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So I was watching a youtube video about complex analysis and the professor wrote the equation on the board:

$e^{i\theta}=e^{i\beta}$

And then he said that:

$\theta=\beta+2\pi k$

For $k\in \Bbb{Z}$

No problems so far.

Then he says, "or in shorthand":

$\theta\in\beta+2\pi\Bbb{Z}$

Two questions:

First what is the name of this notation?

What would be the similar notation for the case where:

$\theta=ae+b\pi$

With $a\in\Bbb{Z}$ and $b\in\Bbb{Z}$?

I want to write:

$\theta\in e\Bbb{Z}+\pi\Bbb{Z}$

But I know that that is wrong because $a$ and $b$ could be different integers.

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  • $\begingroup$ The same concept in functional programming languages is called 'lifting': The first $+$ operator and scalar multiplication are ordinary addition on complex numbers $\mathbb C$. The second $+$ operator and scalar multiplication are 'lifted' versions of those ordinary operations, to subsets of $\mathbb C$, or in other words, lifted to $P(\mathbb C)$. See, e.g., wiki.haskell.org/Lifting and stackoverflow.com/q/2395697/223837. $\endgroup$ – Marnix Klooster Feb 8 '18 at 0:20
  • $\begingroup$ The answers below do a good job, but I might note that "coset" a common name for $\beta + 2\pi \mathbb Z$. That is, it is the translation of the subgroup $2\pi\mathbb Z$ by $\beta$. (There is also "sumset", which is what you have in the second example, but it'd be weird to see in this context) $\endgroup$ – Milo Brandt Feb 8 '18 at 4:14
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In general, if $A$ and $B$ are subsets of some set with an addition structure, you can write $A+B = \{a+b \mid a \in A, b \in B\}$. Formally what the professor wrote is $$\beta + 2\pi \Bbb Z = \{ \beta + 2\pi n \mid n \in \Bbb Z \},$$and what you're proposing is also correct: $$e\Bbb Z + \pi \Bbb Z = \{ae+b\pi \mid a,b \in \Bbb Z \}.$$The point is precisely that $e\Bbb Z + \pi \Bbb Z \neq (e+\pi)\Bbb Z$. I do not think that this notation has any specific name, though.

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    $\begingroup$ It's a Minkowski sum. $\endgroup$ – Henricus V. Feb 8 '18 at 3:54
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    $\begingroup$ I followed the @HenricusV. comment above and found Minkowski addition in Wikipedia. $\endgroup$ – CiaPan Feb 8 '18 at 11:16
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Actually, you are correct: $\theta=ae+b\pi$ for any $a,b\in \mathbf{Z}$ can be written as $\theta\in \mathbf{Z}e+\mathbf{Z}\pi$. This notation simply means that if we choose any integers $n,m$ from $\mathbf{Z}$ in the first and second space, respectively, we have that $\theta=ne+m\pi$. Note that this does not require $n=m$. I'm not sure of a specific name for this notation, but it is often used in algebra.

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