Inverse function

Inverse function of some function \(f\) is sigh function \(g\!=\!f^{-1}\), that in wide range of argument \(z\), the relation

\(f(g(z))\!=\!z\)

holds. In some range, the relation

\(g(f(z))\!=\!z\)

is also valid.

Self-inverse functions

Some functions are their own inverse function. They are called self-inverse.

Here are examples to the self-inverse functions:

\(f(z)=-z\)

\(f(z)=1/z\)

Confusions

Every non-trivial holomorphic functions take the same value at various values of the argument. Hense, the inverse function is not unique. For such a function, notation \(F^{-1}\) cannot be used as definition of function; the branch of the inverse function should be specified.

However, for monotonous real-holomorphid function, there exist natural choice of the branch, such that the inverse function is also real-holomorhic at least in vicinity of some segment of the real axis.

Another confusion can be caused by the system of notations, where the superscript after the name of function is interpreted as combination of function with pow function; \(f^{a}(z)\) is interpreted as \(f(z)^{a}\).

This is often with trigonometric functions, instead of \(\sin(x)^{a}\) one writes \(\sin^a(x)\).

Then, for example, \(\sin(x)^{-1}(x)\) means both \(~\arcsin(x)~\) and \(~\displaystyle \frac{1}{\sin(x)}~\).

In order to avoid such a concision, in TORI combination of some function \(f\) with the power function with parameter \(n\), applied to argument \(z\) is written as \(f(z)^n\) or \(f(z^n)\), dependently on the order of combination, but never as \(f^n(z)\) .

The last notation, \(f^n(z)\), is reserved for the \(n\)th iterate of function \(f\), applied to argument \(z\).

Some confusion may appear with \(n\)th derivative, that is also specified in the superscript. In order to avoid this, the high order derivatives are written as \(~f^{\{n\}}(z)~\) or \(~f^{(n)}(z)~\)

Then,
\(f^{\{0\}}(z)=f(z)\)
\(f^{\{1\}}(z)=f'(z)\)
\(f^{\{2\}}(z)=f''(z)\)
and so on;
\(f^{\{-1\}}(z)=\int_c^zf(t) \,\mathrm dt~\) for some constant \(c\)

Not only c-numbers

Here, the inverse is described for functions of a real argument.

The same term, "inverse", may have many different meanings.

In particular, functions of more complicated objects can be considered too. Inverse, as operation of inversion, can be applied to various objects, from Boolean variables and the Soviet concept of history (sovietism) to operator in quantum mechanics and integral transform of a functions (see, for example, Fourier transform).

References

https://en.wikipedia.org/wiki/Inverse_function

Keywords

Holomorphic function, Inverse function, Iterate, notation,,,,