**From:** Philippe Verdy (*verdy_p@wanadoo.fr*)

**Date:** Sun Apr 17 2011 - 18:29:05 CDT

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2011/4/15 Hans Aberg <haberg-1@telia.com>:

*> Parenthesizes can be semantic, for example f^(k) (f superscripted with (k)) might denote the n-th derivative, while f^k the n-th composite or power of f.
*

*>
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Actually when *extra* parentheses are used in a superscript (i.e.

parentheses that are not needed to correctly evaluate the value of the

inner expression which is already self-contained by the superscript),

this gives a different semantic to those parentheses, which are not

used for their function of grouping, but to change the semantic into a

n-th derivative.

If you transcribe such expression into plain-text only, without

support for superscripts/exponent, you usually use the "^" symbol, but

then nothing can help make the distinction between grouping

parentheses and nth-derivative, because the inner expression is no

longer self-contained. The only way to preserve this, is to use extra

parentheses (exactly like for the case when superscripts are

available). So you should write: f^((n)).

If you write f^(n) only, it's not clear if this is the n-th exponent

or the n-th derivative of function f. This may be implicit in some

contexts, but specifying such formula without expliciting this context

will cause problems. Notably when both interpretations are used in the

same text (n-th derivatives and n-th exponentiations are heavily used

concurrently, for example when working with limited developments, or

with convergence of series, in probabilities...)

I've seen also f^('n), where the prime is denoted by some single quote

or apostrophe at the beginning of the inner expression. This preserves

the parentheses for their usual grouping-only meaning. If the inner

expression is simple enough (such as a single number or variable name,

you can even remove these parentheses and write f^'n directly. This

also preserves the distinction with f'^n (which denotes the n-th

exponent of the first derivative of function f, because the prime

operator is left-associative only).

I've also seen the even shorter notation f('n), without any ^ symbol

which then remains reserved to exponentiation only. (I will leave out

of this discussion the choice of the "correct" character to use for

the prime symbol, but it you cannot use true superscripts, then you're

also likely to use basic text encoding. Other notations like f^[n] or

f^{n} may be found (sometimes without the ^ symbol as well), provided

that square brackets and braces are not used as alternative grouping

parentheses to help reading complex formulas with multiple nesting

levels, or to denote vectors/matrices, or value ranges and sets, or to

denote indices/subscripts for terms of series.

Philippe.

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