Many functions in mathematics are denoted using abbreviated names. However, the capitalisation of these function names seems to be inconsistent to me: functions like $\sin$, $\cos$ and $\exp$ are all lowercase, whereas $\mathrm{Re}$ and $\mathrm{Im}$ (real and imaginary parts of complex numbers) have the first letter capitalised.

To some extent this may be subjective or up to the style guides of a specific journal, but many appear to be quite standard. Is there any rule for when the abbreviation should be capitalised?

  • $\begingroup$ I don't see any clear pattern, but I think most function names are just written lower case. What are some other examples of capitalized function names? $\endgroup$
    – littleO
    Apr 9 at 8:54
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    $\begingroup$ It depends on the country, whether it is capitalist or not. $\endgroup$
    – Jean Marie
    Apr 9 at 8:56
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    $\begingroup$ I would say that the rule of thumb is that the abbreviated functions are never capitalised, unless they are. $\endgroup$
    – user239203
    Apr 9 at 8:59
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    $\begingroup$ @littleO some more examples are $\Pr$, used to denote a probability, and $\mathrm{Ei}$, the exponential integral. There are also the Airy functions $\mathrm{Ai}$ and $\mathrm{Bi}$, but as Airy is a proper noun that's probably sufficient explanation. It does seem like lowercase is the rule and capitalised is the exception, but the capitalisation often seems arbitrary. $\endgroup$
    – DavidH
    Apr 9 at 9:05
  • $\begingroup$ Other examples: $E$ for mathematical expectation, $F$ for cdf (and more generally for primitive functions, for making the distinction with lower-case $f$), $F$ as well for hypergometric functions, etc. Other cases with capital letters of greek alphabet: Euler function $\Gamma$, Laplacian $\Delta$... Yes, a kind of rule could be "capital letters mainly for names that have a single letter"... $\endgroup$
    – Jean Marie
    Apr 9 at 12:21

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