In mathematics, the Neville theta functions, named after Eric Harold Neville, are defined as follows:
θ c ( z , m ) = 2 π q ( m ) 4 ∑ k = 0 ∞ ( q ( m ) ) k ( k + 1 ) cos ( 1 2 ⋅ ( 2 k + 1 ) π z K ( m ) ) 1 K ( m ) 1 m 4 θ d ( z , m ) = 1 / 2 2 π ( 1 + 2 ∑ k = 1 ∞ ( q ( m ) ) k 2 cos ( π z k K ( m ) ) ) 1 K ( m ) θ n ( z , m ) = 1 / 2 π 2 ( 1 + 2 ∑ k = 1 ∞ ( − 1 ) k ( q ( m ) ) k 2 cos ( π z k K ( m ) ) ) 1 1 − m 4 1 K ( m ) θ s ( z , m ) = π 2 q ( m ) 4 ∑ k = 0 ∞ ( − 1 ) k ( q ( m ) ) k ( k + 1 ) sin ( 1 / 2 ( 2 k + 1 ) π z K ( m ) ) 1 1 − m 4 1 m 4 1 K ( m ) where:
K ( m ) = EllipticK ( m ) K ′ ( m ) = EllipticK ( 1 − m ) q ( m ) = e − π K ( m ) / K ′ ( m ) is the elliptic nomeSubstitute z = 2.5, m = 0.3 into the above definitions of Neville theta functions(using Maple) once obtain the following(consistent with results from wolfram math).
θ c ( 2.5 , 0.3 ) = − 0.65900466676738154967 θ d ( 2.5 , 0.3 ) = 0.95182196661267561994 θ n ( 2.5 , 0.3 ) = 1.0526693354651613637 θ s ( 2.5 , 0.3 ) = 0.82086879524530400536 θ c ( z , m ) = θ c ( − z , m ) θ d ( z , m ) = θ d ( − z , m ) θ n ( z , m ) = θ n ( − z , m ) θ s ( z , m ) = − θ s ( − z , m ) NetvilleThetaC[z,m], NevilleThetaD[z,m], NevilleThetaN[z,m], and NevilleThetaS[z,m] are built-in functions of Mathematica No such functions in Maple.
