Question

if f is holomorphic on D[2,7] and takes only pure imaginary values, must f be constant? Prove or disprove

Answer 1

Answer:

Here f(z) is holomorphic. let f = u + iv

Where u = u(x, y), v = v(x, y)

Now given that f(z) takes only purely imaginary value then u = 0,

so f = +iv

By Cauchy Riemann theorem we have

Now u = 0 ,

~

v = constant = c (let)

So f(z) is a constant function.