By Shafarevich I.R. (ed.)

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Example text

26) is equal to zero. Indeed, set p - 0 and q = m/2. 22) takes the form (4 `n2 - n)z 3 = 0, from which it follows that z 1 = z2 = z3 = 0. 12), we substitute -z 40 I. HISTORICAL INTRODUCTION. THE JACOBI INVERSION PROBLEM for z in the integral f3. 12) in a somewhat different notation. §9. Main directions of development of the theory of Abelian functions We can now state Jacobi's inversion problem in the most general form. 1) be an irreducible algebraic equation of genus p and let Z, RI(z, w)dz,...

This explains why singular points of this type are called points of indeterminacy.

Set and yi(u) = x(u)w1(u) Then in a neighborhood of the point a we have f(u) _ o1 (u)/yr1 (u), where c°i and yri are analytic at a but not necessarily entire functions. , f(u) is analytic at this point; (/3) yri(a) = 0 but o1 (a) 0; then a is a singular point and a pole of f(u); (y) yr1(a) = 0 and Api(a) = 0; then a is again a singular point of f(u), called a point of indeterminacy. Here the origin is a point of indeterminacy. If A is any complex number, then at all points of the one-dimensional com- plex manifold u2 - Aui = 0, except at the origin, we have f(u) = A.

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