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A polynomial given by Phi_n(x)=product_(k=1)^n^'(x-zeta_k), (1) where zeta_k are the roots of unity in C given by zeta_k=e^(2piik/n) (2) and k runs over integers relatively ...

Newton's method, also called the Newton-Raphson method, is a root-finding algorithm that uses the first few terms of the Taylor series of a function f(x) in the vicinity of a ...

The probability that a random integer between 1 and x will have its greatest prime factor <=x^alpha approaches a limiting value F(alpha) as x->infty, where F(alpha)=1 for ...

Given a homogeneous linear second-order ordinary differential equation, y^('')+P(x)y^'+Q(x)y=0, (1) call the two linearly independent solutions y_1(x) and y_2(x). Then ...

The Ablowitz-Ramani-Segur conjecture states that a nonlinear partial differential equation is solvable by the inverse scattering method only if every nonlinear ordinary ...

The associated Legendre differential equation is a generalization of the Legendre differential equation given by d/(dx)[(1-x^2)(dy)/(dx)]+[l(l+1)-(m^2)/(1-x^2)]y=0, (1) which ...

The Benney equation in 1+1 dimensions is the nonlinear partial differential equation ...

The third-order ordinary differential equation 2y^(''')+yy^('')=0. This equation arises in the theory of fluid boundary layers, and must be solved numerically (Rosenhead ...

An algorithm which finds rational function extrapolations of the form R_(i(i+1)...(i+m))=(P_mu(x))/(P_nu(x))=(p_0+p_1x+...+p_mux^mu)/(q_0+q_1x+...+q_nux^nu) and can be used ...

This theorem states that, for a partial differential equation involving a time derivative of order n, the solution is uniquely determined if time derivatives up to order n-1 ...