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By analogy with the log sine function, define the log cosine function by C_n=int_0^(pi/2)[ln(cosx)]^ndx. (1) The first few cases are given by C_1 = -1/2piln2 (2) C_2 = ...

A_m(lambda)=int_(-infty)^inftycos[1/2mphi(t)-lambdat]dt, (1) where the function phi(t)=4tan^(-1)(e^t)-pi (2) describes the motion along the pendulum separatrix. Chirikov ...

If a function has a Fourier series given by f(x)=1/2a_0+sum_(n=1)^inftya_ncos(nx)+sum_(n=1)^inftyb_nsin(nx), (1) then Bessel's inequality becomes an equality known as ...

Planck's's radiation function is the function f(x)=(15)/(pi^4)1/(x^5(e^(1/x)-1)), (1) which is normalized so that int_0^inftyf(x)dx=1. (2) However, the function is sometimes ...

Let p_n be the nth prime, then the primorial (which is the analog of the usual factorial for prime numbers) is defined by p_n#=product_(k=1)^np_k. (1) The values of p_n# for ...

A sieving procedure that can be used in conjunction with Dixon's factorization method to factor large numbers n. Pick values of r given by r=|_sqrt(n)_|+k, (1) where k=1, 2, ...

The quadratrix was discovered by Hippias of Elias in 430 BC, and later studied by Dinostratus in 350 BC (MacTutor Archive). It can be used for angle trisection or, more ...

The rectangle function Pi(x) is a function that is 0 outside the interval [-1/2,1/2] and unity inside it. It is also called the gate function, pulse function, or window ...

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 ...

A stadium, also called a discorectangle, obround, or sausage body, is a geometric figure consisting of a rectangle with top and bottom lengths a whose ends are capped off ...