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The series for the inverse tangent, tan^(-1)x=x-1/3x^3+1/5x^5+.... Plugging in x=1 gives Gregory's formula 1/4pi=1-1/3+1/5-1/7+1/9-.... This series is intimately connected ...

Machin's formula is given by 1/4pi=4cot^(-1)5-cot^(-1)239. There are a whole class of Machin-like formulas with various numbers of terms (although only four such formulas ...

By analogy with the sinc function, define the sinhc function by sinhc(z)={(sinhz)/z for z!=0; 1 for z=0. (1) Since sinhx/x is not a cardinal function, the "analogy" with the ...

By analogy with the tanc function, define the tanhc function by tanhc(z)={(tanhz)/z for z!=0; 1 for z=0. (1) It has derivative (dtanhc(z))/(dz)=(sech^2z)/z-(tanhz)/(z^2). (2) ...

The cosecant cscz is the function defined by cscz = 1/(sinz) (1) = (2i)/(e^(iz)-e^(-iz)), (2) where sinz is the sine. The cosecant is implemented in the Wolfram Language as ...

A definite integral is an integral int_a^bf(x)dx (1) with upper and lower limits. If x is restricted to lie on the real line, the definite integral is known as a Riemann ...

The prime number theorem gives an asymptotic form for the prime counting function pi(n), which counts the number of primes less than some integer n. Legendre (1808) suggested ...

A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides ...

The Bernoulli numbers B_n are a sequence of signed rational numbers that can be defined by the exponential generating function x/(e^x-1)=sum_(n=0)^infty(B_nx^n)/(n!). (1) ...

An H-space, named after Heinz Hopf, and sometimes also called a Hopf space, is a topological space together with a continuous binary operation mu:X×X->X, such that there ...