E^2*pi*i, where from?

1. Sep 19, 2010

sirwalle

1. The problem statement, all variables and given/known data
The problem at hand is that I don't understand wherefrom my text book got a certain term(e^(2*pi*i). It doesn't say. At least not as I understand it.
The book says:

2. Relevant equations
e^(z+2*pi*i) = e^z*e^(2*pi*i) = e^z*1 = e^z

From where does e^(2*pi*i) come? I get the stuff leading to the answer, I just can't seem to understand from where that term comes from.

2. Sep 19, 2010

Staff: Mentor

Basic property of exponents:

$$e^{a+b} = e^a e^b$$

3. Sep 19, 2010

Vivamus

I'm not certain as to what exactly your asking, but I hope this helps!

Recall the identity property of exponents:

ea+b=eaeb

a=z
b=2*pi*i

Therefore,
ez+(2*pi*i) = ez*e2*pi*i

4. Sep 19, 2010

sirwalle

Oh, no, I am sorry if I was not clear. I simply don't know wherefrom they get the 2*pi*i from in e^(z+2*pi*i).

The information I get is what I've written. I believe that the 2*pi refers to the period. It just seems kind of abrupt to randomly insert it without any proof or reference to hardly anything..

5. Sep 19, 2010

Bohrok

From calculus one learns that $$e^{i\pi} = -1$$
So, using a certain property of exponents, $$e^{2i\pi} = (e^{i\pi})^2 = (-1)^2 = 1$$

6. Sep 20, 2010

Staff: Mentor

OK. Could be they just added 2*pi*i at random. Why? Because they can

Do you know Euler's formula?

$$e^{ix} = \cos x + i \sin x$$

If you combine it all you see if you insert 2*pi*i into exponent at random, you will not change the result. Sometimes it can be a useful identity.

7. Sep 20, 2010

Mentallic

Actually they could add any integer multiple of $2\pi i$ and still leave the answer unchanged.

$$e^z=e^{z+2\pi i n}$$ where n is any integer.

8. Sep 20, 2010

Hurkyl

Staff Emeritus
We can't directly help you if you don't show us what they were doing up to that point.

9. Sep 20, 2010

Staff: Mentor

Lol, it must have been a senior moment on my side. I intended to write 2*pi*i*n but looked at 2*pi*i and decided there already is an integer (i) in the formula :grumpy:

10. Sep 20, 2010

Mentallic

People seem to find new uses for i each and every day

11. Sep 21, 2010

sirwalle

That's the thing. They aren't doing anything, it has its own little "information box". It says nothing after, nothing before. Just what I've written. All I know is that it has to do with Euler (the chapter is about Euler), if that helps?

12. Sep 21, 2010

Staff: Mentor

So it must be what I told you earlier - they just show an interesting and important property.

It is like asking where did the 2*pi came from in sin(x+n*2*pi) = sin(x) :tongue2: