Inside-Outside Theorem

Let P(z) and Q(z) be univariate polynomials in a complex variable z, and let the polynomial degrees of P and Q satisfy deg(Q)>=deg(P+2). Then

int_gamma(P(z))/(Q(z))dz=2piisum_(a_i in A)Res_(z=a_i)(P(z))/(Q(z))
=-2piisum_(b_i in B)Res_(z=b_i)(P(z))/(Q(z)),

where gamma is a simple closed clockwise-oriented contour, A is the set of roots of Q inside of gamma, and B is the set of roots of Q outside of gamma.

The first equality is an instance of the residue theorem. On the Riemann sphere, the simple closed contour gamma splits the sphere into two regions. After the change of variables w=1/z, the point zero is mapped to infinity and vice versa. What was the "inside" of gamma becomes the outside of gamma in the new coordinate. The second equality is the residue theorem applied to the meromorphic one-form alpha=P/Qdz in the coordinate w, with a minus sign because gamma travels clockwise after the coordinate change. The hypothesis on the degrees of P and Q ensure that alpha does not have a pole at z=infty.


The above diagram shows two different points of view of the contour gamma and the poles of the meromorphic one-form P/Qdz on the Riemann sphere. The usual point of view is centered at z=0, but the role of inside and outside is switched from the point of view of z=infty. The poles inside are labeled blue and outside are green.

The theorem also follows from taking the contour integral at infinity, i.e., a circle of large radius R. The hypothesis on the degree says that this integral tends to zero. Hence it must actually be zero, because at some point the circle contains all of the poles of P/Q. This is a special case of the fact that on a compact Riemann surface, in this case the Riemann sphere, the sum of the complex residues of a meromorphic one-form is zero.

See also

Complex Residue, Contour, Contour Integral, Jacobian, Residue Theorem, Riemann Sphere, Root

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Inside-Outside Theorem." From MathWorld--A Wolfram Web Resource.

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