Question

If f(z) is analytic in a domain d, and if |f| is constant, then f is ________.

Answer 1

If
`\( f(z) \)` is analytic in a domain
`\( D \)`, and if
`\( |f| \)` is constant, then
`\( f \)` is a constant function.

Step-by-step explanation:

Analytic functions are complex functions that can be represented by power series expansions. When
`\( |f| \)` is constant, it implies that the modulus (magnitude) of
`\( f(z) \)` is constant throughout the domain
`\( D \)`. To understand why
`\( f \)` must be a constant function, we can use the Cauchy-Riemann equations.

The Cauchy-Riemann equations state that if
`\( f(z) = u(x, y) + iv(x, y) \)` is analytic, where
`\( u \)` and
`\( v \)` are real-valued functions of
`\( x \)` and
`\( y \)` (the real and imaginary parts of
`\( f \))`, then the partial derivatives of
`\( u \)` and
`\( v \)` satisfy specific relationships. In particular, if
`\( |f| \)` is constant, then the modulus of
`\( f \)` is constant, and this implies that
`\( u \)` and
`\( v \)`satisfy the conditions:

`\[ (\partial u)/(\partial x) = (\partial v)/(\partial y) \quad \text{and} \quad (\partial u)/(\partial y) = -(\partial v)/(\partial x) \]`

When
`\( |f| \)` is constant, the magnitude of
`\( f \)` does not vary with respect to
`\( x \)` and
`\( y \)`, which implies that both
`\( (\partial u)/(\partial x) \`) and
`\( (\partial v)/(\partial y) \)` are zero. Similarly,
`\( (\partial u)/(\partial y) \)` and
`\( -(\partial v)/(\partial x) \)` are also zero. These conditions, combined with the Cauchy-Riemann equations, lead to
`\( u_x = u_y = v_x = v_y = 0 \)`, indicating that
`\( u \)` and
`\( v \)` are constant. Consequently,
`\( f \)` is a constant function in the given domain
`\( D \).`