By I.G. Macdonald

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Frobenius, Ueber die Grundlagen der Theorie der Jacobischen Functionen J. Reine Angew. Math. 97 (1884), 16-48. ag, 1956. CHAPTER II Periodic Functions of Several Complex Variables §1. Divisibility relation for functions analytic at a point It will be convenient to treat points of a p-dimensional complex space C° as p-dimensional complex vectors. A vector u e C° will be denoted by a column matrix u = (ui up)T where u1, ... , up are complex numbers. A complex-valued function f(u) is said to be analytic at a point a = (ai ...

ABELIAN INTEGRALS. ABEL'S THEOREM 37 Let the coefficients of the polynomial o(z , w) (all or only some) be variable and denote the coefficients by a1, ... , a. 12) are functions of the parameters a1 (j = 1 , ... , k). Denote these points by (z1, w1) (i = 1, ... , N). 11) (z0 is not a singular point of w(z)) to the points of intersection indicated above, will be a function of a1 N -1 fZ J R(z, w) dz = yr(a1, ... , ak). 13) Zo Notice that by fixing the values of wo at the point zo we choose one of the nsingle-valued branches of w(z) in the neighborhood of zo and continue it along the integration paths.

4) This is the Cauchy's Integral Theorem for a function on a Riemann surface F. If the deformation mentioned above is impossible without touching the points where the integral and R (z , w) are oo, then the integral is equal to the product of 2ic i and the sum of the residues of the function, where the residues are computed appropriately for the points mentioned. Finally, if A cannot be deformed continuously into a point on F, then the value w, which can be zero or nonzero, of the integral along A remains 32 I.