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A generalization of the helicoid to the parametric equations x(u,v) = avcosu (1) y(u,v) = bvsinu (2) z(u,v) = cu. (3) In this parametrization, the surface has first ...

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, ...

A 1-cusped epicycloid has b=a, so n=1. The radius measured from the center of the large circle for a 1-cusped epicycloid is given by epicycloid equation (◇) with n=1 so r^2 = ...

A normal distribution with mean 0, P(x)=h/(sqrt(pi))e^(-h^2x^2). (1) The characteristic function is phi(t)=e^(-t^2/(4h^2)). (2) The mean, variance, skewness, and kurtosis ...

A factorization algorithm which works by expressing N as a quadratic form in two different ways. Then N=a^2+b^2=c^2+d^2, (1) so a^2-c^2=d^2-b^2 (2) (a-c)(a+c)=(d-b)(d+b). (3) ...

Let f(x,y) be a homogeneous function of order n so that f(tx,ty)=t^nf(x,y). (1) Then define x^'=xt and y^'=yt. Then nt^(n-1)f(x,y) = ...

The w-polynomials obtained by setting p(x)=3x and q(x)=-2 in the Lucas polynomial sequence. Setting f_n(1)=f_n (1) give a Fermat-Lucas number. The first few Fermat-Lucas ...

If f(x) is an even function, then b_n=0 and the Fourier series collapses to f(x)=1/2a_0+sum_(n=1)^inftya_ncos(nx), (1) where a_0 = 1/piint_(-pi)^pif(x)dx (2) = ...

Consider a square wave f(x) of length 2L. Over the range [0,2L], this can be written as f(x)=2[H(x/L)-H(x/L-1)]-1, (1) where H(x) is the Heaviside step function. Since ...

If f(x) is an odd function, then a_n=0 and the Fourier series collapses to f(x)=sum_(n=1)^inftyb_nsin(nx), (1) where b_n = 1/piint_(-pi)^pif(x)sin(nx)dx (2) = ...