Apologies for a question that is specific to one country (but perhaps others find it a curious example of how mathematical notation can vary between countries).

In Finnish calculus texts, if $F$ is an antiderivative of $f$, it is customary to write a definite integral as $$ \int_a^b f(x) \; dx = \bigg/_{\!\!\!\!a}^{\,b} F(x) $$ where the right hand side means $F(b)-F(a)$.

Q: When and where did this Finnish notation originate, and how did it spread?

As far as I know, this notation is not used anywhere outside Finland, and indeed Latex does not support it very well. (Finnish mathematicians simulate it with a big $/$ operator and tweak the spacing of the limits.) Elsewhere the typical notations are $$ F(x) \Big|_{x=a}^b \qquad\text{and}\qquad \Big[ F(x) \Big]_{x=a}^b $$ possibly with the "$x=$" omitted.

The earliest occurrence I know is in Ernst Lindelöf's ''Johdatus korkeampaan analyysiin'' (3rd printing, 1942, page 363).

Lindelöf page 363

Lindelöf simply introduces the notation here, but offers no explanation on whether it is already common. I wonder if this is the first use or if it is in fact older.

  • $\begingroup$ I don't know about the history of this, and have never seen it before, but it certainly does have the virtue of compatibility with the currently-standard "inside-out" order of operations. That is, unlike ${d\over dx}f(x)\Big|^b_a$, where order of operations requires a moment's thought, $\Big|_a^b {d\over dx}f(x)$ is less ambiguous. $\endgroup$ Mar 9, 2022 at 20:48
  • $\begingroup$ Yeah, that's true, it would seem sensible to use prefix operators throughout. I always felt the postfix big-vertical-line is an odd one out in this context. (To be fair, postfix makes sense as "function evaluation at these points", just like taking a function $f$ and appending $(x)$ to evaluate.) $\endgroup$ Mar 10, 2022 at 9:19
  • $\begingroup$ True, over the years postfix notation for functions, as in $(x)f$, has been advocated, so that notation for composition of functions is compatible with reading left-to-right: $(x)fg$ has us apply (left-most) $f$ first... $\endgroup$ Mar 10, 2022 at 16:32


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