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Section 16.7 Dimensions of Step and Delta Functions
Dimensions of the Step Function.
The definition of the step function,
(16.3.1) , i.e.
\begin{equation*}
\Theta(x)=\begin{cases}
0 \amp x\lt 0\\
\frac{1}{2} \amp x=0\\
1 \amp x\gt 0
\end{cases}
\end{equation*}
clearly shows that the value of the step function is either the pure number zero or one, with no dimensions.
In
Figure 16.8 , you saw a figure where the step function was multiplied by some other function
\(f(x)\) so that the step function could be thought of as a switch, that ``turns on’’ the function
\(f(x)\text{.}\) In this case, the product function
\(f(x)\, \Theta(x)\) inherits the dimensions of
\(f(x)\text{.}\)
Dimensions of the Delta Function.
In contrast to the step function, above, the delta function
does have dimensions, but the dimensions change, according to the context. They are determined by the defining equation
(16.4.2) , i.e.
\begin{equation*}
\int_b^c \delta(x)\, dx = 1 \qquad\qquad b\lt 0\lt c
\end{equation*}
Since we can think of the differential \(dx\) as a little piece of \(x\text{,}\) it has the same dimensions as \(x\text{.}\) The integral says to add up a bunch of little pieces that look like \(\delta(x)\, dx\) and end up with the dimensionless number one. That tells us that the delta function \(\delta(x)\) by itself has dimensions that are the inverse of the dimensions of \(dx\text{.}\) For example, if \(x\) has dimensions of length \(L\text{,}\) then \(\delta(x)\) has dimensions of inverse length \(L^{-1}\text{.}\)
