RiemannComplexNumbers

This Julia module gives an alternative to Complex numbers and their operations to give mathematically more sensible results.

The Complex Problem

Standard complex field operations in Julia work fine; the problems begin to arise when dividing by zero. It is logical to extend the real numbers with a positive infinity and a negative infinity. And we have both +Inf and -Inf in Julia. However, there are problems with the implementation of infinite values for Julia Complex numbers. Here are some examples.

# For real numbers, division by 0 gives an infinite result
julia> 1/0
Inf

# This division by 0 for complex numbers is fine
julia> (2+3im)/0
Inf + Inf*im

# But this one doesn't make sense
julia> 2im/0
NaN + Inf*im

# For real numbers we have the following sensible result
julia> (Inf + 3) == (Inf + 2)
true

# But it breaks for complex numbers
julia> (Inf + 3im) == (Inf + 2im)
false

This Solution

This module defines an alternative to Complex numbers in which there is a single infinite value (we call ComplexInfinity). We introduce a new type called RC (an abbreviation for Riemann Complex number). Let's see how the previous calculations work in this new context:

julia> using RiemannComplexNumbers

julia> (2+3IM)/0
ComplexInf

julia> 2IM/0
ComplexInf

julia> Inf + 3IM == Inf + 2IM
true

The constant IM is the replacement for im that can be used to construct Riemann Complex numbers. In general, wrapping values in RC will work:

julia> RC(2)
2 + 0IM

julia> RC(3-im)
3 - 1IM

Dividing by zero gives the following:

julia> (2-3IM)/0
ComplexInf

julia> 3/0IM
ComplexInf

julia> 0/0IM
ComplexNaN

To convert an RC number to a Complex do this:

julia> z = 3.5 - 5IM
3.5 - 5.0IM

julia> Complex(z)
3.5 - 5.0im

Basic arithmetic operations work exactly the same for RC numbers as for Complex but will be slower (to deal with division by zero and operations with ComplexInf and ComplexNaN).

Some basic functions (such as sqrt and exp) are provided. See the functions.jl file in the src directory.

To Do

Some LinearAlgebra operations don't work; I'm not sure why. For example, evaluating the determinant of an RC matrix throws errors.