# Properties of a complex exponential function. [closed]

Does the function $e^{z}=\sum\limits_{n=0}^{\infty}\frac{z^{n}}{n!}$ where $z$ is complex have all of the same properties of the real exponential function?

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## closed as not a real question by Will Hunting, Sasha, Austin Mohr, tomasz, NorbertNov 13 '12 at 22:09

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Which properties would you expect it to not have? Which properties would you want it to have? The two biggest properties are $e^{w+z}=e^{w}e^{z}$ and $e^{0}=1$. It is continuous and differentiable, and its derivative is the same function. – Thomas Andrews Nov 13 '12 at 21:09
Its growth properties are very different. Compare what happens when you go to infinity along the imaginary axis with behaviour along the real axis. And the real version is one-to-one, which the complex one isn't; far from it. – Harald Hanche-Olsen Nov 13 '12 at 21:14
@spernerslemma It's certainly still unbounded on the complex numbers, it just isn't unbounded on the imaginary axis (or any parallel axis.) – Thomas Andrews Nov 13 '12 at 21:19
There some differences. For example $e^x+1$ is alway non-zero. On the other hand $e^z+1$ has infinitely many solutions. – PAD Nov 13 '12 at 21:40

The biggest properties are

• It is well-defined for all $z\in\mathbb C$
• $e^{w+z}=e^{w}e^{z}$
• $e^{0}=1$
• It is continuous and (complex) differentiable, and its derivative is the same function.
• For all $z$, $e^z\neq 0$.

• It is equal to $$\lim_{n\to\infty} (1+\frac{z}{n})^n$$
As other have pointed out in comments, there are a few properties it doesn't have. It is not $1-1$, so its inverse (the natural logarithm) is not as "nice" a function. You have to either leave it undefined or deal with multi-valued functions.
It is also periodc with period $2\pi i$. – Stefan Nov 13 '12 at 21:23
@Stefan Harald's comment is actually somewhat difficult to phrase in complex numbers. On the real line, $e^x$ grows arbitrarily large if you are traveling to the right towards infinity, while it grows arbitrarily close to zero as $x\to -\infty$. But in complex number, you have lots of additional directions to travel. – Thomas Andrews Nov 14 '12 at 16:31