Results 1 to 2 of 2

Thread: complex logarithm

  1. #1
    Super Member Random Variable's Avatar
    Joined
    May 2009
    Posts
    959
    Thanks
    3

    complex logarithm

    Let $\displaystyle z = x+iy $

    I want to show that $\displaystyle \ln(e^{iz}) = 2i \pi \lfloor \frac{1}{2} - \frac{x}{2 \pi} \rfloor + iz$

    $\displaystyle \ln(e^{iz}) = \ln|e^{iz}| + i \tex{Arg} (e^{iz}) $

    $\displaystyle = \ln (e^{-y}) + i \text{Arg} (e^{iz}) $

    $\displaystyle = -y + i \text{Arg} (e^{iz}) $


    EDIT: $\displaystyle \text{Arg} (e^{iz}) $ is some angle in the interval $\displaystyle (-\pi, \pi] $. I don't understand how I can express that with use of the floor function.
    Last edited by Random Variable; Jun 11th 2011 at 04:48 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Dec 2009
    From
    1111
    Posts
    872
    Thanks
    4
    Quote Originally Posted by Random Variable View Post
    Let $\displaystyle z = x+iy $

    I want to show that $\displaystyle \ln(e^{iz}) = 2i \pi \lfloor \frac{1}{2} - \frac{x}{2 \pi} \rfloor + iz$

    $\displaystyle \ln(e^{iz}) = \ln|e^{iz}| + i \tex{Arg} (e^{iz}) $

    $\displaystyle = \ln (e^{-y}) + i \text{Arg} (e^{iz}) $

    $\displaystyle = -y + i \text{Arg} (e^{iz}) $


    EDIT: $\displaystyle \text{Arg} (e^{iz}) $ is some angle in the interval $\displaystyle (-\pi, \pi] $. I don't understand how I can express that with use of the floor function.
    Dear Random Variable,

    Whatever the value of x; $\displaystyle \lfloor \frac{1}{2} - \frac{x}{2 \pi} \rfloor$ is an integer. That is,

    $\displaystyle \lfloor \frac{1}{2} - \frac{x}{2 \pi} \rfloor=n\mbox{ where }n\in{Z}$

    Consider, $\displaystyle \exp{\left( 2i \pi \lfloor \frac{1}{2} - \frac{x}{2 \pi} \rfloor+iz\right)}=\exp{(2\pi ni+iz)}=e^{2\pi ni}.e^{iz}=e^{iz}~;~(e^{2\pi ni}=1)$

    Hence, $\displaystyle \ln(e^{iz}) = 2i \pi \lfloor \frac{1}{2} - \frac{x}{2 \pi} \rfloor + iz$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Complex logarithm question
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: Jun 15th 2011, 03:57 PM
  2. Complex Logarithm Function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 23rd 2011, 05:46 AM
  3. Complex logarithm question
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: May 1st 2011, 03:06 AM
  4. Complex number(logarithm)
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Dec 18th 2009, 04:24 AM
  5. Complex number and Logarithm help me
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Oct 2nd 2006, 05:08 AM

Search Tags


/mathhelpforum @mathhelpforum