1. ## 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...

2. 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 $\displaystyle a=1$.

3. ## 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}?

4. 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.