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The forward and inverse Kontorovich-Lebedev transforms are defined by K_(ix)[f(t)] = int_0^inftyK_(ix)(t)f(t)dt (1) K_(ix)^(-1)[g(t)] = ...

In the theory of special functions, a class of functions is said to be "of the third kind" if it is similar to but distinct from previously defined functions already defined ...

Second and higher derivatives of the metric tensor g_(ab) need not be continuous across a surface of discontinuity, but g_(ab) and g_(ab,c) must be continuous across it.

The kei_nu(z) function is defined as the imaginary part of e^(-nupii/2)K_nu(ze^(pii/4))=ker_nu(z)+ikei_nu(z), (1) where K_nu(z) is a modified Bessel function of the second ...

Although Bessel functions of the second kind are sometimes called Weber functions, Abramowitz and Stegun (1972) define a separate Weber function as ...

If x_0 is an ordinary point of the ordinary differential equation, expand y in a Taylor series about x_0. Commonly, the expansion point can be taken as x_0=0, resulting in ...

The negative derivative S(v)=-D_(v)N (1) of the unit normal N vector field of a surface is called the shape operator (or Weingarten map or second fundamental tensor). The ...

The Weingarten equations express the derivatives of the normal vector to a surface using derivatives of the position vector. Let x:U->R^3 be a regular patch, then the shape ...

The Legendre polynomials, sometimes called Legendre functions of the first kind, Legendre coefficients, or zonal harmonics (Whittaker and Watson 1990, p. 302), are solutions ...

The first solution to Lamé's differential equation, denoted E_n^m(x) for m=1, ..., 2n+1. They are also called Lamé functions. The product of two ellipsoidal harmonics of the ...