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Implicit differentiation is the procedure of differentiating an implicit equation with respect to the desired variable x while treating the other variables as unspecified ...

Kelvin defined the Kelvin functions bei and ber according to ber_nu(x)+ibei_nu(x) = J_nu(xe^(3pii/4)) (1) = e^(nupii)J_nu(xe^(-pii/4)), (2) = e^(nupii/2)I_nu(xe^(pii/4)) (3) ...

The (associated) Legendre function of the first kind P_n^m(z) is the solution to the Legendre differential equation which is regular at the origin. For m,n integers and z ...

The modified Lommel functions of the first and second kind give the solution to the Lommel differential equation with a minus sign in front of the linear term, i.e., ...

Polynomials O_n(x) that can be defined by the sum O_n(x)=1/4sum_(k=0)^(|_n/2_|)(n(n-k-1)!)/(k!)(1/2x)^(2k-n-1) (1) for n>=1, where |_x_| is the floor function. They obey the ...

A solution to the spherical Bessel differential equation. The two types of solutions are denoted j_n(x) (spherical Bessel function of the first kind) or n_n(x) (spherical ...

An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For ...

The self-intersection of a one-sided surface. The word "cross-cap" is sometimes also written without the hyphen as the single word "crosscap." The cross-cap can be thought of ...

A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called ...

A system of curvilinear coordinates in which two sets of coordinate surfaces are obtained by revolving the curves of the elliptic cylindrical coordinates about the y-axis ...