic/‘13: I < l emerge almost uniformly there to 10(3). with rn(o) - o, and let x be a omtinmn containing ,1 - 0 which lies a the lease of |;I< 1 under the transformation In- :°(z). show that for n suf- ficiently lnrge It lies in the hinge of Isl < 1 under the transfomation v =- fn(:). State and prove the Bieherbsch mchensats. 1f the region I) of the plans is star-shaped (with res ct to the origin) and if v-f(l) nape D onto v|<l, then the subregion If s)|<r, 0<r<l, of D is also star-shaped. Show that a doubly connected region bounded by two disjoint Jordan curves can he lapped confomally onto an anmalus. Show by Iethods of potential theory that any simply connected region possessing a Green's function can be nsypsd conformally onto a circle. Sm that an arbitrary moth surface which is topologically equivalent to a sphere can be lapped cmfornally onto a sphere. State the Unifonisstion Theorem and outline its proof. Professor J. L. Walsh 11 Math; 213a (EB Hmuewcrk Due Nov. 7.2, 1965 Land the 11-41982 hrneformation‘which carries 0, i, -i into -1, 0. Show that any four distinct points can be carried by a linear trans- formation into positions I, -1, k, -k where _> ‘ depends on the points. How many solutions? I Find the linear transformation whiyhparries [2; 2 into I2 4- 1| = 1, the point ~2": to the ofirigin,"ancl'theworigin into i. What is the most general tiEnsformation,of :'[z| 2 1 onto itself? of the upper hali p‘ane onto itself? and >— Find a linear transformation which carries Izl = |z — = ‘1: into concentric circles. What is the ratio of the radii? Find the fixed points of z Zz 32-4 z w: 2;-1 ' w‘ '321' ' w‘ 3-1 ' “"— Z-z ' Which of these transformations are elliptic, hyperbolic, or parabolic? l and Iz-1[=4. Find all circles which are orthogonal to In! 8’5 Math. 213 a .‘iomework Due November 29, l965 In the“: exercises all mappings are to be conformal and you are exoocted ‘ . to give an explicit expression for the analytic function which yields the required mapping, I, Map the common partofthe disks ’2] < l and [z -ll «. 1 0hr0 H’e inside of the unit circle. Choose the mapping so that the twa symmetr'? a re pres". rved. 7-. Map the region between = 1 and I .17.- 1 z ’ 2!: on 3 half plane \ 3. Map the complement of the arc = I, y i 0 on the outside of the unit circle so that the points at m correspond tr. each other 4. Map the outside of the parabola ya = 2px on the disk [W] > 1 so that x = 0 and x = «g conesoond to w = l and w = O. 5. Map the outside of the dlipse (ac/a)Z -: (y/b)?' = 1 onto [w] < l with preservation of symmetries. 6. Map the pa rt of the z-plsne to the left of the right—hand branch 2 of the hyperbole: x - ya = 1 on a half plane. (Hint: Ccmsfdu: on use ridc the mapping of the upper half of the . 2 re 1 — - ‘ ' ‘ g onzby w - a, , on the other 2143 .11: mapping .2! re quadrant by w = z; - 32). Math. 2.13a Hour-examination Dec. 3, 1965 .. z + l - . , - l. prand. in partial factions. z (z - l) 2. What are the values of (l + if? For what values oi z is convergent, and what is the Sum? Prove that a continuous function from one metric space to another maps connected sets on connected sets. Find the image of the region l< |z+l| < z underthe mapping (g; 2 Z . .- w _ Ta!— . Is the mappmg one to one? The circle I: — II = l is mapped by w =73“?!- . Where is the center of the image circle? What is the value of ; IzIzdz 7 where -. y is the clock-wise boundary of the first quadrant of In the following integrals C sense. Find zdz I‘ dz 7. a) -—-—-— b) .2 .. c) 1:: dz . Cz-l " L, z:_1' (a - I) What is 1 153d 5‘" = 217i ‘rc £377;— -1 if C is the unit circle (positive sense) and 030") = l: + l’ (Different answers ior [2} < 1 and > 1), I <1. is the circle I2! = 2 in the positive Math. 213!) Homework 1. 3. Due February 7, I966 11 [(2) is analytic in |z| 51 and u If(z)| = 1 when a rational function. 1, show by use of the reflection principle that {(z) is I Prove that lim 2 (l+§)n= e n—>oo uniformly on any compact set. (Use the series expansion of lo 30 + D. Show that 00 (p (s) = Z 1:5 n=1 converges for Re a > 1, that it represents an anyaltic function (known as Riemann's zeta-function) and that .P'(s) can be obtained by te rm-wise differentiation. Prove that (1- 21") ms) = 1‘5 - z" + 3-5 - when Re .5. > I, and that this series converges for Re 5 > 0. (Suggestion: estimate 11'8 - (n+ l)'s)_ Li —l-z— is developed in powers of z - a where a is a l+z real number, what is the radius of convergence. Find the develop- ment. (Suggestion: use partial fractions). Develop log (iii—1) in powers of 2 up to the term 26, Find the first three non-zero terms in the development of tan 2. by dividing the sine and the cosine eerie 8. (Check with the book, 2nd. ed. p. 182, lst. ed. p. 146), Math 2131: l. Homework Due March. 14, 1966 Z The expression £12.). _ 3 sun, {z z} = f’(z) £'(z) is known as the Schwarzian derivative of f. If f has a pole or zero of order m at find the leading term in the Laurent 20, development of {L z}. Find the Taylor series of (log (4 - z))Z about the origin. (Give the general expression for the n th coefficient). Find the Laurent series of term 23. cot 2 about the origin up to the Compare the development in the preceding exercise with what you can get from the partial fraction development of 01; z. Show 00 a? . I ‘ that you can thus find the values of Z and 2 1 n 1 What is the canonical product development of cos (V z), and what is its genus? Show that if {(2) is of genus 0 or 1 with only real zeros, and if £(z) is real for real 2., then all the zeros of f'(z) are real. Hint: use the canonical product and consider Im f'(Z)/f(z). GE Y? MATHEMATICS 213b Professor Walsh June 5, 1965 9:15 a.m. 1. Prove the validity of the Cauchy—Hadamard formula for the radius of convergence of a power series. 2. It is sometimes stated that a power series converges uniformly in its circle of convergence. Correct this state- ment and prove the intended theorem. 3. A function with period 2ni is analytic at every finite point of the plane. Derive a formula for the function. 4. State, and outline the proof of, the Riemann mapping . theorem. 5. Prove that any two elliptic functions with the same periods are connected by an algebraic relation. 6. State and prove a theorem expressing the mean value property as sufficient for the harmonicity of a function. 7. Show that a function harmonic and bounded for all z is identically constant. I", t) OVER in: Mathematics 213b Page 2 8. Define subharmonic function. Show that a continuous function v(z) is subharmonic in a region 0 if and only if we have 2" - v(zo) <. i J“ M2 +re1°)d6 for every disk lz - zol g r in n . 9. Show how an arbitrary region of finite connectivity can be mapped onto an annulus minus a number of circular arcs. 10. Without the use of the Riemann mapping theorem, and by study of the level loci of Green's function G(z,0) for a simply connected region 0 (having at least two boundary ipoints and containing the origin) with pole in the origin, where G(z,0) —> — oo as z a 0, show that the function w - exp[G(z,O) + iH(z)] maps 0 one-to-one and conformally onto lwl < 1. Here H(z) indicates a function conjugate to G(Z.O) in 0- Final Examinati Professor welsh 9/ [thematics 221 May 27, 1963 2:15 p.m. State and mrz‘s Lemma. State explicitly any form of the of Maximum Modulus that you use. state and Bieberbach Area Theorem (Flachensatz). Show thatxction z + azz2 and schlicopen unit disk, then ’32, g 2 . + ... is analytic Show that‘n unit disk is mapped conformally onto a convex mh circle in the disk whose center is the originLonto a convex curve. Let the fw1(z), f2(z),..., analytic and schlicht in a simpld region D , with f1(0) = 0 and 0 in D ,continuously there to a function f(z) not identitant. Show that if a closed bounded set E is in the map of D by f(z) { then E is containmap of D by each fn(z) for n sufficient Outline tthhe Riemann mapping theorem. Show how a functions arise naturally in the con— formal mapjtiply connected regions. (mmm 30 Mathematics 221 8. 9. Show equiva sphere. State the Uniformization Theor May 27, 1963 that a (suitably smooth) surface which is topologically lent to a sphere can be mapped conformally onto a em and outline its proof. 7} ...
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