Exponential

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

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

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

Function \(\exp_b\) refers to the "exponential to 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 i 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 Exponential and Logarithm are qualified as elementary functions.

Superfunction

Exponential is superfunction of multiplication \(T = z\! \mapsto\! \mathrm ez\): \[ \exp(z\!+\!1)= T(\exp(z)) \]

Constant \(\mathrm e\) is called "base of natural logarithm";

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

Any Superfunction of exp is called SuperExponential.

SuperExponential appears as solution \(F\) or 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

Inverse of the superfiunction is called Abelfunction (or, mode formally, Abel function).

Inverse function of tertation 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: \[ \exp^n(z) = \mathrm{tet}\big(n+\mathrm{ate}(z)\big) \]

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

Confusion

Some authors use poor notations, interpreting \(f^n(z)\) as \( f(z)^n \); especially dealing with trigonometric functions.

These notations cause confuses. 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 poor notation, \(\sin^{-1}(z)\) is interpreted as \(\displaystyle \frac{1}{\sin(z)}\).

References

Keywords

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

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