Complex Analysis : Applications of Harmonic Functions

Study concepts, example questions & explanations for Complex Analysis

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Example Questions

Example Question #53 : Complex Analysis

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

Possible Answers:

 is nowhere differentiable

 must be constant throughout 

 is nowhere analytic

None of the other answers

Correct answer:

 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 .

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