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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) ...

If any set of points is displaced by X^idx_i where all distance relationships are unchanged (i.e., there is an isometry), then the vector field is called a Killing vector. ...

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 ...

Stratton (1935), Chu and Stratton (1941), and Rhodes (1970) define the spheroidal functions as those solutions of the differential equation (1) that remain finite at the ...

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 ...