# Origin of the special Finnish notation for difference of antiderivative

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 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.

• 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. Mar 9 at 20:48
• 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.) Mar 10 at 9:19
• 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... Mar 10 at 16:32