# Is there a correct yet more compact way to write these equations?

I've got the following equation which denotes the total absorbance $A$ as determined by the sum of the absorbances of individual molecules:

\begin{equation} A(\nu,c_{1},...,c_{n}) = \sum_{mol=1}^{n} \epsilon_{mol}(\nu)c_{mol}l_{path}, \end{equation}

where $n$ is the amount of molecular species, and $c_{mol}$ and $\epsilon_{mol}$ the concentration and the molar absorptivity for each molecule in the gas. Though I believe this notation is correct, I was wondering if there is a more compact way to write this considering that my next equation continues using the set of concentrations:

\begin{equation} I_o (\nu,c_{1},...,c_{n})= I_i(\nu)10^{-A(\nu,c_{1},...,c_{n})}. \end{equation}

Preferably I'd have the latter equation more like this: \begin{equation} I_o (\nu,C)= I_i(\nu)10^{-A(\nu,C)}, \end{equation} where $C=\{c_{1},...,c_{n}\}$ denotes the set of all concentrations of molecules in the gas. This raises the question of how to write the first equation so as to be consistent in my notation. The most compact way seems to be

\begin{equation} A(\nu,C) = \sum_{mol=1}^{n} \epsilon_{mol}(\nu)c_{mol}l_{path}, \end{equation} but I believe this doesn't make sense as $C$ is not seen explicitly on the right hand side. Hence my question: is there a correct yet more compact way to write these equations?

I think your notation makes sense. You define a vector $C \in \mathbb{R}^n$ as $C= (c_1,c_2,...,c_n)$. The absorbance is a function of $n+1$ variables: $\nu$ and $c_{mol}$ for $mol=1,2,...,n$, than can be expressed in a more conpact way as $A(\nu, C)$. Your last equation it is perfectly valid since you have defined $c_{mol}$ as the $mol$-th component of the vector $C$.
• Thank you for the answer. I'm still wondering if the presence of $C$ only on the left hand side would be considered correct from a strict mathematical-notation point-of-view, and if there are any rules about this. I'm also wondering if there are any established rules concerning the use of a set/vector to indicate the dependency of a function on many elements like this. – OlavRG Feb 18 '15 at 14:26
• If a function depends on many variables, like in your case, thus it is a multivariable function and its image is often reported with an abbreviate notation. Consider for example the function $$f: \mathbb{R}^2 \rightarrow \mathbb{R}$$ defined by $$f(x,y) = 3x +xy^2$$. Using a more compact notation you can define a vector $v=(x,y)$ and write simply $$f(v) = 3x + xy^2$$, this is particularly handy when you have several variables. The two notations are completely equivalent. – NNec Feb 18 '15 at 15:08