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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- Jun 2nd 2006, 12:16 AMnatyHolomorphic 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... - Jun 3rd 2006, 06:21 PMThePerfectHackerQuote:

Originally Posted by**naty**

$\displaystyle f_a:\mathbb{C}^*\to\mathbb{C}^*$

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 $\displaystyle a=1$. - Jun 5th 2006, 06:36 AMnatyholomorphic 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}? - Jun 5th 2006, 11:18 AMThePerfectHackerQuote:

Originally Posted by**naty**