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The constant e is base of the natural logarithm. e is sometimes known as Napier's constant, although its symbol (e) honors Euler. e is the unique number with the property ...

An integral of the form intf(z)dz, (1) i.e., without upper and lower limits, also called an antiderivative. The first fundamental theorem of calculus allows definite ...

The inverse cotangent is the multivalued function cot^(-1)z (Zwillinger 1995, p. 465), also denoted arccotz (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. ...

Consider the average length of a line segment determined by two points picked at random in the interior of an arbitrary triangle Delta(a,b,c) with side lengths a, b, and c. ...

A logarithmic singularity is a singularity of an analytic function whose main z-dependent term is of order O(lnz). An example is the singularity of the Bessel function of the ...

The two-argument Ramanujan function is defined by phi(a,n) = 1+2sum_(k=1)^(n)1/((ak)^3-ak) (1) = 1-1/a(H_(-1/a)+H_(1/a)+2H_n-H_(n-1/a)-H_(n+1/a)). (2) The one-argument ...

Find the tunnel between two points A and B on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of ...

In the Wolfram Language, WignerD[{j, m ,n}, psi, theta, phi] gives the m×n matrix element of a (2j+1)-dimensional unitary representation of SU(2) parametrized by three Euler ...

Fermat's 4n+1 theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number p can be represented in an essentially unique ...

Like the entire harmonic series, the harmonic series sum_(k=1)^infty1/(p_k)=infty (1) taken over all primes p_k also diverges, as first shown by Euler in 1737 (Nagell 1951, ...