Does an Analytic Function Vanishing on a Disc Boundary Vanish Inside?

[Metric_Space]

Homework Statement

If an analytic function vanishes on the boundary of a closed disc in its domain
, show it vanishes on the full disc

Homework Equations

CR equations?

The Attempt at a Solution

Not sure how to start this one.

 

[micromass]

Hi Metric_space!

Have you seen the identity theorem for holomorphic functions. This says

If two holomorphic functions f and g on a domain D agree on a set which has an accumulation point c in D then f = g on all of D.

Did you see something like that? It would immediately give you the answer...

 

[Metric_Space]

Interesting ...I'll reread my notes. Thanks!

 

[Dick]

Interesting ...I'll reread my notes. Thanks!

You could also use the Cauchy Integral Formula, which is almost certainly in your notes even if micromass' theorem isn't.

 

[Metric_Space]

Is it just as simple as applying the Cauchy Integral formula?

ie. it follows directly from the CIF?

 

[HallsofIvy]

Well, what does the Cauchy Integral Formula say? What if you take the integral to be around the boundary of that disk?