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Complex functions with a real variable (graphs)

  1. Feb 14, 2017 #1
    1. The problem statement, all variables and given/known data

    How do the values of the following functions move in the complex plane when t (a positive real number) goes to positive infinity?

    y=t^2

    y=1+i*t^2


    y=(2+3*i)/t

    3. The attempt at a solution

    I thought:

    y=t^2 - along a part of a line that does not pass through the origin

    y=1+i*t^2 - along a part of parabola


    y=(2+3*i)/t - along a part of hyperbola

    Unfortunately everything is wrong. I understand that e.g. y=1+i*t^2 is a line =1 and a parabola but I don't know how to connect it. Could you give me a hint how to visualise this?
    Other possibilities: spirals inward/outward, clockwise/counterclockwise, along a circle, radially inward/outward


     
  2. jcsd
  3. Feb 14, 2017 #2

    PeroK

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    With ##t## as a parameter, you are not plotting ##y## against ##t##, but simply the locus of ##y## in the plane.
     
  4. Feb 14, 2017 #3
    I am trying to imagine it. So e.g. t^2 would move counterclockwise along a circle? (Moving parabola?)
     
  5. Feb 14, 2017 #4
    I got the second one right, moving along part of a line... Wow! I am beginning to understand.
     
  6. Feb 14, 2017 #5

    PeroK

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    Note that for the first locus ##y## is always real.
     
  7. Feb 14, 2017 #6
    Is it simply a parabola in this case?
     
  8. Feb 14, 2017 #7
    As for the third I guess it spirals clockwise inward as t^(-1).
     
  9. Feb 14, 2017 #8

    PeroK

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    Why not simply plot ##y##? You seem to be still thinking that you are plotting real ##y## against real ##t## on a normal 2D graph. That's not the case at all.

    You are plotting a single complex number ##y## as its value changes.

    If ##y## is real then it is confined to the real line and cannot be a parabola!
     
  10. Feb 14, 2017 #9

    PeroK

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    No. In particular, I'm not sure how you got the clockwise motion?
     
  11. Feb 14, 2017 #10

    LCKurtz

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    Perhaps there would be less confusion for the OP if the variable had been called ##z## instead of ##y##. Especially since complex numbers are usually expressed as ##z = x + iy##. Pretty poor notation for the problem if you ask me.
     
  12. Feb 14, 2017 #11
    Thank you very much for your patience. :)
    I tried plotting and got the third right.
     
  13. Feb 14, 2017 #12
    Ok. I got all the three right. Phew. It is very simple in fact. :(
     
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