Question

If \(u+i v\) is analytic, then \(v-i u\) is also analytic.

Short answer

If \( u+iv \) is analytic, then \( v-iu \) is also analytic.

Step-by-step solution

Introduction to Analytic Functions

A function \( f(z) = u(x, y) + i v(x, y) \) is said to be analytic in a domain if it satisfies the Cauchy-Riemann equations in that domain. These equations are: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).

Verify Given Function is Analytic

We know from the problem statement that \( f(z) = u(x, y) + i v(x, y) \) is analytic, meaning that \( u \) and \( v \) satisfy the Cauchy-Riemann equations: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).

Identify the New Function

Now consider the new function \( g(z) = v(x, y) - i u(x, y) \). We need to show that this function is also analytic.

Apply Cauchy-Riemann Equations to the New Function

Let's check if \( g(z) = v - i u \) satisfies the Cauchy-Riemann equations:- The partial derivative of \( v \) with respect to \( x \), \( \frac{\partial v}{\partial x} \), should equal the partial derivative of \( -u \) with respect to \( y \), \( -\frac{\partial u}{\partial y} \).- The partial derivative of \( v \) with respect to \( y \), \( \frac{\partial v}{\partial y} \), should be the negative of the partial derivative of \( -u \) with respect to \( x \), \( \frac{\partial u}{\partial x} \).

Check the First Condition

Calculate \( \frac{\partial v}{\partial x} \) and \( -\frac{\partial u}{\partial y} \). By the original Cauchy-Riemann equations: \(-\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}\). This condition is satisfied.

Check the Second Condition

Calculate \( \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial x} \). According to the original Cauchy-Riemann equations: \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\). This condition is also satisfied.

Conclusion

Since the function \( g(z) = v - i u \) satisfies the Cauchy-Riemann equations, it is analytic. Therefore, we have shown that if \( u+iv \) is analytic, then \( v-iu \) is also analytic.

Key concepts

Cauchy-Riemann Equations

In complex analysis, the Cauchy-Riemann equations are fundamental tools used to determine if a complex function is analytic. Analytic functions are, essentially, functions that are differentiable at every point in their domain. The Cauchy-Riemann equations come into play when the function is expressed in terms of real components.Suppose you have a complex function written as:

• \( f(z) = u(x, y) + iv(x, y) \)

In this representation, \( u \) and \( v \) are real-valued functions.To verify if \( f(z) \) is analytic, these equations must be satisfied:

• \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \)
• \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \)

If both conditions are met, \( f(z) \) is analytic at that point. This pairing of partial derivatives ensures that the changes in the real and imaginary parts of the function align perfectly to allow smooth complex differentiation.

Complex Analysis

Complex analysis is a powerful branch of mathematics that deals with complex numbers and the functions that involve them. Unlike real analysis, complex analysis involves examining functions of a complex variable and utilizes unique properties that only complex functions can have. Key Features:

• Analytic functions: Much of complex analysis focuses on such functions because they are continuously differentiable in a domain.
• Holomorphic functions: A term often used synonymously with analytic functions, meaning those that are complex-differentiable.

Complex analysis introduces surprising results, such as how integrals along paths depend only on their endpoints and not on the path taken—a feature not seen with real functions. The study often includes the exploration of mappings and transformations using these rich properties of complex numbers. Why It Matters:

• Complex analysis solutions often translate elegantly to real-world applications.
• It is integral to fields such as engineering, physics, and applied mathematics, resolving intricate behavior of waves, electrical circuits, and fluid dynamics.

Partial Derivatives

Partial derivatives provide a way to understand how multivariable functions change, focusing on one variable at a time while keeping others constant. In the context of complex functions, they allow us to break down the function into real and imaginary parts to see how each behaves independently.When studying a function like \( f(z) = u(x, y) + iv(x, y) \), where \( z = x + iy \):

• \( \frac{\partial u}{\partial x} \) examines how the real part \( u \) changes in relation to \( x \), holding \( y \) fixed.
• \( \frac{\partial v}{\partial y} \) considers how the imaginary part \( v \) changes with respect to \( y \), maintaining \( x \) constant.

Importance of Partial Derivatives:

• Essential for formulating and solving differential equations in physics and engineering.
• They help find maxima and minima in optimization problems involving multiple variables.

In complex analysis, partial derivatives are crucial in satisfying the Cauchy-Riemann equations, which assure us that a function is smooth and behaves nicely under differentiation.