Search Results for ""

Landau (1911) proved that for any fixed x>1, sum_(0<|I[rho]|<=T)x^rho=-T/(2pi)Lambda(x)+O(lnT) as T->infty, where the sum runs over the nontrivial Riemann zeta function zeros ...

A Bessel function of the second kind Y_n(x) (e.g, Gradshteyn and Ryzhik 2000, p. 703, eqn. 6.649.1), sometimes also denoted N_n(x) (e.g, Gradshteyn and Ryzhik 2000, p. 657, ...

H_n^((2))(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 the second kind. Hankel functions of the second kind ...

_0F_1(;a;z)=lim_(q->infty)_1F_1(q;a;z/q). (1) It has a series expansion _0F_1(;a;z)=sum_(n=0)^infty(z^n)/((a)_nn!) (2) and satisfies z(d^2y)/(dz^2)+a(dy)/(dz)-y=0. (3) It is ...

The spherical Hankel function of the second kind h_n^((1))(z) is defined by h_n^((2))(z) = sqrt(pi/(2x))H_(n+1/2)^((2))(z) (1) = j_n(z)-in_n(z), (2) where H_n^((2))(z) is the ...

An exponent is the power p in an expression of the form a^p. The process of performing the operation of raising a base to a given power is known as exponentiation.

Consider a power series in a complex variable z g(z)=sum_(n=0)^inftya_nz^n (1) that is convergent within the open disk D:|z|<R. Convergence is limited to within D by the ...

The inhomogeneous Helmholtz differential equation is del ^2psi(r)+k^2psi(r)=rho(r), (1) where the Helmholtz operator is defined as L^~=del ^2+k^2. The Green's function is ...

The anticomplementary circle is the circumcircle of the anticomplementary triangle. It has radius R_A=2R, where R is the circumradius of the reference triangle, and center at ...

Approximants derived by expanding a function as a ratio of two power series and determining both the numerator and denominator coefficients. Padé approximations are usually ...