Exponential

\(y=\exp(x)\) and \(y=\exp_{\sqrt{2}}(x) \)

\(u+\mathrm iv=\exp(x\!+\!\mathrm iy) \)

Exponential \(\exp\) is an elementary function that appears as solution of the differential equation \[\exp'(z)=\exp(z)\] with initial condition \[\exp(0)=1\] Explicit plot of exponential is shown with blue curve in figure at right.

The notation \(\exp_b\) refers to the exponential function with base \(b\): \[\exp_b(z)=b^z=\exp\big(\ln(b)\ z\big)\]

By default, exponential refers to "exponential to base \(\mathrm e \approx 2.71\)"; it is called also "natural exponential".

Exponential to base \(\sqrt{2}\) is shown in the top figure with red curve. The top figure is borrowed from book «Superfunctions»; it appears there as Fig.14.3 at page 178.

Complex map of the natural exponential is shown in figure at right.
It is borrowed from the same book, Fig.14.2 at page 177.

Inverse function

The inverse function of exponential is called "logarithm". It is denoted with symbol "log" or "ln"; \[ \exp\!\big(\log(z)\big)=z \]

Inverse function of exponential to base \(b\) is called "logarithm to base \(b\)"; \[ \exp_b\!\big(\log_b(z)\big)=z \]

Both the exponential and the logarithm are considered elementary functions.

Superfunction

The exponential is the superfunction of the multiplication map \(T(z)=\mathrm e z\): \[ \exp(z\!+\!1)= T(\exp(z)) \]

In the equation above, the multiplication is treated as a transfer function.

The constant \(\mathrm e\) is the base of the natural logarithm and has the expansion

\[\mathrm e=\sum_{k=0}^{\infty} \frac{1}{k!} \]

Any Superfunction of exp is called SuperExponential.

SuperExponential appears as solution \(F\) of the Transfer equation \[ F(z+1)=\exp\!\big(F(z)\big)\]

In general, the Superfunction is not unique.
The additional conditions are necessary to provide the uniqueness.

The requirement of the smooth behavior at \( \pm \mathrm i \infty\) and the additional condition \(F(0)=1\) lead to the special kind of real holomorphic SuperExponential called Tetration; this function is denoted with symbol "tet".

Abelfunction

The inverse of a superfunction is called an Abel function

Inverse function of tetration is called ArcTetration; it is denoted with symbol ate: \[ \mathrm{tet}\big(\mathrm{ate}(z)\big) = z \]

It satisfies the Abel equation \[ \mathrm{ate}\big(\exp(z)\big) = \mathrm{ate}(z) + 1 \]

Iterates

The Superfunction and the Abelfunction of exponential allows to express the \(n\)th iterate of exponential as follows: \[ \exp^n(z) = \mathrm{tet}\big(n+\mathrm{ate}(z)\big) \]

Here \(\exp^n\) denotes the \(n\)-fold functional iterate (not exponentiation).

In this expression, number \(n\) of the iterate has no need to be integer; it can be real or even complex.

Confusion

Some authors use ambiguous notation, writing \(f^n(z)\) when they actually mean \(f(z)^n\)

These notations cause confusion. In order to see this, it is sufficient to set \(f=\sin\) and \(n\!=\!-1\).

On the one hand, \(\sin^{-1}(z)= \arcsin(z) \)

On the other hand, in the ambiguous notation, \(\sin^{-1}(z)\) is interpreted as \(\displaystyle \frac{1}{\sin(z)}\).

The correct (unambiguous) notation:

\(\sin⁡(z)^{−1}=1/\sin⁡(z) \) (reciprocal)

\(\sin^{-1}(z)=\arcsin(z) \) (inverse function)

\(\sin^{0}⁡(z)=z \) (identity function)

\(\sin^{1}⁡(z)=\sin(z) \) (just sin)

\(\sin^{2}⁡(z)=\sin(\sin(z)) \) (second iterate of sin)

\(\sin⁡^{3}(z)=\sin(\sin(\sin(z))) \) (third iterate of sin)

and so on . And the same for other functions, including exp.

Acknowledgement

ChatGPT helped to improve this article.

References

https://en.wikipedia.org/wiki/Exponential_function In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. ..

https://en.citizendium.org/wiki/Exponential_function The exponential function of \(z\), denoted by \( \exp(z) \) or \(e^z\), can be defined as the solution of the differential equation \(\exp^{\prime}(z)\equiv \frac{d e^z}{dz}=\exp(z)\) with the additional condition \(\exp(0)=1.\, \)

Keywords

«Elementary function», «Exponential», «Logarithm», «Special function», «Superfunction», «Superfunctions», «Tetration»,

«ate», «arcsin», «exp», «log», «sin», «tet»,