exponential function never vanishes

In the entry exponential functionDlmfDlmfMathworldPlanetmathPlanetmath (http://planetmath.org/ExponentialFunction) one defines for real variable x the real exponential function  expx, i.e. ex, as the sum of power series:


The series form implies immediately that the real exponential function attains only positive values when  x0.  Also for  -1x<0  the positiveness is easy to see by grouping the series terms pairwise.

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


The last sum equals 1.  So, if  -x>0,  then  e-x>0,  whence ex must be positive.

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


From this we see that the absolute valueMathworldPlanetmathPlanetmathPlanetmath of ez is ex, which we above have proved to be positive.  Accordingly, we may write the

Theorem.  The complex exponential function never vanishes.

Title exponential function never vanishes
Canonical name ExponentialFunctionNeverVanishes
Date of creation 2014-11-22 21:23:32
Last modified on 2014-11-22 21:23:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 32A05
Classification msc 30D20
Synonym real exponential function is positive
Related topic ExponentialFunction
Related topic ExponentialFunctionDefinedAsLimitOfPowers
Related topic PeriodicityOfExponentialFunction