I am working my way carefully through an article in Analytical Chemistry on round-off error. I'm a scientist but not a professional mathematician so it is rather slow-reading for me and none of it is intuitively obvious. The article is frustrating to read because there are numerous typographical errors that temporarily stop me in my tracks before I can convince myself that they are indeed errors. More problematic are the occasions when I'm unsure whether I have encountered a typo or simply failed to understand something. Hence the following ...
At one point in the article, a formula is stated and the derivation of the formula is said to rely on three general properties of characteristic functions. The verbatim statement of the third such property is:
If $g(t)$ is the characteristic function of $f(z)$, then the characteristic function of $\int_{-\infty}^{z} f(z) \ dz$ is $-g(t)/it$
where $i = \sqrt{-1}$.
I assume that the denominator for $-g(t)$ is intended to be the whole of $i \times t$ (am I correct?) ... but I don't understand what is going on with $z$ being the apparent limit of the integral as well as being the variable of integration? Is there a typo, and if so, what should the formula say ... or have I misunderstood something? ... If the limit of integration can really be the same as the variable of integration, then what should I read to understand what is going on?
Yes, this is most likely a typo, as you can see since they are using $t$ on the right of your equation. It should rather be $$ \int_{-\infty}^t f(z)\,\mathrm d z = -\,g(t)/(it). $$
After edit: Then it should rather be "the characteristic function of $\int_{-\infty}^z f(w)\,\mathrm dw$ is $-g(t)/(it)$", or any other variable name different from $z$. It is true that the abuse of notation of using the same variable inside and outside the integral is sometimes made. It is however as you noticed ambiguous.