Math Problem Statement

Show that analytic function f z u iv ( ) = + form an Orthogonal system.

Solution

To show that the analytic function f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y) forms an orthogonal system, we need to confirm that the real part u(x,y)u(x, y) and the imaginary part v(x,y)v(x, y) of f(z)f(z) satisfy the Cauchy-Riemann equations and are therefore harmonic functions. The Cauchy-Riemann equations ensure that the level curves of uu and vv are orthogonal, establishing an orthogonal system.

  1. Cauchy-Riemann Equations: For f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y) to be analytic, the partial derivatives of uu and vv with respect to xx and yy must satisfy the following equations: 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. Orthogonality Condition: To show that u(x,y)u(x, y) and v(x,y)v(x, y) form an orthogonal system, we examine the gradients of uu and vv: u=(ux,uy)andv=(vx,vy).\nabla u = \left( \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y} \right) \quad \text{and} \quad \nabla v = \left( \frac{\partial v}{\partial x}, \frac{\partial v}{\partial y} \right). The condition for orthogonality is that the dot product of u\nabla u and v\nabla v is zero: uv=uxvx+uyvy=0.\nabla u \cdot \nabla v = \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \frac{\partial v}{\partial y} = 0.

  3. Verification Using Cauchy-Riemann Equations: Substituting the Cauchy-Riemann equations into the dot product condition, we get: uxvx+uyvy=vyvx+uy(ux)=0.\frac{\partial u}{\partial x} \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \frac{\partial v}{\partial y} = \frac{\partial v}{\partial y} \cdot \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \cdot \left(-\frac{\partial u}{\partial x}\right) = 0. This confirms that u\nabla u and v\nabla v are orthogonal.

  4. Conclusion: Since u(x,y)u(x, y) and v(x,y)v(x, y) satisfy the Cauchy-Riemann equations and the dot product condition holds, uu and vv form an orthogonal system in the complex plane.

Would you like more details on any of these steps?

Here are five related questions to deepen your understanding:

  1. How do the Cauchy-Riemann equations relate to the concept of differentiability in complex analysis?
  2. Why is it significant that analytic functions produce harmonic (i.e., orthogonal) systems in the complex plane?
  3. What are some examples of analytic functions whose real and imaginary parts form notable orthogonal systems?
  4. How can the gradient concept in calculus help visualize orthogonal systems in complex functions?
  5. Can all complex functions be decomposed into orthogonal systems of real and imaginary components?

Tip: Remember that analytic functions, due to the Cauchy-Riemann equations, exhibit smoothness and directional independence, which contribute to the orthogonality of their level curves.

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Math Problem Analysis

Mathematical Concepts

Complex Analysis
Orthogonal Systems
Analytic Functions
Harmonic Functions

Formulas

Cauchy-Riemann Equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
Orthogonality Condition: ∇u · ∇v = 0

Theorems

Cauchy-Riemann Theorem
Orthogonality of Harmonic Functions

Suitable Grade Level

Undergraduate - Complex Analysis