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The parabolic cylinder functions are a class of functions sometimes called Weber functions. There are a number of slightly different definitions in use by various authors. ...

A generalized conformal mapping.

The surface given by the parametric equations x = asinu (1) y = asinv (2) z = asin(u+v). (3) It is a sextic surface with algebraic equation (4) The coefficients of the first ...

A spheroidal harmonic is a special case of an ellipsoidal harmonic that satisfies the differential equation d/(dx)[(1-x^2)(dS)/(dx)]+(lambda-c^2x^2-(m^2)/(1-x^2))S=0 on the ...

The hyperbolic cosine is defined as coshz=1/2(e^z+e^(-z)). (1) The notation chx is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). This function describes the ...

Let f(x,y)=u(x,y)+iv(x,y), (1) where z=x+iy, (2) so dz=dx+idy. (3) The total derivative of f with respect to z is then (df)/(dz) = ...

Minkowski space is a four-dimensional space possessing a Minkowski metric, i.e., a metric tensor having the form dtau^2=-(dx^0)^2+(dx^1)^2+(dx^2)^2+(dx^3)^2. Alternatively ...

A modified spherical Bessel function of the first kind (Abramowitz and Stegun 1972), also called a "spherical modified Bessel function of the first kind" (Arfken 1985), is ...

A modified spherical Bessel function of the second kind, also called a "spherical modified Bessel function of the first kind" (Arfken 1985) or (regrettably) a "modified ...

The geodesic on an oblate spheroid can be computed analytically, although the resulting expression is much more unwieldy than for a simple sphere. A spheroid with equatorial ...