Question

Prove that f(z)=3x+y+i(3y−x) is analytic in C, hamely: entire b) Prove that if f(z) and f(z) are analytic in a domain Ω, then f is a constant function.

Answer 1

To prove that the function f(z) = 3x + y + i(3y - x) is analytic in the complex plane C, we need to show that the Cauchy-Riemann equations are satisfied.

Step-by-step explanation:

To prove that the function f(z) = 3x + y + i(3y - x) is analytic in the complex plane C, we need to show that the Cauchy-Riemann equations are satisfied. The Cauchy-Riemann equations are:

• ∂u/∂x = ∂v/∂y
• ∂u/∂y = -∂v/∂x

Let's break down the function into its real and imaginary parts:

• u = 3x + y
• v = 3y - x

We can see that the partial derivatives of u and v with respect to x and y satisfy the Cauchy-Riemann equations. Therefore, f(z) = 3x + y + i(3y - x) is analytic in C.