Search Results for ""

The exponential transform is the transformation of a sequence a_1, a_2, ... into a sequence b_1, b_2, ... according to the equation ...

The Hankel functions of the first kind are defined as H_n^((1))(z)=J_n(z)+iY_n(z), (1) where J_n(z) is a Bessel function of the first kind and Y_n(z) is a Bessel function of ...

A product involving an infinite number of terms. Such products can converge. In fact, for positive a_n, the product product_(n=1)^(infty)a_n converges to a nonzero number iff ...

Given a Taylor series f(x)=f(x_0)+(x-x_0)f^'(x_0)+((x-x_0)^2)/(2!)f^('')(x_0)+... +((x-x_0)^n)/(n!)f^((n))(x_0)+R_n, (1) the error R_n after n terms is given by ...

Lucas's theorem states that if n>=3 be a squarefree integer and Phi_n(z) a cyclotomic polynomial, then Phi_n(z)=U_n^2(z)-(-1)^((n-1)/2)nzV_n^2(z), (1) where U_n(z) and V_n(z) ...

Let a line in three dimensions be specified by two points x_1=(x_1,y_1,z_1) and x_2=(x_2,y_2,z_2) lying on it, so a vector along the line is given by v=[x_1+(x_2-x_1)t; ...

The geometric centroid of a polyhedron composed of N triangular faces with vertices (a_i,b_i,c_i) can be computed using the curl theorem as x^_ = ...

A generalization of the binomial coefficient whose notation was suggested by Knuth, |_n; k]=(|_n]!)/(|_k]!|_n-k]!), (1) where |_n] is a Roman factorial. The above expression ...

For a connection A and a positive spinor phi in Gamma(V_+), Witten's equations (also called the Seiberg-Witten invariants) are given by D_Aphi = 0 (1) F_+^A = ...

Trigonometry