Understanding Complex Numbers and Their Geometric Representation

[terbed]

Homework Statement

$$z=1-i$$
$$e^{iz} = ?$$
I have to solve this problem and than picture it.

Homework Equations

The Attempt at a Solution

$$e^{iz} =e^{i(1-i)}=e^{i+1}=e^i*e$$
I don't really understand how to picture this result. I assume their is an other way, in which the result has a geometric meaning.

 

[scurty]

Do you know how to graph a point in the polar form $z = re^{i\theta}$?

 

[terbed]

Yes I know! But there is "i+1" which fustrate. But I know how to graph for example $$e^i$$

 

[CaptainHammer]

Does the order of resolution matter?

If not, do the graphic first. z will be in the fourth quadrant with coordinates (1, -i)

e^(iz)=e^[i(1-i)]=e.e^i

edit: Think of e as the radius r.

 

[HallsofIvy]

In the standard graph of the complex plane, the point (a, b) represents the complex number a+ bi. The complex number i+ 1 (= 1+ i, of course) is represented by the point (1, 1). That strikes me as being easier than $e^i$!

But it is not difficult to go from one to the other. The distance from (0, 0) to (a, b) is $\sqrt{a^2+ b^2}$ and the angle the line from (0, 0) to (a, b) makes with the x-axis is arctan(b/a). a+ bi is the same as $\sqrt{a^2+ b^2}e^{arctan(b/a)}$. 1+ i has a= b= 1 so $\sqrt{1^2+ 1^2}= \sqrt{2}$ and $arctan(1/1)= \pi/4$. $1+ i= \sqrt{2}e^{\pi i/4}$.

 

[terbed]

Thanks!

 

[CaptainHammer]

Sometimes the simplest, dumbest thing is the correct one.

 

[HallsofIvy]

That's always been my plan!

 

[haruspex]

In the standard graph of the complex plane, the point (a, b) represents the complex number a+ bi. The complex number i+ 1 (= 1+ i, of course) is represented by the point (1, 1). That strikes me as being easier than $e^i$!

Yes, but you're not suggesting it equals $e^i$ are you?
terbed, do you know how to write $e^{iθ}$ in a + ib form? What does that give in this case?

 

[terbed]

I don't really know your problem! Yes I can represent it in a+bi form.

 

[haruspex]

I don't really know your problem! Yes I can represent it in a+bi form.

I thought you were trying to get a handle on what eⁱ looked like, so I expected at some point an answer in a+ib form. But I don't see that anywhere in the thread. What did you get for that?