Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to show a function is analytic?

  1. Sep 6, 2010 #1
    how to show a function is analytic??

    I know that to show a function is analytic I need to prove its differentiable, but for a fuction like log(z-i), how could i show it is analytic???
     
  2. jcsd
  3. Sep 6, 2010 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Re: how to show a function is analytic??

    Use the "Cauchy-Riemann equations which should be mentioned early in any book on "functions of a complex variable". A function f(x+ iy)= u(x,y)+ iv(x,y) is analytic at [itex]z_0= x_0+ iy_0[/itex] if and only if the partial derivatives, [itex]\partial u/\partial x[/itex], [itex]\partial u/\partial y[/itex], [itex]\partial v/\partial x[/itex], and [itex]\partial v/\partial y[/itex] are continous at the point and
    [tex]\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}[/tex]
    and
    [tex]\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}[/tex]
     
  4. Sep 6, 2010 #3
    Re: how to show a function is analytic??

    yes, i know wat u mean, but i dont know how to seperate log(z-i) into u + iv form
     
  5. Sep 6, 2010 #4
    Re: how to show a function is analytic??

    Why not just differentiate it and then show the derivative exists in a region surrounding a point then it is analytic in that region so:

    [tex]\frac{d}{dz} \log(z-i)=\frac{1}{z-i}[/tex]

    and that derivative exists everywhere except at z=i.
     
  6. Sep 8, 2010 #5
    Re: how to show a function is analytic??

    Or you could integrate the function over a closed line and show the integral is zero.
     
  7. Mar 17, 2012 #6
    Re: how to show a function is analytic??

    use log(z) = log(|z|) + i (arg(z))
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: How to show a function is analytic?
  1. Analytic Function (Replies: 7)

  2. Analytic functions (Replies: 3)

Loading...