Results 1 to 2 of 2

Math Help - Complex Variables

  1. #1
    Senior Member vincisonfire's Avatar
    Joined
    Oct 2008
    From
    Sainte-Flavie
    Posts
    469
    Thanks
    2
    Awards
    1

    Complex Variables

    Let  f (z) be an entire function such that for all x, y,
     |f (x +iy)| \neq  0 and  arg(f (x + iy)) = xy . Find f .

    First I thought that working using Euler's notation would make the thing simpler. Let  f (r,\theta)=\rho e^{i\phi}
    The condition are thus  |f(r,\theta)|=\rho \neq  0
    and  arg(f (x + iy)) = \phi = r^2cos(\theta)sin(\theta) .
    Then  f(r,\theta) = \rho e^{ir^2cos(\theta)sin(\theta)} where
     \rho \neq  0 . Since this is a composition of entire function it should be an entire function.
    I'm not comfortable with complex variables so I ask for some help. The question suggest there might be a single  f (z) but I [think I] found many. Does what is above makes any sense?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Oct 2008
    Posts
    156
    Quote Originally Posted by vincisonfire View Post
    Let  f (z) be an entire function such that for all x, y,
     |f (x +iy)| \neq 0 and  arg(f (x + iy)) = xy . Find f .

    First I thought that working using Euler's notation would make the thing simpler. Let  f (r,\theta)=\rho e^{i\phi}
    The condition are thus  |f(r,\theta)|=\rho \neq 0
    and  arg(f (x + iy)) = \phi = r^2cos(\theta)sin(\theta) .
    Then  f(r,\theta) = \rho e^{ir^2cos(\theta)sin(\theta)} where
     \rho \neq 0 . Since this is a composition of entire function it should be an entire function.
    I'm not comfortable with complex variables so I ask for some help. The question suggest there might be a single  f (z) but I [think I] found many. Does what is above makes any sense?



    I think you are correct. It says for all  x,y ,  |f(x+iy)| \neq 0 and  \text{arg}(f(x+iy))  = xy . So the argument is not constant. Nor is the modulus. Also polynomials of the form  f(x+yi) are analytic/entire.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Complex Variables
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: February 5th 2010, 04:12 AM
  2. help with complex variables
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: February 5th 2010, 04:12 AM
  3. complex variables
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: January 28th 2010, 12:28 PM
  4. Complex Variables
    Posted in the Calculus Forum
    Replies: 6
    Last Post: January 13th 2009, 07:27 PM
  5. Complex Variables
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 9th 2008, 08:28 AM

Search Tags


/mathhelpforum @mathhelpforum