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Math Help - Any two, non zero, complex numbers.

  1. #1
    Junior Member universalsandbox's Avatar
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    Any two, non zero, complex numbers.

    Log is the principal branch of the log function.
    If two non-zero complex numbers z_{1},z_{2}
    Then show the following:
    Log( z_{1}z_{2}) = Log( z_{1}) + Log( z_{2}) - 2N*pi*i, for N = 0 or N = 1


    Thanks. Your help is appreciated.
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  2. #2
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    Quote Originally Posted by universalsandbox View Post
    Log is the principal branch of the log function.
    If two non-zero complex numbers z_{1},z_{2}
    Then show the following:
    Log( z_{1}z_{2}) = Log( z_{1}) + Log( z_{2}) - 2N*pi*i, for N = 0 or N = 1


    Thanks. Your help is appreciated.
    Let z_1 = r_1e^{i\theta_1},z_2=r_2e^{i\theta_2} where \theta_1,\theta_2 \in (-\pi,\pi].
    Then, z_1z_2 = r_1r_2 e^{i(\theta_1+\theta_2)}

    Therefore, \arg (z_1z_2) = \theta_1 + \theta_2 + 2\pi N where N =0\text{ or }-1 \text{ or }1.

    But, \log z_1 = \ln r_1 + i \theta_1 \text{ and }\log_2 = \ln z_2 + i \theta_2.
    And, \log z_1z_2 = \ln r_1r_2  + i (\theta_1+\theta_2) + 2i\pi N

    From here it follows that \log z_1z_2 = \log z_1 + \log z_2 + 2\pi i N
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