Complex Analysis simple Mapping question

[RJLiberator]

Homework Statement

Find the image of the rectangle with four vertices A=0, B= pi*i, C= -1+pi*i, D = -1 under the function f(z)=e^x

2. The attempt at a solution
So, the graph of the original points is obvious.
Now I have to map them to the new function.
Seems easy enough, but I am not getting one step. Can someone help me get started?

We have f(z) = e^x

So is it as simple as letting A = 0 = z = e^0 aka 1.
B = pi*i so z = pi+i so f(z) = e^(pi*i) aka cos(pi)+i*sin(pi) = -1
C = -1*pi*i so z = -1+pi*i and thus e^(-1+pi*i) = (1/e)*e^(pi*i) = -1/e
D = -1 so z = -1 and e^(-1) = 1/e

Thus we'd have points (0, 1) = A, (pi*i, -1) = B, (-1+pi*i, -1/e) = C, (-1, 1/e) = D

Not sure how i'd graph C tho :/

 

[FactChecker]

Yes, you have mapped the corners correctly. Now consider how the exponential changes along the 4 straight lines between the corners A,B,C,D. Example: For z going from A to B only the angle of e^z changes. The modulus remains constant. So that line AB gets mapped in a circular arc.

 

[FactChecker]

Thus we'd have points (0, 1) = A, (pi*i, -1) = B, (-1+pi*i, -1/e) = C, (-1, 1/e) = D

Up to this point, I agree. What do you mean here? A = 0 => e^A = e⁰ = 1; You just need to graph the points 1, -1, -1/e, and 1/e and map the lines between them. For instance, the line AB gets mapped to an arc of constant radius=1 that is centered at 0 and goes counter clockwise from 1 to -1. Do the same thinking for the other 3 lines.

 

[RJLiberator]

Hm. Well, I am very happy to hear my early calculations are correct.

I'm not quite sure how to graph it.

From your second post, A goes to (1, 0) ?? instead of (0, 1)?

B goes to (-1, 0) ?? And then it's a circle with radius one connecting them so it's a semi-circle above the x-axis? Then what happens to the other lines?

 

[FactChecker]

And then it's a circle with radius one connecting them so it's a semi-circle above the x-axis?

Yes.

Then what happens to the other lines?

Just remember that e^z = e^x+iy = e^xe^iy. So if x=Re(z) is constant while y=Im(z) changes, then the modulus is constant and only the angle changes. Similarly, if y=Im(z) is constant while x=Re(z) changes, then the angle is constant while the modulus changes. So the 4 edges of the rectangle should be simple to map. Just keep track of what is changing and if it is increasing or decreasing.

 

[RJLiberator]

So C to D would also be a semi circle from -1/e to 1/e. Hm, this doesn't seem right.
Your explanations are clear, but I am still missing something on a basic level it seems.

Not sure what B to C and D to A would be like.

A to B is decreasing
B to C is increasing
C to D is increasing
D to A is increasing

 

[FactChecker]

A=0+i0 to B=0+iπ : real is constant 0 while imaginary increases from 0 to iπ => maps to an arc of radius 1 and angle from 0 increasing to π
B=0+iπ to C=-1+iπ : imaginary is constant iπ while real decreases from 0 to -1 => maps to a line of constant angle π whose modulus decreases from 1 to e^-1

Do similar logic for C to D and for D to A

 

[RJLiberator]

Hm. Lights are starting to flicker.

So A to b increased from 0 to pi in angle with an arc of radius 1
B to C is a line that goes down from 0 to -1 in the imaginary axis
C to D is another angle dependent one that goes from pi to 0 degrees
and then D to A is a straight line back to the origin

Similar to polar coordinates it seems.

I really appreciate your help, time for me to go to work now. You have been wonderful and I plan on re-reading what you wrote again later to fully understand the problem.

Cheers.

 

[FactChecker]

Ok. Be sure to be very precise in your notation and thinking. Talk about the real and imaginary parts of A, B, C, and D. Talk about the modulus and argument (angle) of e^A, e^B, e^C, and e^D. Don't mix them up. You seem to be doing that in your writing, and maybe in your thinking.

 

[RJLiberator]

Hey mate,

I just wanted to follow up and thank you for your help. I understand now what you were trying to tell me.

So it's a semi-circle above the x-axis that goes from 1 to -1, and then comes into -1/e and back with a new radius to 1/e.
It makes sense to me. Still hard for me to explain mathematically, but it makes sense.

Thanks again for the help.