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Complex Analysis simple Mapping question

  1. Jun 21, 2015 #1

    RJLiberator

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    1. The problem statement, all variables and given/known data

    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 :/
     
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  3. Jun 21, 2015 #2

    FactChecker

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    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 ez changes. The modulus remains constant. So that line AB gets mapped in a circular arc.
     
  4. Jun 21, 2015 #3

    FactChecker

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    Up to this point, I agree. What do you mean here? A = 0 => eA = 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.
     
  5. Jun 21, 2015 #4

    RJLiberator

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    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?
     
  6. Jun 21, 2015 #5

    FactChecker

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    Yes.
    Just remember that ez = ex+iy = exeiy. 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.
     
  7. Jun 21, 2015 #6

    RJLiberator

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    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
     
  8. Jun 21, 2015 #7

    FactChecker

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    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
     
  9. Jun 21, 2015 #8

    RJLiberator

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    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.
     
  10. Jun 21, 2015 #9

    FactChecker

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    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 eA, eB, eC, and eD. Don't mix them up. You seem to be doing that in your writing, and maybe in your thinking.
     
  11. Jun 23, 2015 #10

    RJLiberator

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    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 in to -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.
     
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