MHB What could we change so that EXP(C) is a ring?

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Hey! :o

What could we change at the following definition so that $\text{EXP}(\mathbb{C})$ is a ring?
"We define EXP($\mathbb{C}$) to be the the set of expressions
\begin{equation}\label{a}
a=\alpha _0+\alpha _1e^{\mu_1z}+\dots +\alpha _Ne^{\mu_Nz}
\end{equation}
(beyond the `zero function', $0$, which we will consider to be also an element of EXP($\mathbb{C}$)),
where $\alpha_0, \alpha _1,\dots, \alpha_N\in \mathbb{C}\setminus \{ 0\}$ and $\mu_i\in \mathbb{C}\setminus \{ 0\}$; in writing such an expression we will always assume that the $\mu_i$ are pairwise distinct." Do we maybe have to assume that $\alpha_0, \alpha _1,\dots, \alpha_N\in \mathbb{C}$ and not $\alpha_0, \alpha _1,\dots, \alpha_N\in \mathbb{C}\setminus \{ 0\}$ ?
 
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Why isn't $\text{EXP}(\mathbb{C})$ a ring as it is?
 
As it is now we have that $$1+e^{z} \in \text{EXP}(\mathbb{C}) \text{ and } -1+e^z \in \text{EXP}(\mathbb{C})$$ but $$(1+e^z)+(-1+e^z)=2e^z\notin \text{EXP}(\mathbb{C})$$

So, does the definition have to be as follows?

"We define EXP($\mathbb{C}$) to be the the set of expressions
\begin{equation}
a=\alpha _1e^{\mu_1z}+\dots +\alpha _Ne^{\mu_Nz}
\end{equation}
where $\alpha_i \in \mathbb{C}$ and $\mu_i\in \mathbb{C}$; in writing such an expression we will always assume that the $\mu_i$ are pairwise distinct."
 
mathmari said:
"We define EXP($\mathbb{C}$) to be the the set of expressions
\[
a=\alpha _0+\alpha _1e^{\mu_1z}+\dots +\alpha _Ne^{\mu_Nz}
\]
(beyond the `zero function', $0$, which we will consider to be also an element of EXP($\mathbb{C}$)),
where $\alpha_0, \alpha _1,\dots, \alpha_N\in \mathbb{C}\setminus \{ 0\}$ and $\mu_i\in \mathbb{C}\setminus \{ 0\}$
I think the definition does not require that all terms $\alpha _0$, $\alpha _1e^{\mu_1z}$, ..., $\alpha _Ne^{\mu_Nz}$ have to be present. If any of them is present, then its coefficient $\alpha_i$ has to be different from 0.
 
Evgeny.Makarov said:
I think the definition does not require that all terms $\alpha _0$, $\alpha _1e^{\mu_1z}$, ..., $\alpha _Ne^{\mu_Nz}$ have to be present. If any of them is present, then its coefficient $\alpha_i$ has to be different from 0.
But according to the definition of post #1, does $e^z$ belong to $\text{EXP}(\mathbb{C})$ ? We have that $\alpha_0=0$.
 
Evgeny.Makarov said:
I think the definition does not require that all terms $\alpha _0$, $\alpha _1e^{\mu_1z}$, ..., $\alpha _Ne^{\mu_Nz}$ have to be present. If any of them is present, then its coefficient $\alpha_i$ has to be different from 0.

mathmari said:
But according to the definition of post #1, does $e^z$ belong to $\text{EXP}(\mathbb{C})$ ? We have that $\alpha_0=0$.

Hey mathmari! (Wave)

I believe the definition, as it is now, explicitly states that all $\alpha_i$ up to some $\alpha_N$ have to be distinct from zero.

So what can we change? Indeed, drop that requirement and allow all $\alpha_i$ to be zero.
Moreover, $0$ is then automatically included instead of having to make an exception for it.

So the question is, will that get us that for any $a,b,c \in R$:
$$a+b\in R, (a+b)+c=a+(b+c), a+0=a, \exists (-a):a+(-a)=0, a+b=b+a$$
$$ab\in R, a(bc)=(ab)c, a\cdot 1=1\cdot a =a$$
$$a(b+c)=ab+ac, (a+b)c=ac+bc$$
? (Wondering)
 
I agree that the original definition seems to require that $\alpha_0\ne0$. Whether this is intentional or not is hard to say. It is possible that the idea was to have elements of the form $\sum_{i=1}^n\alpha_ie^{\mu_iz}$ where $n\ge0$, $\alpha_i\ne0$ and $\mu_i\ne\mu_j$ for $i\ne j$. When $n=0$ this expression, by definition, is $0$. One of $\mu_i$ may be equal to 0; then this term is a non-zero constant. The benefit of such definition is uniqueness of representation. This uniqueness is lost if we allow all $\alpha_i$ to equal 0.
 
Evgeny.Makarov said:
I agree that the original definition seems to require that $\alpha_0\ne0$. Whether this is intentional or not is hard to say. It is possible that the idea was to have elements of the form $\sum_{i=1}^n\alpha_ie^{\mu_iz}$ where $n\ge0$, $\alpha_i\ne0$ and $\mu_i\ne\mu_j$ for $i\ne j$. When $n=0$ this expression, by definition, is $0$. One of $\mu_i$ may be equal to 0; then this term is a non-zero constant. The benefit of such definition is uniqueness of representation. This uniqueness is lost if we allow all $\alpha_i$ to equal 0.

Ah. Indeed!
So it would also suffice if we only allow $\alpha_0$ to be $0$, or if we allow $\mu_i$ to be $0$ (in which case we can leave out $\alpha_0$ completely).
All other $\alpha_i$ can remain distinct from $0$.
 
Evgeny.Makarov said:
The benefit of such definition is uniqueness of representation. This uniqueness is lost if we allow all $\alpha_i$ to equal 0.

Turns out it's not unique either way.
For instance $2\cdot e^{i(\pi/2)z} = -2\cdot e^{-i(\pi/2) z}$.
 
  • #10
Why do the elements have to be represented in an unique way? (Wondering)
 
  • #11
mathmari said:
Why do the elements have to be represented in an unique way? (Wondering)

It's not required for a ring.
But it would help if we want to deduce stuff based on a specific $\mu_i$.
Then the powers should be independent.
 
  • #12
I like Serena said:
Turns out it's not unique either way.
For instance $2\cdot e^{i(\pi/2)z} = -2\cdot e^{-i(\pi/2) z}$.

Nevermind. I made a mistake with the $z$. They are not equal after all.

mathmari said:
Why do the elements have to be represented in an unique way? (Wondering)

We need it to be able to say that if for any $z \in \mathbb C$ and $\alpha_i,\beta_i \in \mathbb C$ we have:
$$\sum \alpha_i e^{\mu_i z} = \sum \beta_j e^{k_j z}$$
that then:
$$\begin{cases}
\alpha_i = \beta_j &\text{if } \mu_i = k_j\\
\alpha_i = 0 &\text{if } \forall j: \mu_i \ne k_j\\
\beta_j = 0 &\text{if } \forall i: \mu_i \ne k_j\\
\end{cases}$$

And anyway, it's just neat to have a unique representation for every element in the ring.
 

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