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If xsinalpha=sin(2beta-alpha), then (1+x)int_0^alpha(dphi)/(sqrt(1-x^2sin^2phi))=2int_0^beta(dphi)/(sqrt(1-(4x)/((1+x)^2)sin^2phi)).

For an arbitrary not identically constant polynomial, the zeros of its derivatives lie in the smallest convex polygon containing the zeros of the original polynomial.

The index of a vector field with finitely many zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold.

Polynomial identities involving sums and differences of like powers include x^2-y^2 = (x-y)(x+y) (1) x^3-y^3 = (x-y)(x^2+xy+y^2) (2) x^3+y^3 = (x+y)(x^2-xy+y^2) (3) x^4-y^4 = ...

A generalization of the Gaussian sum. For p and q of opposite parity (i.e., one is even and the other is odd), Schaar's identity states ...

The transformation T(x) = frac(1/x) (1) = 1/x-|_1/x_|, (2) where frac(x) is the fractional part of x and |_x_| is the floor function, that takes a continued fraction ...

The curlicue fractal is a figure obtained by the following procedure. Let s be an irrational number. Begin with a line segment of unit length, which makes an angle phi_0=0 to ...

If a is an arbitrary integer relatively prime to n and g is a primitive root of n, then there exists among the numbers 0, 1, 2, ..., phi(n)-1, where phi(n) is the totient ...

The complete elliptic integral of the first kind K(k), illustrated above as a function of the elliptic modulus k, is defined by K(k) = F(1/2pi,k) (1) = ...

The Gegenbauer polynomials C_n^((lambda))(x) are solutions to the Gegenbauer differential equation for integer n. They are generalizations of the associated Legendre ...