As a first approximation, neglect lower order terms

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As a first approximation, neglect lower order terms

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[QUICK DESCRIPTION]

Often in analysis, one is faced with a complicated expression (e.g. an integral or a partial differential equation) involving many different terms, and it may not be immediately clear how to proceed to simplify or otherwise control the expression.  However, in many cases one can assign different terms a different "order" which measures how large or how troublesome the term is expected to be.  Then one way to proceed is to pretend that one can automatically ignore all but the "highest order" or "worst" terms, and solve that simplified problem first.  After the highest order terms are controlled adequately, one can then insert back in the next highest order terms into the argument, and control them also; and continue in this fashion until one returns to the original problem.  This way, one only deals with one level of difficulty at a time, rather than all of them at once.  (Also, there is usually very little point working very hard on the lower order terms unless the higher order terms are already under control.)

The precise meaning of "higher order" and "lower order" terms varies from context to context. But here are some examples:

* If there is an important large parameter $N$ in play, then terms which are quadratic in $N$ (e.g. $aN^2$) could be considered higher order than those that are linear in $N$, terms which are cubic in $N$ would be considered higher order still, and so forth.  Terms which are exponential in $N$ (e.g. $e^{cN}$) would be higher order than all polynomial terms, while terms which are logarithmic in $N$ (e.g. $\log^k N$) would be lower than all polynomial terms, and so forth.
* Similarly, if there is an important small parameter $\varepsilon$ in play, terms that are independent of $\varepsilon$ would be "higher order" than those that are linear in $\varepsilon$, which in turn would be higher order than those that are quadratic in $\varepsilon$, etc.  (In microlocal analysis and quantum mechanics, the symbol $h$ or $\hbar$ is often used for the small parameter.)  Note that in this case, the smaller powers of $\varepsilon$ are often referred to as "lower order" terms rather than "higher order" terms.
* If there is a low regularity function $u$ in play, then terms involving first derivatives $\nabla u$ of $u$ can be considered higher order than those involving zeroth derivatives $u$ of $u$, but are in turn lower order than terms involving second derivatives $\nabla^2 u$ of $u$.  This type of situation is particularly common in nonlinear PDE.
* If there is a nilpotent group $G$ in play, elements in $G$ can be considered higher order than elements in the first commutator group $G_2 = [G,G]$, which in turn are higher order than elements in the second commutator group $G_3 = [G,G_2]$, etc.  (See also "[[First pretend that a normal subgroup is trivial]]")

[PREREQUISITES]

Analysis; partial differential equations; group theory

[EXAMPLE]

(Energy estimates exploiting cancellations for some nonlinear PDE)

(Some example of a computation which is easier to follow if one first "ignores log factors")

Suggestions welcome!

[GENERAL DISCUSSION]

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As a first approximation, neglect lower order terms

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