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

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250 Appendix by Khovanskii A.13 Functions of several complex variables representable by quadratures and generalized quadratures The multi-dimensional case is more complicated than the one-dimensional case. We have to reformulate the basic definitions and, in particular, to change slightly the definition of representability of functions by quadratures and by generalized quadratures. In this section we give a new formulation of the problem. Suppose there have been fixed a class of basic functions and a set of allowed operations. Is a given function (being, for instance, the solution of a given algebraic or differential equation, or the result of one of the other allowed operations) representable in terms of the basic functions by means of the allowed operations? First of all, we are interested in exactly this problem but we give to it a slightly different meaning. We consider the distinct single-valued branches of a multi-valued function as singlevalued functions on different domains: we consider also every multi-valued function as the set of its single-valued branches. We apply the allowed operations (such as the arithmetic operations or the composition) only to the single-valued branches on different domains. Since our functions are analytic, it suffices to consider as domains only small neighbourhoods of points. The problem now is the following: is it possible to express a given germ of a function at a given point in terms of the germs of the basic functions by means of the allowed operations ? Of course, here the answer depends on the choice of the single-valued germ of the multi-valued function at that point. However, it happens that (for the class of basic functions we are interested in) either the representation sought does not exist for any germ of the single-valued function at any point, or, on the contrary, all germs of the given multi-valued function are expressed by the same representation at almost all points. In the former case we say that no branches of the given multi-valued function can be expressed in terms of the branches of the basic functions by means of the allowed operations; in the latter case we say that this representation exists. First of all, observe the difference between this formulation of the problem and that of the problem expounded in §A.1. For analytic functions of a single variable there exists amongst the allowed operations, in fact, the operation of analytic continuation. Consider the following example. Let be an analytic function, de-
Solvability of Equations 251 fined in a domain U of the plane which cannot be continued beyond the boundaries of a domain U, and let be the analytic function in the domain U defined by the equation According to the definition in §A.1, the zero function is representable in the form for all the values of the argument. From this new point of view the equation is satisfied only inside the domain U, not outside it. Previously we were not interested in the existence of a unique domain in which all required properties should hold on the single-valued branches of the multi-valued function: a result of an operation could hold on a domain, another result in another domain on the analytic continuations of the functions obtained. For the of a single variable one can obtain the needed topological limitations even with this extended notion of the operation on analytic multi-valued functions. For functions of several variables we can no longer use this extended notion and we must adopt a new formulation with some restrictions, which can seem less (but which is perhaps more) natural. Let us start by giving precise definitions. Fix the standard space with coordinates system DEFINITION. 1) The germ of a function at a point can be expressed in terms of the germs of the functions at the point a by means of the integration if it satisfies the equation where For the germs of the given functions the germ exists if and only if the 1-form is closed. The germ is thus defined up to additive constants. 2) The germ of a function at the point can be expressed in terms of the germs of the functions at the point a by means of the exponential and of integrations if it satisfies the equation where For the germs of the given functions the germ exists if and only if the 1-form is closed. The germ is thus defined up to multiplicative constants. 3) The germ of a function at a point can be expressed in terms of the germs of the functions at the point a by means of a solution of an algebraic equation if the germ does not vanish and satisfies the equation DEFINITION. 1) The class of germs of functions in representable by quadratures (over the field of the constants) is defined by the following
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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
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Solutions 137 150. See 57. Suppose
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Solutions 139 subgroup. Hence if a
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Solutions 141 is commutative. Since
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Solutions 143 180. Since the lower
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Solutions 145 FIGURE 43 190. The pe
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Solutions 147 permutations of type
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Solutions 149 is a field. 195. We h
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Solutions 151 necessarily that and
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Solutions 153 Let C be the field of
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Solutions 155 in as numerators and
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Solutions 157 c) d) e) where 226. W
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Solutions 159 function of real argu
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Solutions 161 i.e., Consider the re
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Solutions 163 Choose as the smalles
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Solutions 165 Since the function is
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Solutions 167 FIGURE 55 FIGURE 56 F
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Solutions 169 FIGURE 60 257. Suppos
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Solutions 171 265. 266. 267. If the
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Solutions 173 following way. If the
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Solutions 175 for all such that and
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Solutions 177 b) In order for to va
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Solutions 179 to the condition coin
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Solutions 181 are continuous functi
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Solutions 183 turn around the other
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Solutions 185 sought is shown in Fi
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Solutions 187 FIGURE 86 FIGURE 87 p
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Solutions 189 sum of multi-valued f
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Solutions 191 316. a) Let and be th
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Solutions 193 by the formal method,
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Solutions 195 The correct scheme is
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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
Page 255 and 256: Solvability of Equations 239 The gr
Page 257 and 258: Solvability of Equations 241 FIG. 1
Page 259 and 260: Solvability of Equations 243 are ob
Page 261 and 262: Solvability of Equations 245 corres
Page 265: Solvability of Equations 249 up to
Page 269 and 270: Solvability of Equations 253 the qu
Page 271 and 272: Solvability of Equations 255 the no
Page 273 and 274: Solvability of Equations 257 THEORE
Page 277 and 278: Bibliography [1] [2] [3] [4] [5] [6
Page 279 and 280: [23] [24] [25] [26] [27] [28] 263 V
Page 281 and 282: Appendix (V.I. Arnold) The topologi
Page 283 and 284: Index Abel’s theorem, 6, 103 addi
Page 285: pack of sheets, 96 parametric equat
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Abel's theorem in problems and solutions - School of Mathematics

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