Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question

closed as not a real question by Will Hunting, Sasha, Austin Mohr, tomasz, no identity Nov 13 '12 at 22:09

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

    
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
3  
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
2  
@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

1 Answer 1

up vote 2 down vote accepted

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 $z$, $e^z\neq 0$.

Additionally:

  • It is equal to $$\lim_{n\to\infty} (1+\frac{z}{n})^n$$

What other properties might you want?

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.

share|improve this answer
1  
It is also periodc with period $2\pi i$. –  Stefan Nov 13 '12 at 21:23
    
I read somewhere about them having almost the same properties, but I just wanted to know wich ones were different. Maybe something like what @Harald Hanche-Olsen said. –  Ivan Lerner Nov 13 '12 at 22:29
    
@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

Not the answer you're looking for? Browse other questions tagged or ask your own question.