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J_n(z) = 1/(2pi)int_(-pi)^pie^(izcost)e^(in(t-pi/2))dt (1) = (i^(-n))/piint_0^pie^(izcost)cos(nt)dt (2) = 1/piint_0^picos(zsint-nt)dt (3) for n=0, 1, 2, ..., where J_n(z) is ...

The integral transform (Kf)(x)=int_0^inftysqrt(xt)K_nu(xt)f(t)dt, where K_nu(x) is a modified Bessel function of the second kind. Note the lower limit of 0, not -infty as ...

Model completion is a term employed when existential closure is successful. The formation of the complex numbers, and the move from affine to projective geometry, are ...

For R[nu]>-1/2, J_nu(z)=(z/2)^nu2/(sqrt(pi)Gamma(nu+1/2))int_0^(pi/2)cos(zcost)sin^(2nu)tdt, where J_nu(z) is a Bessel function of the first kind, and Gamma(z) is the gamma ...

P(Z)=Z/(sigma^2)exp(-(Z^2+|V|^2)/(2sigma^2))I_0((Z|V|)/(sigma^2)), where I_0(z) is a modified Bessel function of the first kind and Z>0. For a derivation, see Papoulis ...

A k-matrix is a kind of cube root of the identity matrix (distinct from the identity matrix) which is defined by the complex matrix k=[0 0 -i; i 0 0; 0 1 0]. It satisfies ...

An Eisenstein series with half-period ratio tau and index r is defined by G_r(tau)=sum^'_(m=-infty)^inftysum^'_(n=-infty)^infty1/((m+ntau)^r), (1) where the sum sum^(') ...

As proved by Sierpiński (1960), there exist infinitely many positive odd numbers k such that k·2^n+1 is composite for every n>=1. Numbers k with this property are called ...

Let n>=0 and alpha_1, alpha_2, ...be the positive roots of J_n(x)=0, where J_n(z) is a Bessel function of the first kind. An expansion of a function in the interval (0,1) in ...

The second-order ordinary differential equation (1-x^2)y^('')-2(mu+1)xy^'+(nu-mu)(nu+mu+1)y=0 (1) sometimes called the hyperspherical differential equation (Iyanaga and ...