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Homework Help: Find the maximum value of this complex function

  1. Feb 13, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the maximum value of f(z) = exp(z) over | z - (1 + i) | ≤ 1

    2. Relevant equations
    |f(z)| yields the maximum value

    3. The attempt at a solution
    f(z) = exp(x) ( cosy + i siny)

    Unfortunately that's all i've got. I've seen examples with polynomials, but not with trigonometric functions. Please, any help?
  2. jcsd
  3. Feb 13, 2015 #2


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    What is the maximum value of just the real part if you have |real(z) - 1 | < 1?
    Does the imaginary part of the exponential function impact its modulus? i.e. what is | e^ix | ?
    ## e^{a+ib} = e^a e^{ib}##
    Start with those...you should get a sense of what needs to happen.
  4. Feb 13, 2015 #3
    The maximum value of [itex]f(z)[/itex] doesn't make sense since [itex]f(z)[/itex] is a complex number. In general, can one complex number be considered less than another complex number? e.g. is [itex]3i[/itex] "less than" [itex]4i[/itex]? These numbers cannot be compared this way however real numbers can.

    Have you learned the maximum modulus principle? If yes, you can parametrize the circle, plug that into the function and then evaluate the modulus. The maximum value will then become clear.
  5. Feb 14, 2015 #4

    Ray Vickson

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    Try writing ##z = 1+i+w##, with ##w = x + i y## and where ##x,y## are real. What do your restrictions on ##z## become in the new variables ##w## or ##x,y##? What does ##|f(z)|## look like in the new variables?
  6. Feb 15, 2015 #5

    I think the most convenient way to start is to imagine the triangle inequality for your boundary. As someone has pointed out already, what would the modulus for exp(ix) be? This will help because the largest modulus of z that you can find will also happen to be the largest modulus for exp(z).

    I think a good place to start is to somehow use the triangle inequality on this rather simple identity:

    |z| = |z - (1+i) + (1+i)|, and see if you can make use of the transitive property for inequalities.
  7. Feb 16, 2015 #6


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    Do you not know that "if a complex valued function is analytic on a set, then it takes on maximum and minimum values only on the boundary of the set.''
  8. Feb 16, 2015 #7

    Ray Vickson

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    If the OP pays attention to post #4, the problem becomes elementary and needs little, if any, deep properties.
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