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A linear functional defined on a subspace of a vector space V and which is dominated by a sublinear function defined on V has a linear extension which is also dominated by ...

An integral equation of the form f(x)=int_a^xK(x,t)phi(t)dt, where K(x,t) is the integral kernel, f(x) is a specified function, and phi(t) is the function to be solved for.

An integral equation of the form phi(x)=f(x)+int_a^xK(x,t)phi(t)dt, where K(x,t) is the integral kernel, f(x) is a specified function, and phi(t) is the function to be solved ...

Lehmer's formula is a formula for the prime counting function, pi(x) = (1) where |_x_| is the floor function, a = pi(x^(1/4)) (2) b = pi(x^(1/2)) (3) b_i = pi(sqrt(x/p_i)) ...

Number Theory

A Lambert series is a series of the form F(x)=sum_(n=1)^inftya_n(x^n)/(1-x^n) (1) for |x|<1. Then F(x) = sum_(n=1)^(infty)a_nsum_(m=1)^(infty)x^(mn) (2) = ...

J_n(x)=1/piint_0^picos(ntheta-xsintheta)dtheta, where J_n(x) is a Bessel function of the first kind.

If m is an integer, then for every residue class r (mod m), there are infinitely many nonnegative integers n for which P(n)=r (mod m), where P(n) is the partition function P.

A discontinuity is point at which a mathematical object is discontinuous. The left figure above illustrates a discontinuity in a one-variable function while the right figure ...

A basis, form, function, etc., in two or more variables is said to be multilinear if it is linear in each variable separately.