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Let phi(n) be any function, say analytic or integrable. Then int_0^inftyx^(s-1)sum_(k=0)^infty(-1)^kx^kphi(k)dx=(piphi(-s))/(sin(spi)) (1) and ...

The sum c_q(m)=sum_(h^*(q))e^(2piihm/q), (1) where h runs through the residues relatively prime to q, which is important in the representation of numbers by the sums of ...

If f:D->Y is a map (a.k.a. function, transformation, etc.) over a domain D, then the range of f, also called the image of D under f, is defined as the set of all values that ...

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 ...

The consideration of the result of a numerical calculation as a function of an adjustable parameter (usually the step size). The function can then be fitted and evaluated at ...

A powerful numerical integration technique which uses k refinements of the extended trapezoidal rule to remove error terms less than order O(N^(-2k)). The routine advocated ...

When computing the sample variance s numerically, the mean must be computed before s^2 can be determined. This requires storing the set of sample values. However, it is ...

The Schroeder stairs, also called Schouten's staircase (Bool et al. 1982, p. 147), are an ambiguous figure that depicts two different staircases at the same time: one going ...

There are at least two Siegel's theorems. The first states that an elliptic curve can have only a finite number of points with integer coordinates. The second states that if ...

For a braid with M strands, R components, P positive crossings, and N negative crossings, {P-N<=U_++M-R if P>=N; P-N<=U_-+M-R if P<=N, (1) where U_+/- are the smallest number ...