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Abel's Curve Theorem

Abel's curve theorem states that the sum of the values of an integral of the "first" or "second" sort

int_(x_0,y_0)^(x_1,y_1)(Pdx)/Q+...+int_(x_0,y_0)^(x_N,y_N)(Pdx)/Q=F(z)

and

(P(x_1,y_1))/(Q(x_1,y_1))(dx_1)/(dz)+...+(P(x_N,y_N))/(Q(x_N,y_N))(dx_N)/(dz)=(dF)/(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.

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References

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

Referenced on Wolfram|Alpha

Abel's Curve Theorem

Cite this as:

Weisstein, Eric W. "Abel's Curve Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbelsCurveTheorem.html

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