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Section 8.2 Dimensions 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?

Answer

Consider the logarithm rule

\begin{equation} \ln(ab)=\ln(a)+\ln(b)\label{logrule}\tag{8.2.2} \end{equation}

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.