RiemannSphereOperations

Maths

This package provides some operations on the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$.

The following are the calculation rules with the point at infinity $\infty$.

• $a + \infty = \infty \quad (a\in\mathbb{C})$
• $a \cdot \infty = \infty \quad (a\in\mathbb{C}\setminus\{0\})$
• $\infty \cdot \infty = \infty$
• $1/0 = \infty$
• $1/\infty = 0$

Note that the following cannot be defined.

• $0 / 0$
• $0 \cdot \infty$
• $\infty / \infty$
• $\infty + \infty$
• $\infty - \infty$

Package implementation

In this package, a value z which satisfies isinf(z) and !isnan(z) is used to represent the infinity $\infty$. Thus, all of complex(Inf,2.0), complex(Inf,-Inf), complex(1//0,1//1) are treated as the same point $\infty$ in this package.

All of the exported functions from this package has ′ (\prime<tab>) suffix which represents a modified version of the function on the Riemann sphere.

julia> using RiemannSphereOperations
julia> inv(complex(0,0))  # 1/0 should be ∞NaN + NaN*im
julia> inv′(complex(0,0))  # correct 😎Inf - Inf*im
julia> complex(1) * Complex(Inf,-Inf)  # 1⋅∞ should be ∞NaN + NaN*im
julia> complex(1) *′ Complex(Inf,-Inf)  # correct 😎Inf - Inf*im
julia> Inf + Inf  # ∞+∞ cannot be definedInf
julia> Inf +′ Inf  # correct 😎NaN

This package has more functions such as isinfty′, +′, /′ etc. See the documentation page for more information.

Related information

• My post on Julia discourse
  • This explains why I made this package.
• RiemannComplexNumbers.jl
  • This package has the same motivation, but its implementation is different.
  • RiemannComplexNumbers.jl defines original type RC <: Number which is used for the complex numbers on the Riemann sphere.
• Original issue on inv(0+0im)
  • This issue explains why inv(0+0im) should return NaN + NaN*im
• Related issue on NaN vs DomainError
  • This related issue is still open.