Problem 49 - Calculus with Concepts in Calculus

Question

Show that the functions \(u\) and \(v\) satisfy the Cauchy-Riemann equations \(u_{x}=v_{y}\) and \(u_{y}=-v_{x}\). $$ u=x^{2}-y^{2}, v=2 x y $$

Step-by-Step Solution

Answer

The functions satisfy the Cauchy-Riemann equations.

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

The function given is \(u(x, y) = x^2 - y^2\). First, find the partial derivative of \(u\) with respect to \(x\).\[\frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(x^2 - y^2) = 2x\]

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

The function given is \(v(x, y) = 2xy\). Now, find the partial derivative of \(v\) with respect to \(y\).\[\frac{\partial v}{\partial y} = \frac{\partial}{\partial y}(2xy) = 2x\]

Step 3: Verify First Cauchy-Riemann Equation

Check if the first Cauchy-Riemann equation \(u_x = v_y\) holds.\[2x = 2x\]The equation holds true.

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

Now calculate the partial derivative of \(u\) with respect to \(y\).\[\frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(x^2 - y^2) = -2y\]

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

Next, find the partial derivative of \(v\) with respect to \(x\).\[\frac{\partial v}{\partial x} = \frac{\partial}{\partial x}(2xy) = 2y\]

Step 6: Verify Second Cauchy-Riemann Equation

Check if the second Cauchy-Riemann equation \(u_y = -v_x\) holds.\[-2y = -2y\]The equation holds true.

Key Concepts

Complex Analysis

Complex analysis is a branch of mathematics focusing on functions of complex numbers. It deals with complex-valued functions defined in a complex plane. The usefulness of complex analysis comes from the property that functions within this field exhibit higher regularity and symmetry than their real counterparts.
At its core, complex analysis leverages the rich structure of complex numbers, which include a real part and an imaginary part. A function of a complex variable, often denoted as \(f(z)\), where \(z = x + iy\) and \(x, y\) are real numbers, is differentiated and integrated in complex form.
Key tools in complex analysis include:

• Analytic functions - these have derivatives at every point in their domains.
• Cauchy-Riemann equations - a set of two partial differential equations that provide conditions under which a function is analytic.

It serves as a foundational element for many areas of mathematics and engineering, particularly in electromagnetics and fluid dynamics.

Partial Differentiation

Partial differentiation is a technique used to find the derivative of a function with several variables with respect to one variable, keeping all others constant. It plays a crucial role in understanding functions of multiple variables, like those encountered in complex analysis.
In partial differentiation, you'll often see notation like \(\frac{\partial u}{\partial x}\), indicating the derivative of \(u\) with respect to \(x\). This process evaluates how the function changes as one specific variable changes, ignoring changes in other variables.
Why is partial differentiation important? Consider a function, such as \(u(x, y) = x^2 - y^2\). By partially differentiating \(u\) with respect to \(x\), you get \(2x\), showing how \(u\) changes as \(x\) changes alone. Similarly, partial derivatives with respect to \(y\) help in examining function behavior along the \(y\) direction.
The concept is essential in fields such as economics, physics, and engineering, allowing us to model systems influenced by various independent variables. In complex analysis, it's used to confirm whether functions satisfy specific criteria like the Cauchy-Riemann conditions.

Cauchy-Riemann Conditions

The Cauchy-Riemann conditions are two equations in complex analysis that a function must satisfy to be considered analytic. They allow us to verify if a complex function has derivatives everywhere within a region, making it conformal and enabling further analysis.
These conditions are represented as:

• First Equation: \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\)
• Second Equation: \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\)

Here, \(u(x, y)\) is the real part of \(f(z)\) and \(v(x, y)\) is the imaginary part. For example, in the exercise accompanying these conditions, the functions \(u(x, y) = x^2 - y^2\) and \(v(x, y) = 2xy\) satisfy both conditions.

Checking these conditions involves calculating the partial derivatives as shown:
- \(\frac{\partial u}{\partial x} = 2x\), \(\frac{\partial v}{\partial y} = 2x\)- \(\frac{\partial u}{\partial y} = -2y\), \(\frac{\partial v}{\partial x} = 2y\)
Both equalities confirm the Cauchy-Riemann equations, hence confirming the function is analytic in the region where these conditions hold. Mastering these principles is vital for exploring deeper analytical properties of functions in complex analysis.