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Math Help - a few complex calculus questions

  1. #1
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    a few complex calculus questions

    http://www.math.ualberta.ca/~runde/files/ass411-2.pdf
    I believe I got #1, 4, 5, 6i)
    for 2, i parametrized f(t) = t+it , t belongs to [0,1]
    the its
    integral from 0 to 1 of (2it)e^(2i(t^2)) dt
    from there I don't know what to do
    and I don't know if that is right either

    3 I think is partially done for me in my notes, I'll see if I can do it
    but hints are appreciated

    4 I used the unit circle at (1,1) showed the integral wasn't 0 and then said the function wasn't analytical since the integral wasn't 0 on a closed curve <- can someone confirm this is right?

    6ii) I'm not sure what to do there especially since my proof for i) isn't that great
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  2. #2
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    Quote Originally Posted by jbpellerin View Post
    for 2, i parametrized f(t) = t+it , t belongs to [0,1]
    the its
    integral from 0 to 1 of (2it)e^(2i(t^2)) dt
    from there I don't know what to do
    and I don't know if that is right either
    Hint: The function you are integrating has a primitive.

    3 I think is partially done for me in my notes, I'll see if I can do it
    but hints are appreciated
    Let X be the set of all points in D which can be polygonally connected and Y be the set of all points in D which cannot be polygonally connected. Prove that X and Y are open sets. Now since X\cap Y = \emptyset and X\cap Y = D it follows that X= D or Y=D by definition of connectdness of D.

    Note: This argument can be used to show that there is a polygonal path consisting of only vertical and horizontal segments.

    4 I used the unit circle at (1,1) showed the integral wasn't 0 and then said the function wasn't analytical since the integral wasn't 0 on a closed curve <- can someone confirm this is right?
    If there is f:\mathbb{C}\to \mathbb{C} with f'(z) = \bar z then f is twice differenciable too (in fact it is infinitely differenciable) and this would mean that the mapping z\mapsto \bar z is differenciable - but that is false - it is not differenciable at 0.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Hint: The function you are integrating has a primitive.
    so is the integral simply e^{(2i)}-1

    cool I just learned how to write some nice looking math on this site haha
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