Exponential Function

$\begin{figure}\begin{center}\BoxedEPSF{ExponentialFunction.epsf scaled 700}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{ExpReIm.epsf scaled 750}\end{center}\end{figure}$

The exponential function is defined by

$\begin{displaymath} \mathop{\rm exp}\nolimits (x) \equiv e^x, \end{displaymath}$

(1)

where e is the constant 2.718.... It satisfies the identity

$\begin{displaymath} \mathop{\rm exp}\nolimits (x+y)=\mathop{\rm exp}\nolimits (x)\, \mathop{\rm exp}\nolimits (y). \end{displaymath}$

(2)

If $z\equiv x+iy$ ,

$\begin{displaymath} e^z = e^{x+iy} = e^xe^{iy} = e^x(\cos y+i\sin y). \end{displaymath}$

(3)

$\begin{displaymath} a+bi=e^{x+iy}, \end{displaymath}$

(4)

then

$\displaystyle y$	$\textstyle =$	$\displaystyle \tan^{-1}\left({b\over a}\right)$	(5)
$\displaystyle x$	$\textstyle =$	$\displaystyle \ln\left\{{b\csc \left[{\tan^{-1}\left({b\over a}\right)}\right]}\right\}$
	$\textstyle =$	$\displaystyle \ln\left\{{a\sec \left[{\tan^{-1}\left({b\over a}\right)}\right]}\right\}.$	(6)

$\begin{figure}\begin{center}\BoxedEPSF{ExpInvReIm.epsf scaled 800}\end{center}\end{figure}$

The above plot shows the function $e^{1/z}$ .

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Exponential Function.'' §4.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 69-71, 1972.

Fischer, G. (Ed.). Plates 127-128 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 124-125, 1986.

Spanier, J. and Oldham, K. B. ``The Exponential Function $\mathop{\rm exp}\nolimits (bx+c)$ '' and ``Exponentials of Powers $\mathop{\rm exp}\nolimits (-ax^\nu)$ .'' Chs. 26-27 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 233-261, 1987.

Yates, R. C. ``Exponential Curves.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 86-97, 1952.