# When should the first letter of an abbreviated function be capitalised?

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?

• 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? Apr 9 at 8:54
• It depends on the country, whether it is capitalist or not. Apr 9 at 8:56
• I would say that the rule of thumb is that the abbreviated functions are never capitalised, unless they are.
– user239203
Apr 9 at 8:59
• @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. Apr 9 at 9:05
• 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"... Apr 9 at 12:21