# Recognizing a “form” in a function

While teaching integrals, when seeing

``````f(x) = 8/(8x+1)
``````

we can say:

"We recognize a u'/u form, thus F(x) = ln(...)" (translation from what I'd say in French)

Another similar situation:

"In order to compute an integral invovling the function g(x)=(2x+1)^4, we need to put this function in the form / transform the function into g(x) = k u' u^n. We use the well-known form u' u^n, then ..."

Sorry for the math (off-topic) part, but my main concern is the english wording. Is the word "form" appropriated here? What is the natural wording you'd use in these 2 siutations?

I feel that the word formula ("we recognize a u'/u formula") is not exactly what I'm looking for.
Would "pattern" be better than "form"?

• From the point of view of usage, my recollection from math classes was that the word "form" was commonly used in cases like these. I don't recall professors using "pattern". Also very common was the expression "an equation of the form ..." – Al Maki Oct 20 '17 at 16:01
• yep, forms – Phil Sweet Oct 20 '17 at 16:56

I think what you have is totally appropriate! I'd maybe expand a little bit to something like

"We recognize that this equation has a u'/u form, which implies that the integral can be expressed"

But mathematics allows itself to be communicated in terse terms! If I saw this in a textbook I wouldn't think that it was out of place. If you are really hurting for a different word, pattern can stand in for form in all of those examples.

• I am used to see "form" as a formular in web design, so I wasn't really sure if it's frequent to see "form" as the translation of the french "forme" when you recognize a pattern... – Basj Oct 20 '17 at 14:27
• They are certainly spelled the same! In fact there's one more meaning of form that allows us to say things like "The form of the form helped me form an opinion". So we have 3 forms: (1) shape or general charather (2) thing that you fill out, as might be found on a website (3) verb meaning to shape. – mcnutt Oct 20 '17 at 15:16

Speaking as a software developer, I would suggest you use pattern instead of form.

Pattern
noun

1.2. A regular and intelligible form or sequence discernible in the way in which something happens or is done.

The dictionary lists a complicated and vague definition, but what you call a "form" falls under this listed definition.

A less abstract definition is found in on the Wikipedia page for software design patterns:

In software engineering, a software design pattern is a general reusable solution to a commonly occurring problem within a given context in software design.

Patterns have an underlying mathematical structure; indeed, mathematics can be seen as the search for regularities, and the output of any function is a mathematical pattern.

Mathematics is sometimes called the "Science of Pattern", in the sense of rules that can be applied wherever needed.

If I understand you correctly, when you say:

We recognize a u'/u form.

You actually mean

We can recognize the generalized u'/u problem, which we can solve using the generalized solution for **u'/u*.

Thus suggesting that it can be solved in the same way that you solve any u'/u problem.

If that is correct, then pattern is the correct word here:

We recognize a u'/u pattern.

Maybe it's easier to understand with a more general mathematical example:

All odd integers follow the `2n+1` pattern.

This example nicely proves the case why pattern fits for what you call a "form".