The constant of integration in all the formulas of this chapter is also omitted. Therefore, the equality sign (=) means that the functions on the left and right of this symbol differ by a constant.ELSEVIERTable of Integrals, Series, and Products (Sixth Edition)
CV. On some definite integrals involving Legendre functionsA few definite integrals involving more than two Legendre functions in the integrand have been considered by Ferrers, Adams, Dougall, Nicholson and Bailey. We take for example the following integrals.doi:10.1017/S0013091500024172N. G. Shabde...
Lommel functionstrigonometric integralsmonotonicity propertiesinequalitieszerosWe prove thatfor all \\(\\mu \\ge 1/2\\) and \\(x\\ge \\pi \\) , where \\(s_{\\mu ,\\frac{1}{2}}(x)\\) is the Lommel function of the first kind. This refines a result of Steinig published in ...
Legendre functions of order n are solutions to Legendre's differential equation(1−x2)d2ydx2−2xdydx+n(n+1)y=0 for n>0 and |x|<1. For n∈N0, Legendre polynomials Pn(x) are examples of Legendre functions, and may be defined via the Rodrigues formula(3)Pn(x)=12nn!dndxn(x2...
The definite integral of a function is closely related to the antiderivative and indefinite integral of a function. The primary difference is that the indefinite integral, if it exists, is a real number value, while the latter two represent an infinite number of functions that differ only by a...
Because Mathcad’s integration method divides the interval into four subintervals and then successively doubles the number of points, it can return incorrect answers for periodic functions with having periods 1/2n times the length of the interval. To avoid this problem, divide the interval into two...
associated Legendre functionscomplete elliptic integrals33C1033C5533C0533E05An earlier method for obtaining indefinite integrals of special function from the second-order linear equations which define them has been reformulated in terms of Riccati equations, which are nonlinear and first-order. For this...
Möbius (Square-free) Variant. Reduction of the summation to k-almost primes with k distinct prime factors defines a signed variant of the prime zeta functions: Defi...Richard J. Mathar, Twenty digits of the Erd˝os-Lebensold constant, arXiv:0811.4739 [math.NT (2008)....
If still not successful, try G-functions irrespective of the limits. The option meijerg=True, False, None can be used to, respectively: always use G-function methods and no others, never use G-function methods, or use all available methods (in order as described above). It defaults to No...
Nature of problem: Triple-K integrals were introduced in [2] as a convenient tool for the analysis of conformal 3-point functions in momentum space. All 3-point functions of scalar operators, conserved currents and stress tensor can be expressed in terms of triple-K integrals. Furthermore, a...