https://file.aiquickdraw.com/mathbot/file/1727074054682dr0ub164.jpg

The question asks whether \( u + iv \) is analytic, where \( u \) and \( v \) are defined as follows:

\[
u = \frac{\partial \psi}{\partial y} - \frac{\partial \phi}{\partial x}
\]
\[
v = \frac{\partial \phi}{\partial x} + \frac{\partial \psi}{\partial y}
\]

Here, \( \phi \) and \( \psi \) are functions of \( x \) and \( y \).

To determine if \( u + iv \) is analytic, we need to check if the Cauchy-Riemann equations are satisfied. The Cauchy-Riemann equations in terms of partial derivatives are:

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

### Step 1: Compute the partial derivatives of \( u \) and \( v \)
1. \( u = \frac{\partial \psi}{\partial y} - \frac{\partial \phi}{\partial x} \)
   - \( \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}\left(\frac{\partial \psi}{\partial y}\right) - \frac{\partial^2 \phi}{\partial x^2} \)
   - \( \frac{\partial u}{\partial y} = \frac{\partial^2 \psi}{\partial y^2} - \frac{\partial}{\partial y}\left(\frac{\partial \phi}{\partial x}\right) \)

2. \( v = \frac{\partial \phi}{\partial x} + \frac{\partial \psi}{\partial y} \)
   - \( \frac{\partial v}{\partial x} = \frac{\partial^2 \phi}{\partial x^2} +

Let ϕ and ψ be functions of x and y satisfying appropriate conditions. If u = ∂ψ/∂y − ∂ϕ/∂x and v = ∂ϕ/∂x + ∂ψ/∂y, is then u + iv analytic?

Explore whether the function u + iv is analytic, based on Cauchy-Riemann equations and partial derivatives. This detailed solution covers the necessary conditions for analyticity in the context of complex functions. Suitable for college-level complex analysis students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation