# New PDF release: Abel’s Theorem in Problems & Solutions By V. B. Alekseev

Translated via Sujit Nair

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C) z2 (t) = z0 · z1 (t), where |z0 | = 1 d) z2 (t) = z0 · z1 (t) where z0 is a fixed complex number Problem-250 Let z1 (t) be a parametric equation of the curve C. What curve is described by the equation z2 (t) if z2 (t) = z1 (1 − t)? Problem-251 Let z1 (t) and z2 (t) be parametric equations of the curves C1 and C2 and let z1 (1) = z2 (0). What curve is described by the equation z3 (t) if:  1  z1 (2t) for 0 ≤ t ≤ 2 z3 (t) = 1  z2 (2t − 1) for < t ≤ 1? 2 Problem-252 Let z(t) = cos πt + i sin πt (Fig.

The number of inversions characterizes the disorder in this line with respect to the usual order 1, 2, . . , n. Problem-180 Find the number of inversions in line 3, 2, 5, 4, 1. From now on, we will not be interested in the number of inversions in a line, but in its parity. Problem-181 Prove that the parity of the number of inversions in a line changes if we interchange the position two arbitrary numbers. Definition 21 The permutation 12 . . n i1 i2 . . in is called an even or odd depending on whether there are even or odd number of inversions in the lower line.

Problem-205 Prove that the complex numbers form a commutative group under addition. Which is the identity element (zero) of this group? From now on, complex numbers will be denoted by one letter for convenience, for example z (or w). e. z1 · z2 = z2 · z1 and (z1 · z2 ) · z3 = z1 · (z2 · z3 ) for any complex numbers z1 , z2 , z3 . It is easy to verify that (a, b) · (1, 0) = (1, 0) · (a, b) = (a, b) for any complex number (a, b). Thus, the complex number (1, 0) is the identity element in the set of the complex numbers under multiplication.