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An equation derived by Kronecker: where r(n) is the sum of squares function, zeta(z) is the Riemann zeta function, eta(z) is the Dirichlet eta function, Gamma(z) is the gamma ...

As Lagrange showed, any irrational number alpha has an infinity of rational approximations p/q which satisfy |alpha-p/q|<1/(sqrt(5)q^2). (1) Furthermore, if there are no ...

An inverse function of an Abelian integral. Abelian functions have two variables and four periods, and can be defined by Theta(v,tau;q^'; ...

Let P be the set of primes, and let Q_p and Z_p(t) be the fields of p-adic numbers and formal power series over Z_p=(0,1,...,p-1). Further, suppose that D is a "nonprincipal ...

The sum of the reciprocals of roots of an equation equals the negative coefficient of the linear term in the Maclaurin series.

The Riemann sphere C^*=C union {infty}, also called the extended complex plane. The notation C^^ is sometimes also used (Krantz 1999, p. 82). The notation C^* is also used to ...

The complex plane C with the origin removed, i.e., C-{0}. The punctured plane is sometimes denoted C^* (although this notation conflicts with that for the Riemann sphere C-*, ...

A sum which includes both the Jacobi triple product and the q-binomial theorem as special cases. Ramanujan's sum is ...

In response to a letter from Goldbach, Euler considered sums of the form s_h(m,n) = sum_(k=1)^(infty)(1+1/2+...+1/k)^m(k+1)^(-n) (1) = ...

A reflection relation is a functional equation relating f(-x) to f(x), or more generally, f(a-x) to f(x). Perhaps the best known example of a reflection formula is the gamma ...