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Thread: Complex Variables

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    Senior Member vincisonfire's Avatar
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    Complex Variables

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

    First I thought that working using Euler's notation would make the thing simpler. Let $\displaystyle f (r,\theta)=\rho e^{i\phi} $
    The condition are thus $\displaystyle |f(r,\theta)|=\rho \neq 0 $
    and $\displaystyle arg(f (x + iy)) = \phi = r^2cos(\theta)sin(\theta) $.
    Then $\displaystyle f(r,\theta) = \rho e^{ir^2cos(\theta)sin(\theta)} $ where
    $\displaystyle \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 $\displaystyle f (z) $ but I [think I] found many. Does what is above makes any sense?
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    Quote Originally Posted by vincisonfire View Post
    Let $\displaystyle f (z) $ be an entire function such that for all x, y,
    $\displaystyle |f (x +iy)| \neq 0 $ and $\displaystyle arg(f (x + iy)) = xy $. Find f .

    First I thought that working using Euler's notation would make the thing simpler. Let $\displaystyle f (r,\theta)=\rho e^{i\phi} $
    The condition are thus $\displaystyle |f(r,\theta)|=\rho \neq 0 $
    and $\displaystyle arg(f (x + iy)) = \phi = r^2cos(\theta)sin(\theta) $.
    Then $\displaystyle f(r,\theta) = \rho e^{ir^2cos(\theta)sin(\theta)} $ where
    $\displaystyle \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 $\displaystyle f (z) $ but I [think I] found many. Does what is above makes any sense?



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