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Thread: Holomorphic automorphism

  1. #1
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    Lightbulb Holomorphic automorphism

    I need to prove, that f is holomorphic automorphism C*=C\{0}, when
    it's formation is
    fa(z)=az
    or
    fa(z)=a/z

    for some non-zero complex number a.

    Can anybody help me?
    Thank you...
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  2. #2
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    Quote Originally Posted by naty
    I need to prove, that f is holomorphic automorphism C*=C\{0}, when
    it's formation is
    fa(z)=az
    or
    fa(z)=a/z

    for some non-zero complex number a.

    Can anybody help me?
    Thank you...
    The function,
    $\displaystyle f_a:\mathbb{C}^*\to\mathbb{C}^*$
    Defined as $\displaystyle f_a(z)=az$ is not a homomorphism because,
    $\displaystyle f_a(xy)=f_a(x)f_b(y)$
    thus,
    $\displaystyle a(xy)=(ax)(ay)$
    thus,
    $\displaystyle axy=a^2xy$
    Only true when $\displaystyle a=1$. Thus it is not an automorphism.

    Similarily,
    $\displaystyle f_a(xy)=f_a(x)f_a(y)$
    Gives,
    $\displaystyle \frac{a}{xy}=\frac{a^2}{xy}$
    Only true for $\displaystyle a=1$.
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  3. #3
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    Question holomorphic automorphism C*=C\{0}

    I need to prove, that f is holomorphic automorphism C*=C\{0}
    ONLY when it's formation is
    f_a(z)=az
    or
    f_a(z)=a/z

    for some non-zero complex number a.

    How do I get to this formation of f if I know that f is holomorphic automorphism C*=C\{0}?
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  4. #4
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    Quote Originally Posted by naty
    I need to prove, that f is holomorphic automorphism C*=C\{0}
    ONLY when it's formation is
    f_a(z)=az
    or
    f_a(z)=a/z

    for some non-zero complex number a.

    How do I get to this formation of f if I know that f is holomorphic automorphism C*=C\{0}?
    $\displaystyle f_a$ is only a automorphism when $\displaystyle a=1$. When it is that value then the first case is holomorphic.
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