Abel's theorem in problems and solutions - School of Mathematics

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188 Problems of Chapter 2 CD. Since the function has a finite number of branch points then one can choose the curves AB and CD sufficiently short and the curves CA and BD sufficiently close to the cut, in such a way that in the region bounded by the curve CABDC there are no branch points of the function In this case one can evidently transform the curve CABD into the curve CD without passing through any branch point. Since the function possesses the monodromy property, the function at the point D is uniquely defined by continuity along the curves CD and CABD. Starting from the branch and covering the curve CABD, moving first on the branch, one passes later to the branch, moving finally on it. In this way, along the curve CABD and therefore also along the curve CD, one moves from the branch to the branch, exactly as along the curve AB. FIGURE 89 311. a) Hint. Answer. (Here is the positive value of the square root) b) ±1/2; c) 0; d) e) 0. 312. Hint. It suffices to prove that the functions and possess the properties stated by the problem, and that if the functions and possess these properties then the functions (where is an integer) also possess these properties. Solution. 1) If then The curve sought is the curve with the parametric equation where is the parametric equation of the curve C. 2) If then and the required curve is the curve with the equation (degenerated to a point). 3) Suppose that and that the statement of the problem is true for the functions and By the definition of the
Solutions 189 sum of multi-valued functions we have where is one of the values of and is one of the values of Since for the functions and the statement of the problem holds, there exist two continuous images and starting respectively from the points and If and are the parametric equations of the curves and then the function (which is continuous, being the sum of continuous functions) is the parametric equation of the required curve because In an identical way one considers the case in which (in this last case the continuous function sought is because by hypothesis the curve C does not pass through the points at which the function is not defined, and consequently 4) Suppose that and that for the statement of the problem is true. By the definition of the function we have where is one of the values of The mapping can be considered as the composition of two mappings, and Since for the function the statement of the problem holds, there exists at least one continuous image of the curve C under the mapping beginning at the point By virtue of the result of Problem 293, there exists at least one continuous image of the curve under the mapping beginning at the point The curve is the curve required. 313. At the point chosen arbitrarily, the function takes values: where Since the sum of continuous functions is a continuous function, the single-valued continuous branches of the function are the following functions: where 314. a) See Figure 90. Hint. Use the schemes of the Riemann surfaces of the functions and (cf., 288, 299). b) See Figure 91. Hint. Cf., 304, 307. c) See Figure 92. Hint. Cf., 288, 292. d) See Figure 93 (the Riemann surface is shown in Figure 127). Hint. Draw first the schemes of the Riemann surfaces of the functions and
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ABEL’S THEOREM IN PROBLEMS AND SO
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Abel’s Theorem in Problems and So
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Contents Preface for the English ed
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A.10.3 The integrable case A.11 Top
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Preface to the English edition by V
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of the differential equations. Unfo
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Preface In high school algebraic eq
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Introduction We begin this book by
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Introduction 3 where by is indicate
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Introduction 5 By Viète’s theore
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Introduction 7 We will be able to p
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Chapter 1 Groups 1.1 Examples In ar
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Groups 11 the triangle ABC into its
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Groups 13 7. Find all symmetries of
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Groups 15 notations (see the soluti
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Groups 17 element, which we will de
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Groups 19 35. Suppose that is the m
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Groups 21 If a group is isomorphic
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Groups 23 tude); (rotation by 180°
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Groups 25 Hence left cosets generat
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Groups 27 as a permutation of the t
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Groups 29 102. Let be the order of
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Groups 31 111. Find all normal subg
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Groups 33 FIGURE 8 127. Prove that
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Groups 35 137. Prove that ker is a
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Groups 37 We now observe what happe
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Groups 39 FIGURE 12 vertices; 4) ro
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Groups 41 Every permutation of degr
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Groups 43 180. Prove that by multip
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Chapter 2 The complex numbers When
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The complex numbers 47 195. Prove t
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The complex numbers 49 where is the
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The complex numbers 51 Consequently
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The complex numbers 53 For complex
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The complex numbers 55 think that n
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The complex numbers 57 214. Let us
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The complex numbers 59 219. Prove t
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The complex numbers 61 functions (s
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The complex numbers 63 of continuit
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The complex numbers 65 erations of
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The complex numbers 67 a) b) c) d)
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The complex numbers 69 REMARK. If a
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The complex numbers 71 2.8 Images o
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The complex numbers 73 where all th
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The complex numbers 75 in such func
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The complex numbers 77 FIGURE 26 no
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The complex numbers 79 C from to th
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The complex numbers 81 DEFINITION.
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The complex numbers 83 FIGURE 32 cu
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The complex numbers 85 to infinity
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The complex numbers 87 property is
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The complex numbers 89 means will b
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The complex numbers 91 For example,
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The complex numbers 93 of the Riema
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The complex numbers 95 in Figure 38
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The complex numbers 97 permutation
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The complex numbers 99 2.13 Monodro
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The complex numbers 101 First we pr
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The complex numbers 103 and prove t
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Chapter 3 Hints, Solutions, and Ans
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Solutions 107 positive integer unde
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Solutions 109 on the left and by on
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Solutions 111 39. a) See Table 8; b
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Solutions 113 of the real numbers i
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Solutions 115 to all and therefore
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Solutions 117 is a subgroup of the
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Solutions 119 hypothesis. The group
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Solutions 121 The given group is th
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Solutions 123 of symmetries of the
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Solutions 125 i.e., there exists an
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Solutions 127 Answer. The normal su
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Solutions 129 In this way the quoti
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Solutions 131 sends every into itse
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Solutions 133 ement of the commutan
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Solutions 135 143. Let be the three
Page 153 and 154: Solutions 137 150. See 57. Suppose
Page 155 and 156: Solutions 139 subgroup. Hence if a
Page 157 and 158: Solutions 141 is commutative. Since
Page 159 and 160: Solutions 143 180. Since the lower
Page 161 and 162: Solutions 145 FIGURE 43 190. The pe
Page 163 and 164: Solutions 147 permutations of type
Page 165 and 166: Solutions 149 is a field. 195. We h
Page 167 and 168: Solutions 151 necessarily that and
Page 169 and 170: Solutions 153 Let C be the field of
Page 171 and 172: Solutions 155 in as numerators and
Page 173 and 174: Solutions 157 c) d) e) where 226. W
Page 175 and 176: Solutions 159 function of real argu
Page 177 and 178: Solutions 161 i.e., Consider the re
Page 179 and 180: Solutions 163 Choose as the smalles
Page 181 and 182: Solutions 165 Since the function is
Page 183 and 184: Solutions 167 FIGURE 55 FIGURE 56 F
Page 185 and 186: Solutions 169 FIGURE 60 257. Suppos
Page 187 and 188: Solutions 171 265. 266. 267. If the
Page 189 and 190: Solutions 173 following way. If the
Page 191 and 192: Solutions 175 for all such that and
Page 193 and 194: Solutions 177 b) In order for to va
Page 195 and 196: Solutions 179 to the condition coin
Page 197 and 198: Solutions 181 are continuous functi
Page 199 and 200: Solutions 183 turn around the other
Page 201 and 202: Solutions 185 sought is shown in Fi
Page 203: Solutions 187 FIGURE 86 FIGURE 87 p
Page 207 and 208: Solutions 191 316. a) Let and be th
Page 209 and 210: Solutions 193 by the formal method,
Page 211 and 212: Solutions 195 The correct scheme is
Page 213 and 214: Solutions 197 branch of the functio
Page 215 and 216: Solutions FIGURE 110 FIGURE 111 329
Page 217 and 218: Solutions 201 It follows that and b
Page 219 and 220: Solutions 203 340. To every sheet o
Page 221 and 222: Solutions 205 Since by hypothesis t
Page 223 and 224: Solutions 207 moving along a curve
Page 225 and 226: Drawings of Riemann surfaces 209 Th
Page 227 and 228: FIGURE 119 211
Page 237 and 238: Appendix by A. Khovanskii: Solvabil
Page 239 and 240: Solvability of Equations 223 functi
Page 241 and 242: Solvability of Equations 225 Suppos
Page 243 and 244: Solvability of Equations 227 where
Page 245 and 246: Solvability of Equations 229 finds
Page 247 and 248: Solvability of Equations 231 quadra
Page 249 and 250: Solvability of Equations 233 More p
Page 251 and 252: Solvability of Equations 235 group.
Page 253 and 254: Solvability of Equations 237 Every
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Solvability of Equations 239 The gr
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Solvability of Equations 241 FIG. 1
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Solvability of Equations 243 are ob
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Solvability of Equations 245 corres
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Solvability of Equations 247 where
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Solvability of Equations 249 up to
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Solvability of Equations 251 fined
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Solvability of Equations 253 the qu
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Solvability of Equations 255 the no
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Solvability of Equations 257 THEORE
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Solvability of Equations 259 where
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Bibliography [1] [2] [3] [4] [5] [6
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[23] [24] [25] [26] [27] [28] 263 V
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Appendix (V.I. Arnold) The topologi
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Index Abel’s theorem, 6, 103 addi
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pack of sheets, 96 parametric equat
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