Problem 51 - Calculus with Concepts in Calculus

Question

Show that the functions \(u\) and \(v\) satisfy the Cauchy-Riemann equations \(u_{x}=v_{y}\) and \(u_{y}=-v_{x}\). $$ u=e^{x} \cos y, v=e^{x} \sin y $$

Step-by-Step Solution

Answer

The Cauchy-Riemann equations are satisfied.

Step 1: Calculate Partial Derivative of u with Respect to x

First, we differentiate the function \( u = e^{x} \cos y \) with respect to \( x \). Treating \( \cos y \) as a constant, we get: \[u_{u}{x} = \frac{\partial}{\partial x}(e^{x} \cos y) = e^{x} \cos y\]

Step 2: Calculate Partial Derivative of v with Respect to y

Next, we differentiate \( v = e^{x} \sin y \) with respect to \( y \), considering \( e^{x} \) as a constant:\[u_{v}{y} = \frac{\partial}{\partial y}(e^{x} \sin y) = e^{x} \cos y\]

Step 3: Check the Cauchy-Riemann Equation for u_x = v_y

Examine if \( u_{x} = v_{y} \): From Steps 1 and 2:\[u_{u}{x} = e^{x} \cos y\] and \[u_{v}{y} = e^{x} \cos y\]Both are equal, satisfying \( u_{x} = v_{y} \).

Step 4: Calculate Partial Derivative of u with Respect to y

Differentiate \( u = e^{x} \cos y \) with respect to \( y \) this time. Treat \( e^{x} \) as a constant:\[u_{u}{y} = \frac{\partial}{\partial y}(e^{x} \cos y) = -e^{x} \sin y\]

Step 5: Calculate Partial Derivative of v with Respect to x

Now, differentiate \( v = e^{x} \sin y \) with respect to \( x \), keeping \( \sin y \) constant:\[u_{v}{x} = \frac{\partial}{\partial x}(e^{x} \sin y) = e^{x} \sin y\]

Step 6: Check the Cauchy-Riemann Equation for u_y = -v_x

We need to check if \( u_{y} = -v_{x} \):From Steps 4 and 5:\[u_{u}{y} = -e^{x} \sin y\]\[u_{v}{x} = e^{x} \sin y\]It holds that \( -v_{x} = -e^{x} \sin y = u_{y} \).

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, especially when exploring functions of several variables. In the context of Cauchy-Riemann equations, we examine how these derivatives help determine if a complex function is differentiable. When finding a partial derivative, we differentiate with respect to one variable while keeping other variables constant. This operation reveals how the function changes locally along the axis of the chosen variable.

For example, if we have a function like \( u = e^{x} \cos y \), finding \( u_x \) means differentiating \( u \) with respect to \( x \). Here, \( \cos y \) is treated as a constant. Similarly, to find \( u_y \), \( e^{x} \) is kept constant, and the differentiation is performed with respect to \( y \).

• Why it's important: Understanding partial derivatives allows us to break down complex functions into simpler components, which is essential for checking conditions like the Cauchy-Riemann equations.
• Application: These derivatives are crucial in physics, engineering, and economics where multi-dimensional changes are analyzed.

Complex Functions

Complex functions extend the idea of functions to the complex plane, where inputs and outputs are complex numbers. Such functions are at the heart of complex analysis. They encompass both real and imaginary parts, which themselves are functions of real variables.

In our exercise, \( u(x,y) = e^{x} \cos y \) and \( v(x,y) = e^{x} \sin y \) are real-valued components of a complex function \( f(z) = u(x,y) + iv(x,y) \), where \( z = x + iy \). The behavior of complex functions can be much richer than that of real functions.

• Real and Imaginary Parts: Any complex function can be broken down into its real component \( u(x, y) \) and its imaginary component \( v(x, y) \).
• Importance in Analysis: When these parts satisfy the Cauchy-Riemann equations, the function is holomorphic, meaning it is differentiable at all points in its domain.

Understanding and analyzing these components ensures that we can work efficiently with complex functions across domains of mathematics and physics.

Differentiation

Differentiation is the process of finding the rate at which a function changes as its input changes. In the realm of complex functions, differentiation involves the Cauchy-Riemann equations—a set of two conditions that a function must satisfy to be differentiable in the complex sense.

The process feels familiar: compute the derivative with respect to each variable. However, for a complex function \( f(z) = u(x, y) + iv(x, y) \) to be differentiable, it must satisfy:

• \( u_x = v_y \)
• \( u_y = -v_x \)

These equations ensure that locally, the function behaves like a complex linear function, akin to a line in the real plane.

Checking these conditions involves leveraging our skills in computing partial derivatives. If they hold, differentiation tells us that the function is smooth and well-behaved across its domain, making it a powerful tool for understanding the function's properties and behaviors.