# exponential function never vanishes

In the entry exponential function (http://planetmath.org/ExponentialFunction) one defines for real variable $x$ the real exponential function  $\exp{x}$, i.e. $e^{x}$, as the sum of power series:

 $e^{x}\;=\;\sum_{k=0}^{\infty}\frac{x^{k}}{k!}$

The series form implies immediately that the real exponential function attains only positive values when  $x\geqq 0$.  Also for  $-1\leqq x<0$  the positiveness is easy to see by grouping the series terms pairwise.

In to study the sign of $e^{x}$ for arbitrary real $x$, we may multiply the series of $e^{x}$ and $e^{-x}$ using Abel’s multiplication rule for series (http://planetmath.org/AbelsMultiplicationRuleForSeries).  We obtain

 $e^{x}e^{-x}\;=\;\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\!\sum_{k=0}^{\infty}(-1)^{% k}\frac{x^{k}}{k!}\;=\;\sum_{n=0}^{\infty}\!\sum_{j=0}^{n}(-1)^{j}\frac{x^{n}}% {j!(n\!-\!j)!}\;=\;\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\!\sum_{j=0}^{n}\!{n% \choose j}(-1)^{j}\;=\;\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\cdot 0^{n}.$

The last sum equals 1.  So, if  $-x>0$,  then  $e^{-x}>0$,  whence $e^{x}$ must be positive.

Let us now consider arbitrary complex value  $z=x\!+\!iy$  where $x$ and $y$ are real.  Using the addition formula of complex exponential function and the Euler relation, we can write

 $e^{z}\;=\;e^{x+iy}\;=\;e^{x}e^{iy}\;=\;e^{x}(\cos{y}+i\sin{y}).$

From this we see that the absolute value of $e^{z}$ is $e^{x}$, which we above have proved to be positive.  Accordingly, we may write the

The complex exponential function never vanishes.

Title exponential function never vanishes ExponentialFunctionNeverVanishes 2014-11-22 21:23:32 2014-11-22 21:23:32 pahio (2872) pahio (2872) 8 pahio (2872) Theorem msc 32A05 msc 30D20 real exponential function is positive ExponentialFunction ExponentialFunctionDefinedAsLimitOfPowers PeriodicityOfExponentialFunction