# Complex Analysis : Applications of Harmonic Functions

## Example Questions

### Example Question #52 : Complex Analysis

If  and  are both analytic through a domain , which of the following is true?

must be constant throughout

is nowhere differentiable

is nowhere analytic

must be constant throughout

Explanation:

Since  is analytic, the Cauchy-Riemann Equations give us

Since  is analytic, the Cauchy-Riemann Equations give us .

Putting these two results together gives us

which implies that  is constant. Similar reasoning gives is that  is constant. This implies that must be constant throughout .