Thread: 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...

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,

Defined as is not a homomorphism because,

thus,

thus,

Only true when . Thus it is not an automorphism.

Similarily,

Gives,

Only true for .

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}?

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}?

is only a automorphism when . When it is that value then the first case is holomorphic.