prove that analytic function with constant modulus is constant in easy methood with explanation



To prove that an analytic function with a constant modulus is constant, we can use the following simple method based on the **Cauchy-Riemann equations** and **Liouville's theorem**. Here's a step-by-step explanation:

### Step 1: Understanding the Problem
Let \( f(z) \) be an analytic function on an open connected domain \( D \subset \mathbb{C} \). Suppose the modulus \( |f(z)| \) of the function is constant for all \( z \in D \). We need to prove that \( f(z) \) must be a constant function.

### Step 2: Apply the Cauchy-Riemann Equations
Since \( f(z) \) is analytic, it satisfies the Cauchy-Riemann equations. Write \( f(z) \) in terms of its real and imaginary parts:

\[
f(z) = u(x, y) + iv(x, y)
\]

where \( u(x, y) \) and \( v(x, y) \) are real-valued functions representing the real and imaginary parts of \( f(z) \), and \( z = x + iy \).

The Cauchy-Riemann equations are:

\[
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
\]

### Step 3: Differentiate the Constant Modulus Condition
Given that \( |f(z)| = \text{constant} \), we have:

\[
|f(z)| = \sqrt{u(x, y)^2 + v(x, y)^2} = C
\]

for some constant \( C \). Squaring both sides:

\[
u(x, y)^2 + v(x, y)^2 = C^2
\]

### Step 4: Differentiate with Respect to \( x \) and \( y \)
Now, differentiate both sides of this equation with respect to \( x \):

\[
2u \frac{\partial u}{\partial x} + 2v \frac{\partial v}{\partial x} = 0
\]

Since the Cauchy-Riemann equations give us \( \frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y} \), we can substitute:

\[
2u \frac{\partial u}{\partial x} - 2v \frac{\partial u}{\partial y} = 0
\]

Similarly, differentiate with respect to \( y \):

\[
2u \frac{\partial u}{\partial y} + 2v \frac{\partial v}{\partial y} = 0
\]

Substituting from the Cauchy-Riemann equations \( \frac{\partial v}{\partial y} = \frac{\partial u}{\partial x} \), we get:

\[
2u \frac{\partial u}{\partial y} + 2v \frac{\partial u}{\partial x} = 0
\]

### Step 5: Analyze the Equations
We now have the system:

\[
u \frac{\partial u}{\partial x} = v \frac{\partial u}{\partial y}, \quad u \frac{\partial u}{\partial y} = -v \frac{\partial u}{\partial x}
\]

The only way these equations can hold for all \( x \) and \( y \) (in a connected domain) is if:

\[
\frac{\partial u}{\partial x} = \frac{\partial u}{\partial y} = 0
\]

This implies \( u(x, y) \) is constant, and similarly, \( v(x, y) \) is constant. Therefore, \( f(z) = u + iv \) must be a constant function.

### Step 6: Conclusion
Since \( f(z) \) is analytic and \( |f(z)| \) is constant, we have shown that \( f(z) \) must be constant.

### Summary
An analytic function with constant modulus must be constant. This was proven using the Cauchy-Riemann equations, which show that both the real and imaginary parts of the function must be constant if the modulus is constant.

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### Related Questions:
1. What are the Cauchy-Riemann equations and their significance in complex analysis?
2. How does Liouville's theorem relate to bounded entire functions?
3. Can you provide an example of a non-constant analytic function and discuss its modulus?
4. How does the concept of holomorphic functions differ from analytic functions?
5. What is the significance of connected domains in complex analysis?
6. Can the constant modulus condition be relaxed to non-constant bounded modulus? What changes?
7. How do we extend the proof to non-open domains?
8. What role does the maximum modulus principle play in complex analysis?

### Tip:
When proving properties of analytic functions, always consider whether the Cauchy-Riemann equations and the implications of the function's behavior (like modulus) can simplify the problem.

Learn how to prove that an analytic function with a constant modulus is constant using the Cauchy-Riemann equations and Liouville's theorem. This problem explores complex analysis concepts in depth.

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