## Section8.8Dimensions in Power Series

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

$$\sin z = z -\frac{1}{3!} z^3 + \frac{1}{5!} z^5 + \dots\tag{8.8.1}$$

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?

$$\ln(ab)=\ln(a)+\ln(b)\tag{8.8.2}$$
if $$a$$ has non-trivial dimensions, but $$ab$$ is dimensionless. Then the power series expansion of the left-hand side of (8.8.2) will be a sum of dimensionless terms, as expected, but the power series expansion of the right-hand side of (8.8.2) 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.