Question

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

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

Homework Answers

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.

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