## Section8.2Dimensions in Power Series

When we consider a power series expansion of a special function such as

\begin{equation} \sin z = z -\frac{1}{3!} z^3 + \frac{1}{5!} z^5 + \dots\tag{8.2.1} \end{equation}

we can notice an interesting fact. If the variable $z$ were to have any kind dimensions (e.g. length, $L$) then the power series expansion of that special function would add together terms with different dimensions ($L\text{,}$ $L^3\text{,}$ $L^5\text{,}$ etc.). Since this is impossible, it implies that the argument of such special functions must always be dimensionless. This fact provides a quick check in many long algebraic manipulations. Look for expressions like $\sin\frac{x}{a}$ rather than $\sin x\text{,}$ where $a$ is a physical parameter with the same dimensions as $x\text{.}$

Is the logarithm function an exception to this rule?

if $a$ has non-trivial dimensions, but $ab$ is dimensionless. Then the power series expansion of the left-hand side of (8.2.2) will be a sum of dimensionless terms, as expected, but the power series expansion of the right-hand side of (8.2.2) might appear to have terms of different dimensions. The resolution is that the expansion parameters $a$ and $b$ are really $\frac{a}{1}$ and $\frac{b}{1}\text{,}$ where the factors of $1$ have the appropriate dimensions to make the expansion parameters dimensionless.