Complex numbers - polar representation

[jonnburton]

I have just begun studying this topic, and don't quite understand the complex exponential function.


The book says polar representation makes use of the complex exponential function, defined by:

\(\displaystyle e^z = 1 + \frac{z^2}{2!} + \frac{z^3}{3!} +...\)

It goes on to say that it immediately follows that for \(\displaystyle z = i\theta\), \(\displaystyle \theta\) real,

\(\displaystyle e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2!} - \frac{i\theta^3}{3!}+...\)


But I don't see how this is arrived at.

Given that \(\displaystyle i =\sqrt{-1}\), and therefore \(\displaystyle i^2 =-1\), I would come to the following:


\(\displaystyle e^{i\theta} = 1 + i\theta + \frac{i^2\theta^2}{2!} + \frac{i^3\theta^2}{3!}+\frac{i^4\theta^2}{4!}\)


Which, when the i is taken care of, I think comes out as:

\(\displaystyle 1 + i\theta -\frac{\theta^2}{2!}+\frac{\sqrt1\theta^3}{3ª}-\frac{\theta^4}{4!}...\)


(In my reasoning, as \(\displaystyle i=\sqrt{-1}\), \(\displaystyle i^2 = -1\); \(\displaystyle i^3=\sqrt 1\); \(\displaystyle i^4 = 1\), and \(\displaystyle i^5 = \sqrt{-1}\) etc.

But clearly I am failing to see something; can anyone tell me why my expansion here is wrong?

 

[DrPhil]

I have just begun studying this topic, and don't quite understand the complex exponential function.


The book says polar representation makes use of the complex exponential function, defined by:

\(\displaystyle e^z = 1 + \frac{z^2}{2!} + \frac{z^3}{3!} +...\)

It goes on to say that it immediately follows that for \(\displaystyle z = i\theta\), \(\displaystyle \theta\) real,

\(\displaystyle e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2!} - \frac{i\theta^3}{3!}+...\)


But I don't see how this is arrived at.

Given that \(\displaystyle i =\sqrt{-1}\), and therefore \(\displaystyle i^2 =-1\), I would come to the following:


\(\displaystyle e^{i\theta} = 1 + i\theta + \frac{i^2\theta^2}{2!} + \frac{i^3\theta^2}{3!}+\frac{i^4\theta^2}{4!}\)


Which, when the i is taken care of, I think comes out as:

\(\displaystyle 1 + i\theta -\frac{\theta^2}{2!}+\frac{\sqrt1\theta^3}{3ª}-\frac{\theta^4}{4!}...\)


(In my reasoning, as \(\displaystyle i=\sqrt{-1}\), \(\displaystyle i^2 = -1\); \(\displaystyle i^3=\sqrt 1\); \(\displaystyle i^4 = 1\), and \(\displaystyle i^5 = \sqrt{-1}\) etc.

But clearly I am failing to see something; can anyone tell me why my expansion here is wrong?

Consider the (geometric) series,

\(\displaystyle \displaystyle i^0, i^1, i^2, i^3, i^4, . . . = 1, i, -1, -i, 1, . . .\)

Where you have gone astray is \(\displaystyle i^3 = i^2\cdot i = -1 \cdot i = -i\). Notice that each power of \(\displaystyle i\) moves you 90° CCW around the unit circle, so this sequence of signs repeats every 4th power.

Does that work better?

 

[jonnburton]

Thanks DrPhil, that makes perfect sense!

Consider the (geometric) series,

\(\displaystyle \displaystyle i^0, i^1, i^2, i^3, i^4, . . . = 1, i, -1, -i, 1, . . .\)

Where you have gone astray is \(\displaystyle i^3 = i^2\cdot i = -1 \cdot i = -i\). Notice that each power of \(\displaystyle i\) moves you 90° CCW around the unit circle, so this sequence of signs repeats every 4th power.

Does that work better?