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The spherical harmonics form a complete orthogonal system, so an arbitrary real function f(theta,phi) can be expanded in terms of complex spherical harmonics by ...

The Lebesgue integral is defined in terms of upper and lower bounds using the Lebesgue measure of a set. It uses a Lebesgue sum S_n=sum_(i)eta_imu(E_i) where eta_i is the ...

The word multiplicity is a general term meaning "the number of values for which a given condition holds." For example, the term is used to refer to the value of the totient ...

In the equianharmonic case of the Weierstrass elliptic function, corresponding to invariants g_2=0 and g_3=1, the corresponding real half-period is given by omega_2 = ...

The omega constant is defined as W(1)=0.5671432904... (1) (OEIS A030178), where W(x) is the Lambert W-function. It is available in the Wolfram Language using the function ...

Consider solutions to the equation x^y=y^x. (1) Real solutions are given by x=y for x,y>0, together with the solution of (lny)/y=(lnx)/x, (2) which is given by ...

Let the two-dimensional cylinder function be defined by f(x,y)={1 for r<R; 0 for r>R. (1) Then the Radon transform is given by ...

Suppose that in some neighborhood of x=0, F(x)=sum_(k=0)^infty(phi(k)(-x)^k)/(k!) (1) for some function (say analytic or integrable) phi(k). Then ...

Closed forms are known for the sums of reciprocals of even-indexed Lucas numbers P_L^((e)) = sum_(n=1)^(infty)1/(L_(2n)) (1) = sum_(n=1)^(infty)1/(phi^(2n)+phi^(-2n)) (2) = ...

The second Steiner circle (a term coined here for the first time) is the circumcircle of the Steiner triangle DeltaS_AS_BS_C. Its center has center function ...