Transforming a complex exponentiation into a complex rectangular notation

The following exponentiation:

3e^(1+pi i)

Needs to be written in rectangular notation:

a + ib

Where "i" is the imaginary unit.

This is what I've found out myself so far:

I know that in order to find "a" and "b" I can use the argument "v" in the exponential notation:

re^(iv)

a = cos(v)

b = sin(v)

I also know that e^(iv1+iv2) = e^(iv1) * e^(iv2)

So if I try to apply these rules I get:

3e^(1+pi i) = 3e^1 * 3e^pi

Now calculating "a" I get:

a = 3*cos(1) * 3*cos(pi) = 3*cos(1) * -3

And for "b":

b = 3*sin(1) * 3*sin(pi) = 0

But I get the feeling that 3*cos(1) is wrong.

Re: Transforming a complex exponentiation into a complex rectangular notation

$\displaystyle e^{1} \ne \cos(1)$

$\displaystyle e^{1+i\pi}\;=\;e^{1}\cdot e^{i\pi}\;=\;e\cdot (\cos(\pi)+i\cdot\sin(\pi))\;=\;e\cdot (-1 + 0)\;=\;-e$

Re: Transforming a complex exponentiation into a complex rectangular notation

So e^1 simply equals e?

Does that mean that 3e^1 simply equals 3e?

And shouldn't it be 3*cos(pi) and 3*sin(pi)?

Re: Transforming a complex exponentiation into a complex rectangular notation

Yes. That "i" is more important than you were thinking. Without it, it's just Real Numbers.

Re: Transforming a complex exponentiation into a complex rectangular notation

But here is what I don't understand:

3e^1 = 3e

3e^pi i = 3*cos(pi) + i 3*sin(pi)

3e * (3*cos(pi) + i 3*sin(pi)) = 3e * (-3+0) = -9e

But my facit says "-3e"

Re: Transforming a complex exponentiation into a complex rectangular notation

Note that:

$\displaystyle 3e^{1+i\pi}\neq 3e\cdot 3e^{i\pi}$

$\displaystyle 3e^{1+i\pi}=3e\cdot e^{i\pi}$