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The angles mpi/n (with m,n integers) for which the trigonometric functions may be expressed in terms of finite root extraction of real numbers are limited to values of m ...

A hypergeometric identity discovered by Ramanujan around 1910. From Hardy (1999, pp. 13 and 102-103), (1) where a^((n))=a(a+1)...(a+n-1) (2) is the rising factorial (a.k.a. ...

The Gudermannian function is the odd function denoted either gamma(x) or gd(x) which arises in the inverse equations for the Mercator projection. phi(y)=gd(y) expresses the ...

The inverse function of the logarithm, defined such that log_b(antilog_bz)=z=antilog_b(log_bz). The antilogarithm in base b of z is therefore b^z.

Let a_(n+1) = 1/2(a_n+b_n) (1) b_(n+1) = (2a_nb_n)/(a_n+b_n). (2) Then A(a_0,b_0)=lim_(n->infty)a_n=lim_(n->infty)b_n=sqrt(a_0b_0), (3) which is just the geometric mean.

The geometric mean is smaller than the arithmetic mean, (product_(i=1)^Nn_i)^(1/N)<=(sum_(i=1)^(N)n_i)/N, with equality in the cases (1) N=1 or (2) n_i=n_j for all i,j.

Cis(x) is another name for the complex exponential, Cis(x)=e^(ix)=cosx+isinx. (1) It has derivative d/(dz)Cis(z)=ie^(iz) (2) and indefinite integral intCis(z)dz=-ie^(iz). (3)

The logarithm of the reciprocal of a number, equal to the negative of the logarithm of the number itself, cologx=log(1/x)=-logx.

x^(2n)+1=[x^2-2xcos(pi/(2n))+1] ×[x^2-2xcos((3pi)/(2n))+1]×...× ×[x^2-2xcos(((2n-1)pi)/(2n))+1].

Ein(z) = int_0^z((1-e^(-t))dt)/t (1) = E_1(z)+lnz+gamma, (2) where gamma is the Euler-Mascheroni constant and E_1 is the En-function with n=1.