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Abel's Curve Theorem

The sum of the values of an Integral of the ``first'' or ``second'' sort

\int_{x_0,y_0}^{x_1,y_1} {P\,dx\over Q}+\ldots+\int_{x_0,y_0}^{x_N,y_N} {P\,dx\over Q}=F(z)


{P(x_1,y_1)\over Q(x_1,y_1)} {dx_1\over dz}+\ldots+{P(x_N,y_N)\over Q(x_N,y_N)}{dx_N\over dz}={dF\over dz},

from a Fixed Point to the points of intersection with a curve depending rationally upon any number of parameters is a Rational Function of those parameters.


Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 277, 1959.

© 1996-9 Eric W. Weisstein