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I'm trying to find useful notation for inverse functions that isn't too much in conflict with other notation already in use, but I'm wondering if this notation will come back and bite me in the future. Approximately how bad would using an overline for an inverse be? My motivation for this is because I'd find it nice to have a general way to express an inverse.

$$\begin{align}\overline{f}(x)&= f^{-1}(x) &&\text{ superscript}\\ \overline{\sin} x &= \arcsin x &&\text{ prefix}\\[0.4em] \overline{\exp} x &= \log x &&\text{ different symbol} \end{align}$$

Examples:

$$\begin{align}&(\text{i}) &&f^3\circ \overline{f^2}(x) = f^{3-2}(x) = f(x)\\ &(\text{ii}) &&\sin x = 1 \iff x = \overline{\sin}1\end{align}$$

Overlines are already used for negation operators in set theory, and is similar to the minus and division symbols, which are common inverse operations. But in which contexts might this notation just cause more confusion than help?

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    $\begingroup$ Overlines are conspicuously used to mean complex conjugate. Introducing a new notation is not inherently a bad thing, but it requires the introducer to define it each time it is used. Frequently a paper will define a lot of notation, some old some new, so this is not an insurmountable burden. $\endgroup$ – hardmath Nov 29 '14 at 13:26
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    $\begingroup$ In addition to conjugation, $\overline{f}$ is also frequently used to denote an extension of a function to a larger domain. $\endgroup$ – Daniel Fischer Nov 29 '14 at 13:28
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    $\begingroup$ or in even more basic levels, they sometimes denote mean or average value in some contexts. $\endgroup$ – user160738 Nov 29 '14 at 13:29
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    $\begingroup$ @fvel: They can be. The inverse of a cumulative distribution function comes up fairly often. $\endgroup$ – hardmath Nov 29 '14 at 13:45
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    $\begingroup$ @fvel: The mean (average) is also the expected value, so $E(X)$ can be used if overline is being reserved for another purpose. Notations are flexible to the needs of a presenter. $\endgroup$ – hardmath Nov 29 '14 at 14:08
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Yes, it can cause more confusion!

In the context of complex analysis overlining usually denotes pointwise complex conjugation: $$\overline{f}(z):=\overline{f(z)}$$ Same time depending on the problem the inverse may have different meanings: $$f^{-1}(z)f(z)=f(z)f^{-1}(z)\equiv1$$ $$f^{-1}(f(z))=f(f^{-1}(z))\equiv z$$

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  • $\begingroup$ So it kinda wouldn't make things more complex, considering $f^{-1}$ already has different meanings? $\endgroup$ – Frank Vel Nov 29 '14 at 13:36
  • $\begingroup$ What do you mean with "So it kinda wouldn't make think more complex"? Still the minus one is frequently used as it is clear from the context, for example when the algebra of complex functions is considered then one has pointwise multiplication and so the inverse is clearly with respect to this multiplication. Another example is the algebra of operators on a Banach space where multiplication is taken to be composition then inverse are meant w.r.t. to that multiplication. $\endgroup$ – C-Star-W-Star Nov 29 '14 at 13:43
  • $\begingroup$ @fvel: Your Question did not actually give much illustration of the notion that $f^{-1}$ "already has different meanings", but rather illustrates that in different contexts a different notation might be preferred to express the functional inverse. $\endgroup$ – hardmath Nov 29 '14 at 13:43
  • $\begingroup$ @Freeze_S think->things typo. Anyways thanks for your answer, I thought the last example was related to complex conjugations at first. I guess I'll have to be careful with my notation then... $\endgroup$ – Frank Vel Nov 29 '14 at 13:52

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