# Complex exponentiation: 1^x ≈ 2

**URL:** <https://discourse.julialang.org/t/complex-exponentiation-1-x-2/107994>\
**Category:** Numerics\
**Created:** [December 24, 2023, 4:22am UTC](https://discourse.julialang.org/t/complex-exponentiation-1-x-2/107994 "2023-12-24T04:22:39Z")\
**Posts on this page:** 7\
**Page:** 1

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**Author:** ![mkitti](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mkitti/32/12459_2.png) [@mkitti](https://discourse.julialang.org/u/mkitti)\
**Post date:** [December 24, 2023, 4:22am UTC](https://discourse.julialang.org/t/complex-exponentiation-1-x-2/107994/1 "2023-12-24T04:22:39Z")

</div>

I have a mind worm, and I’m trying to figure out how we might express this in Julia.

Examine the following REPL log including complex exponentiation.

```julia-repl
julia> x = -1im * log(2)/(2π)
 0.0 - 0.1103178000763258im
 
julia> 1^x
 1.0 - 0.0im
 
julia> 2^(1/x)
 1.0 - 1.133107779529596e-15im

julia> 1 ≈ 2^(1/x)
 true
 
julia> 1^x ≈ 2
 false

julia> 1 ≈ exp(1im*2π)
true

julia> exp(1im*2π*x)
2.0 + 0.0im

julia> 1^x ≈ exp(1im*2π*x)
false

julia> 2 ≈ exp(1im*2π*x)
true

```

The problem is that we only are reporting the [principal value](https://en.wikipedia.org/wiki/Principal_value#%3A%7E%3Atext%3DIn_mathematics%2C_specifically_complex_analysis%2Cof_a_positive_real_number.?wprov=sfla1). For example for `sqrt(4)` we only report `2` and not `-2`.

I was hoping that a package such as ComplexValues.jl by @tobydriscoll might help express this better, but I’m not sure how to do so yet.

What I want is some kind of complex polar repesentation that has a notion of equivalence classes. Does this have to be symbolic?

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**Author:** ![Mason](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mason/32/2423_2.png) [@Mason](https://discourse.julialang.org/u/Mason)\
**Post date:** [December 24, 2023, 6:37pm UTC](https://discourse.julialang.org/t/complex-exponentiation-1-x-2/107994/2 "2023-12-24T18:37:23Z")

</div>

if you want to find all the different values, you could do something like

```julia
(^̃)(a, b) = let (;num, den) = rationalize(b)
    [exp((num * log(a) + 2*im*n*π)/den) for n ∈ 0:den-1]
end

```

and then we have

```julia
julia> 1 ^̃ (1/2)
2-element Vector{ComplexF64}:
  1.0 + 0.0im
 -1.0 + 1.2246467991473532e-16im

julia> 1 ^̃ (1/10)
10-element Vector{ComplexF64}:
                  1.0 + 0.0im
   0.8090169943749475 + 0.5877852522924731im
  0.30901699437494745 + 0.9510565162951535im
 -0.30901699437494734 + 0.9510565162951536im
  -0.8090169943749473 + 0.5877852522924732im
                 -1.0 + 1.2246467991473532e-16im
  -0.8090169943749475 - 0.587785252292473im
 -0.30901699437494756 - 0.9510565162951535im
  0.30901699437494723 - 0.9510565162951536im
   0.8090169943749473 - 0.5877852522924734im

```

Extending this to complex exponents is left as an exercise to the reader.

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<div class="post-metadata">

**Author:** ![empet](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/empet/32/221303_2.png) [@empet](https://discourse.julialang.org/u/empet)\
**Post date:** [December 24, 2023, 11:05pm UTC](https://discourse.julialang.org/t/complex-exponentiation-1-x-2/107994/3 "2023-12-24T23:05:55Z")

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The function, z^{m/n}, is a power function. +1 for its definition. But as I understood, the OP is interested in the exponential function, f(z)=a^z, a\in\mathbb{C}\setminus\{0\}, which is defined by f(z)=exp(z\*log(a)).  
While z^{m/n} is multivaluated, the exponential function, f(z)=a^z, is a univaluated function, because in its definition one takes a fixed branch of logarithm.

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**Author:** ![nsajko](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/nsajko/32/221187_2.png) [@nsajko](https://discourse.julialang.org/u/nsajko)\
**Post date:** [December 24, 2023, 11:10pm UTC](https://discourse.julialang.org/t/complex-exponentiation-1-x-2/107994/4 "2023-12-24T23:10:12Z")

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I think that’s kind of Mason’s point?

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<div class="post-metadata">

**Author:** ![empet](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/empet/32/221303_2.png) [@empet](https://discourse.julialang.org/u/empet)\
**Post date:** [December 24, 2023, 11:36pm UTC](https://discourse.julialang.org/t/complex-exponentiation-1-x-2/107994/5 "2023-12-24T23:36:26Z")

</div>

I wanted to point out that in complex function theory, the exponential function f(z)=a^z is univaluated, defined by a fixed value of log(a).

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<div class="post-metadata">

**Author:** ![Benny](https://avatars.discourse-cdn.com/v4/letter/b/49beb7/32.png) [@Benny](https://discourse.julialang.org/u/Benny)\
**Post date:** [December 24, 2023, 11:38pm UTC](https://discourse.julialang.org/t/complex-exponentiation-1-x-2/107994/6 "2023-12-24T23:38:15Z")

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> [@Mason](#):
>
> `[exp((num * log(a) + 2*im*n*π)/den) for n ∈ 0:den-1]`

I’m guessing mkitti is imagining lazier types to represent an output of multivalued functions, so they can compose with existing functions. Just like how we need to do `sqrt(Complex(-4))` to avoid a type-stability-preserving `DomainError`, we probably need convert inputs before passing into functions. Not sure if one `Multivalues` type can do it for outputs of all multivalued functions, my naive thought is that ±2 would look very different from an infinite set of complex numbers.

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<div class="post-metadata">

**Author:** ![nsajko](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/nsajko/32/221187_2.png) [@nsajko](https://discourse.julialang.org/u/nsajko)\
**Post date:** [December 24, 2023, 11:38pm UTC](https://discourse.julialang.org/t/complex-exponentiation-1-x-2/107994/7 "2023-12-24T23:38:47Z")

</div>

> [@mkitti](#):
>
> What I want is some kind of complex polar repesentation that has a notion of equivalence classes.

The equivalence class is represented by a polynomial (equation), whose roots are the elements of the equivalence class. You could use PolynomialRoots.jl (warning, it’s buggy) or AMRVW.jl to find the roots of the polynomial.

For example, given multivalued x = {a^{\frac{m}{n}}}, you want the roots of {x^n} - {a^m}. If the polynomial is just {x^e} - c, you can use Mason’s method for calculating all the roots, instead of reaching out for one of the above packages.