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The complete elliptic integral of the second kind, illustrated above as a function of k, is defined by E(k) = E(1/2pi,k) (1) = ...

A series of the form sum_(k=1)^infty(-1)^(k+1)a_k (1) or sum_(k=1)^infty(-1)^ka_k, (2) where a_k>0. A series with positive terms can be converted to an alternating series ...

The integral int_0^1x^p(1-x)^qdx, called the Eulerian integral of the first kind by Legendre and Whittaker and Watson (1990). The solution is the beta function B(p+1,q+1).

The Fransén-Robinson constant F is defined by F=int_0^infty(dx)/(Gamma(x))=2.8077702420... (OEIS A058655) where Gamma(x) is the gamma function. No closed-form expression in ...

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.

A type of integral named after Henstock and Kurzweil. Every Lebesgue integrable function is HK integrable with the same value.

Topological lower bounds in terms of Betti numbers for the number of critical points form a smooth function on a smooth manifold.

The image of the path gamma in C under the function f is called the trace. This usage of the term "trace" is unrelated to the same term applied to matrices or tensors.

j_n(z)=(z^n)/(2^(n+1)n!)int_0^picos(zcostheta)sin^(2n+1)thetadtheta, where j_n(z) is a spherical Bessel function of the first kind.

rho_(2s)(n)=(pi^s)/(Gamma(s))n^(s-1)sum_(p,q)((S_(p,q))/q)^(2s)e^(2nppii/q), where S_(p,q) is a Gaussian sum, and Gamma(s) is the gamma function.