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The second-order ordinary differential equation y^('')+[(alphaeta)/(1+eta)+(betaeta)/((1+eta)^2)+gamma]y=0, where eta=e^(deltax).

The second singular value k_2, corresponding to K^'(k_2)=sqrt(2)K(k_2), (1) is given by k_2 = tan(pi/8) (2) = sqrt(2)-1 (3) k_2^' = sqrt(2)(sqrt(2)-1). (4) For this modulus, ...

Harmonic coordinates satisfy the condition Gamma^lambda=g^(munu)Gamma_(munu)^lambda=0, (1) or equivalently, partial/(partialx^kappa)(sqrt(g)g^(lambdakappa))=0. (2) It is ...

The interior product is a dual notion of the wedge product in an exterior algebra LambdaV, where V is a vector space. Given an orthonormal basis {e_i} of V, the forms ...

The second-order ordinary differential equation y^('')+[alpha/(cosh^2(ax))+betatanh(ax)+gamma]y=0.

Let alpha and beta be any ordinal numbers, then ordinal exponentiation is defined so that if beta=0 then alpha^beta=1. If beta is not a limit ordinal, then choose gamma such ...

A path gamma is a continuous mapping gamma:[a,b]|->C^0, where gamma(a) is the initial point, gamma(b) is the final point, and C^0 denotes the space of continuous functions. ...

An elliptic integral is an integral of the form int(A(x)+B(x)sqrt(S(x)))/(C(x)+D(x)sqrt(S(x)))dx, (1) or int(A(x)dx)/(B(x)sqrt(S(x))), (2) where A(x), B(x), C(x), and D(x) ...

The duplication formula for Rogers L-function follows from Abel's functional equation and is given by 1/2L(x^2)=L(x)-L(x/(1+x)).

A convex body in Euclidean space that is centrally symmetric with center at the origin is determined among all such bodies by its brightness function (the volume of each ...