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Math Help - 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,
    f_a:\mathbb{C}^*\to\mathbb{C}^*
    Defined as f_a(z)=az is not a homomorphism because,
    f_a(xy)=f_a(x)f_b(y)
    thus,
    a(xy)=(ax)(ay)
    thus,
    axy=a^2xy
    Only true when a=1. Thus it is not an automorphism.

    Similarily,
    f_a(xy)=f_a(x)f_a(y)
    Gives,
    \frac{a}{xy}=\frac{a^2}{xy}
    Only true for 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}?
    f_a is only a automorphism when a=1. When it is that value then the first case is holomorphic.
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