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Topological lower bounds in terms of Betti numbers for the number of critical points form a smooth function on a smooth manifold.

The image of the path gamma in C under the function f is called the trace. This usage of the term "trace" is unrelated to the same term applied to matrices or tensors.

j_n(z)=(z^n)/(2^(n+1)n!)int_0^picos(zcostheta)sin^(2n+1)thetadtheta, where j_n(z) is a spherical Bessel function of the first kind.

rho_(2s)(n)=(pi^s)/(Gamma(s))n^(s-1)sum_(p,q)((S_(p,q))/q)^(2s)e^(2nppii/q), where S_(p,q) is a Gaussian sum, and Gamma(s) is the gamma function.

A^n+B^n=sum_(j=0)^(|_n/2_|)(-1)^jn/(n-j)(n-j; j)(AB)^j(A+B)^(n-2j), where |_x_| is the floor function and (n; k) is a binomial coefficient.

Let E be a set of expressions representing real, single-valued partially defined functions of one real variable. Let E^* be the set of functions represented by expressions in ...

alpha(x) = 1/(sqrt(2pi))int_(-x)^xe^(-t^2/2)dt (1) = sqrt(2/pi)int_0^xe^(-t^2/2)dt (2) = 2Phi(x) (3) = erf(x/(sqrt(2))), (4) where Phi(x) is the normal distribution function ...

Pre-Calculus

A continuous statistical distribution which arises in the testing of whether two observed samples have the same variance. Let chi_m^2 and chi_n^2 be independent variates ...

Let the sum of squares function r_k(n) denote the number of representations of n by k squares, then the summatory function of r_2(k)/k has the asymptotic expansion ...