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Complex Analysis: Largest set where f(z) is analytic

  1. Feb 16, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the largest set D on which f(z) is analytic and find its derivative. (If a branch is not specified, use the principal branch.)

    f(z) = Log(iz+1) / (z^2+2z+5)

    2. Relevant equations


    3. The attempt at a solution
    Not sure how to even attempt this solutions but I wrote down that
    iz+1 ∉ (-∞,0]. This is where I get confused! Not sure if I have to put z in x+iy form.

    For the denominator, z^2+2z+5 ≠ 0 implies z = +/- 1-2i.

    So my incomplete solution would be D = C\ { +/- 1-2i } υ { ?? } and the derivative is 1/(iz+1)?
     
  2. jcsd
  3. Feb 16, 2015 #2

    RUber

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    For the log, the restriction is on the real part.
    For your derivative, it seems like you lost the contribution from the denominator.
     
  4. Feb 16, 2015 #3
    Okay, so if it's just the real part, iz+1 = i(x+iy) + 1 = ix - y +1 so the restriction would just be -y+1, where y ≠ 1?

    I'm unsure what to do for a derivative, in my class notes it states that [log z ]' = 1/z so would it include the whole f(z) function, ie. ((z^2 + 2z + 5) / (iz+1))
     
  5. Feb 16, 2015 #4

    RUber

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    This would be either the product rule or the quotient rule.
    ##[\frac{g(z)}{f(z)}]'= \frac{fg'-gf'}{[f(z)]^2}##
     
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