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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

\mathop{\rm exp}\nolimits (x) \equiv e^x,
\end{displaymath} (1)

where e is the constant 2.718.... It satisfies the identity
\mathop{\rm exp}\nolimits (x+y)=\mathop{\rm exp}\nolimits (x)\, \mathop{\rm exp}\nolimits (y).
\end{displaymath} (2)

If $z\equiv x+iy$,
e^z = e^{x+iy} = e^xe^{iy} = e^x(\cos y+i\sin y).
\end{displaymath} (3)

\end{displaymath} (4)

$\displaystyle y$ $\textstyle =$ $\displaystyle \tan^{-1}\left({b\over a}\right)$ (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 $e^{1/z}$.

See also Euler Formula, Exponential Ramp, Fourier Transform--Exponential Function, Sigmoid Function


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.

© 1996-9 Eric W. Weisstein