Question

Suppose f is entire, with real and imaginary parts u and v satisfying u(x, y) v(x, y) = 3 for

all z = x + iy. Show that f is constant.

Be clearly, please. Do not upload same answers from others on Chegg. THANKS

Answer 1

Suppose,

f ( z) = u(x,y) + iv(x,y) is entire and u(x,y)v(x,y) =3 i.e. uv is constant

Consider (f(z))² ,

As f(z) is entire (f(z))² is entire

and (f(z))²= (u(x,y) + iv(x,y))²

= (u(x,y))² - (v(x,y))²  + 2i u(x,y)v(x,y)

= (u(x,y))² - (v(x,y))² + 6i

As u(x,y) v(x,y) is Constant and (f(z))² is entire

therefore By CR equations ,

(u(x,y))² - (v(x,y))2 is constant

  (f(z))² is constant

Hence f(z) is constant.