From: Philippe Verdy (verdy_p@wanadoo.fr)
Date: Sun Apr 17 2011 - 18:29:05 CDT
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.
>
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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