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Math Help - Complex Analysis: Finding Roots

  1. #1
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    Complex Analysis: Finding Roots

    Find all roots tanh(z)=i

    tanhz = [e^(z)-e^(-z)]/[e^(z)+e^(-z)]

    where do i go from here?
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  2. #2
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    Quote Originally Posted by CarmineCortez View Post
    Find all roots tanh(z)=i

    tanhz = [e^(z)-e^(-z)]/[e^(z)+e^(-z)]

    where do i go from here?
    Here's a start. Re-arrange:

    e^z - e^{-z} = i \left( e^z + e^{-z}\right)

    \Rightarrow (1 - i) e^z = (1 + i) e^{-z}

    \Rightarrow e^{2z} = \frac{1 + i}{1 - i} = i.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Here's a start. Re-arrange:

    e^z - e^{-z} = i \left( e^z + e^{-z}\right)

    \Rightarrow (1 - i) e^z = (1 + i) e^{-z}

    \Rightarrow e^{2z} = \frac{1 + i}{1 - i} = i.
    I don't understand the factoring in you second line, what is that?
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  4. #4
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    Quote Originally Posted by CarmineCortez View Post
    I don't understand the factoring in you second line, what is that?
    At this level of mathematical studies you should have seen the following steps (*):

    e^z - e^{-z} = i \left( e^z + e^{-z}\right)

    * \Rightarrow e^z - e^{-z} = i e^z + i e^{-z}

    * \Rightarrow e^z - i e^z = e^{-z} + i e^{-z}

    \Rightarrow (1 - i) e^z = (1 + i) e^{-z}
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