Math Problem Statement

explain this proof

If f(z) is analytic on a domain D, and if f'(z) = 0 on D, then f(z) is constant.

Solution

This proof is a demonstration of the fact that if a complex function f(z)f(z) is analytic on a domain DD and its derivative f(z)=0f'(z) = 0 throughout DD, then f(z)f(z) must be a constant function on DD. Here’s a detailed explanation of the steps:

Problem Statement

The problem states:

  • Let f(z)f(z) be analytic on a domain DD.
  • If f(z)=0f'(z) = 0 on DD, show that f(z)f(z) is constant on DD.

Solution Explanation

  1. Analyticity and Cauchy-Riemann (C-R) Equations:

    • Since f(z)f(z) is analytic, it must satisfy the Cauchy-Riemann equations in the domain DD. This implies that f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y), where uu and vv are the real and imaginary parts of f(z)f(z), respectively.
    • The Cauchy-Riemann equations state that: ux=vyanduy=vx.\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.
  2. Condition f(z)=0f'(z) = 0:

    • The derivative f(z)=ux+ivxf'(z) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}.
    • Given f(z)=0f'(z) = 0, we have: ux+ivx=0.\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = 0.
    • This implies that both the real and imaginary parts must be zero: ux=0andvx=0.\frac{\partial u}{\partial x} = 0 \quad \text{and} \quad \frac{\partial v}{\partial x} = 0.
  3. Using the Cauchy-Riemann Equations:

    • From the first part, we already have ux=vy=0\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} = 0 and vx=uy=0\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y} = 0.
    • This implies that all partial derivatives ux\frac{\partial u}{\partial x}, uy\frac{\partial u}{\partial y}, vx\frac{\partial v}{\partial x}, and vy\frac{\partial v}{\partial y} are zero.
  4. Implication of Zero Partial Derivatives:

    • Since all partial derivatives of uu and vv are zero, both uu and vv must be constant functions (as they do not vary with xx or yy).
    • Consequently, f(z)=u+ivf(z) = u + iv is also constant because both uu and vv are constants.

Conclusion

Thus, if f(z)f(z) is analytic on DD and f(z)=0f'(z) = 0 on DD, then f(z)f(z) must be constant on DD.

Would you like more details on any specific part, or do you have any questions?

Here are some related questions to explore:

  1. What are the Cauchy-Riemann equations, and why are they significant in complex analysis?
  2. How does analyticity of a function in a domain differ from differentiability?
  3. Can you give an example of a non-constant analytic function that does not satisfy f(z)=0f'(z) = 0?
  4. Why do zero partial derivatives imply that a function is constant?
  5. What are the implications of the Cauchy-Riemann equations in the context of conformal mappings?

Tip: For any complex function, if you know it’s analytic and its derivative is zero, it’s immediately a constant function due to the properties of analytic functions.

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Math Problem Analysis

Mathematical Concepts

Complex Analysis
Analytic Functions
Cauchy-Riemann Equations

Formulas

Cauchy-Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
Derivative of f(z): f'(z) = ∂u/∂x + i∂v/∂x

Theorems

Cauchy-Riemann Equations
Constant Function Theorem in Complex Analysis

Suitable Grade Level

Undergraduate (College Level)