Compute the Neumann factor \(\nu_{n}\)

The Neumann factor, denoted \(\nu_{n}\), is a simple multiplicative constant commonly used in spherical or spheroidal function theory. It accounts for the symmetry of cosine terms or the duplication of even-order contributions in integrals.

Usage

neumann(x)

Arguments

x

An integer iterator.

Value

A numeric vector of the same length as x, containing values of \(\eta_x\), each equal to 1 or 2.

Details

Formally, the Neumann factor is defined as: $$ \eta_n = \begin{cases} 1, & n = 0, \\ 2, & n > 0. \end{cases} $$

This factor frequently appears in the normalization of spherical Bessel functions, Legendre expansions, and spheroidal wave functions. It is not related to the "von Neumann ordinals" used in set theory.

Examples

neumann(0) # should return 1
#> [1] 1
neumann(1) # should return 2
#> [1] 2
neumann(2) # should return 2
#> [1] 2

# Vectorized use:
neumann(0:4)
#> [1] 1 2 2 2 2