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Use the Definition of a Limit to Find a Complex Limit (z->i)

  1. Sep 30, 2014 #1
    1. The problem statement, all variables and given/known data
    The lim(z->i) of [z^2+(1+i)z+2] using the epsilon-delta proof.
    2. Relevant equations
    z=x+iy
    Triangle Inequality: |z+w|<or=|z|+|w|
    3. The attempt at a solution
    For every epsilon>0, there exists a delta>0 such that
    |(z^2+(1+i)z+2)-(i)|<epsilon whenever 0<|z-i|<delta

    I'm not sure how to change z^2+(1+i)z+2 into z-i. Or am I approaching the situation wrong?
     
  2. jcsd
  3. Sep 30, 2014 #2

    Mark44

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    I think you're on the right track. z2 + (1 + i)z + 2 - i factors into (z - i)(z + 1 + 2i). I came up with this by long division.
     
  4. Oct 1, 2014 #3
    When I try the long division I get a remainder of i. Is that okay? I haven't ever done long division with complex numbers.
     
  5. Oct 1, 2014 #4

    vela

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    Are you dividing ##z^2+(1+i)z+2## by ##z-i##? Mark divided ##z^2+(1+i)z+2-i## by ##z-i##.
     
  6. Oct 1, 2014 #5

    Ray Vickson

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    Substitute ##z = i + t## into the original expression ##f(z) = z^2 + (1+i) z + 2## and expand it all out to get a new, equivalent, expression ##q(t)##. Now take ##t \to 0##. Believe me when I tell you it is much easier that way.
     
  7. Oct 1, 2014 #6
    Ahh I see what I did wrong with the long division. I definitely forgot that -i in there. So now I have:
    |(z+i)(z+1+2i)|<epsilon for 0<|z-i|<delta
    |z+i||z+1+2i|<epsilon but what can I do with z+1+2i to isolate z+1?
     
  8. Oct 1, 2014 #7

    Mark44

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    No. You should have (z - i) as one of the factors, not (z + i). For the other factor, you'll need to establish some bounds on it.
     
  9. Oct 1, 2014 #8
    Oh sorry that was a typo. I'm very dyslexic. On my paper I have it written down as |z-i||z+1+2i|<epsilon
     
  10. Oct 1, 2014 #9
    What do you mean by bounds on it?
     
  11. Oct 1, 2014 #10
    trying the modulus wouldn't do anything for me right now since I still have that pesky z there in z+1+2i, can I expand z out as z=a+ib?
     
  12. Oct 1, 2014 #11

    vela

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    You might want to review how you did this kind of problem when you were working with real functions. How did you find a bound when doing those limits?
     
  13. Oct 1, 2014 #12

    Mark44

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    You want to establish the largest and smallest values it (|z + 1 + 2i|) can have.
     
  14. Oct 1, 2014 #13

    Mark44

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    That's no help because you would be trading one unknown, z, for two unknowns, a and b.
     
  15. Oct 1, 2014 #14
    Oh like making an upper or lower bound on it by taking one of the factors to be less than 1 then plugging it back into the other inequality??
     
  16. Oct 1, 2014 #15

    Mark44

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    Something like that. vela's recommendation to review some similar limits with real-valued function is good advice.

    BTW, this thread is probably a better fit in the Calc section, so I moved it.
     
  17. Oct 1, 2014 #16
    Oh okay, thank you. So if I want to make an upper bound I first want to take 0<|z+1+2i|<1 but by definition of absolute value it turns into -1<z+1+2i<1 and if I want to take this and make it z-i I have to add -i-1-2i to each side which gives -i-2-2i<z-i<-i-2i. So |z-i|<-i-2i
     
  18. Oct 1, 2014 #17

    Mark44

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    No, it doesn't work this way with complex numbers, since you can't compare complex numbers using < or >. For instance, if |z| < 1, all that means is that z is a complex number inside the unit circle with center at the origin.
     
  19. Oct 1, 2014 #18

    Ray Vickson

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    Why don't you try out the suggestion I made in Post #5?
     
  20. Oct 1, 2014 #19

    Mark44

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    Ray, I don't see how this is helpful. Melissa already has the limit (which is i), and now she needs to prove that this is the limit using a delta-epsilon proof.
     
  21. Oct 1, 2014 #20
    My professor wants us to follow the definition the way he defined it in class but never gives us any examples. How do you put bounds on complex numbers then? Does anyone have any good resources for Complex Analysis? All I have for a textbook is that old Ian Stewart and David Tall's Complex Analysis: Hitchhikers guide to the plane. There's no examples in it either and the practice questions don't have answers. I never know if I'm doing anything right and clearly I'm not.
     
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