Problem 91 - Calculus Early Transcendentals

Question

In the advanced subject of complex variables, a function typically has the form \(f(x, y)=u(x, y)+i v(x, y),\) where \(u\) and \(v\) are real-valued functions and \(i=\sqrt{-1}\) is the imaginary unit. A function \(f=u+i v\) is said to be analytic (analogous to differentiable) if it satisfies the Cauchy-Riemann equations: \(u_{x}=v_{y}\) and \(u_{y}=-v_{x}\) a. Show that \(f(x, y)=\left(x^{2}-y^{2}\right)+i(2 x y)\) is analytic. b. Show that \(f(x, y)=x\left(x^{2}-3 y^{2}\right)+i y\left(3 x^{2}-y^{2}\right)\) is analytic. c. Show that if \(f=u+i v\) is analytic, then \(u_{x x}+u_{y y}=0\) and \(v_{x x}+v_{y y}=0\)

Step-by-Step Solution

Answer

In conclusion: a. The function \(f(x, y)=\left(x^{2}-y^{2}\right)+i(2xy)\) is analytic because it satisfies the Cauchy-Riemann equations. b. The function \(f(x, y)=x\left(x^{2}-3 y^{2}\right)+i y\left(3 x^{2}-y^{2}\right)\) is also analytic as it satisfies the Cauchy-Riemann equations. c. If a function \(f = u + iv\) is analytic, its second-order partial derivatives follow Laplace's equation, that is, \(u_{xx} + u_{yy} = 0\) and \(v_{xx} + v_{yy} = 0\).

Step 1: a. Verify the analyticity of \(f(x, y)=\left(x^{2}-y^{2}\right)+i(2xy)\)

First, let us find the partial derivatives of the real and imaginary parts: \(u(x,y) = x^2 - y^2\) \(v(x,y) = 2xy\) \(\frac{\partial u}{\partial x} = 2x\) \(\frac{\partial u}{\partial y} = -2y\) \(\frac{\partial v}{\partial x} = 2y\) \(\frac{\partial v}{\partial y} = 2x\) Now, we will check if the Cauchy-Riemann equations are satisfied: \(u_x = v_y \Rightarrow 2x = 2x\) \(u_y = -v_x \Rightarrow -2y = -2y\) Both equations are satisfied, so the function is analytic.

Step 2: b. Verify the analyticity of $f(x, y)=x\left(x^{2}-3 y^{2}\right)+i y\left(3 x^{2}-y^{2}\right)$

Let's find the partial derivatives of the real and imaginary parts: \(u(x,y) = x(x^2 - 3y^2)\) \(v(x,y) = y(3x^2 - y^2)\) \(\frac{\partial u}{\partial x} = 3x^2 - 9y^2\) \(\frac{\partial u}{\partial y} = -6xy\) \(\frac{\partial v}{\partial x} = 6xy\) \(\frac{\partial v}{\partial y} = 3x^2 - 9y^2\) Now, we will check if the Cauchy-Riemann equations are satisfied: \(u_x = v_y \Rightarrow 3x^2 - 9y^2 = 3x^2 - 9y^2\) \(u_y = -v_x \Rightarrow -6xy = -6xy\) Both equations are satisfied, so the function is analytic.

Step 3: c. Show that \(u_{xx} + u_{yy} = 0\) and \(v_{xx} + v_{yy} = 0\) if \(f = u + iv\) is analytic

We know that the function satisfies the Cauchy-Riemann equations: \(u_x = v_y\) \(u_y = -v_x\) Now, let's differentiate these equations with respect to \(x\) and \(y\): \(\frac{\partial^2 u}{\partial x^2} = \frac{\partial v_y}{\partial x}\) (1) \(\frac{\partial^2 u}{\partial y^2} = -\frac{\partial v_x}{\partial y}\) (2) Adding (1) and (2): \(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}\) Since the right side of the equation is the partial derivative of \(v\) with respect to \(y\) minus the partial derivative of \(v\) with respect to \(x\), and these derivatives are equal because of the Cauchy-Riemann equations, we get: \(u_{xx} + u_{yy} = 0\) Similarly, we can differentiate the Cauchy-Riemann equations for \(v\): \(\frac{\partial^2 v}{\partial x^2} = \frac{\partial u_y}{\partial x}\) (3) \(\frac{\partial^2 v}{\partial y^2} = -\frac{\partial u_x}{\partial y}\) (4) Adding (3) and (4): \(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y}\) Again, since the right side of the equation is the partial derivative of \(u\) with respect to \(y\) minus the partial derivative of \(u\) with respect to \(x\), and these derivatives are equal because of the Cauchy-Riemann equations, we get: \(v_{xx} + v_{yy} = 0\)

Key Concepts

Cauchy-Riemann Equations

The Cauchy-Riemann equations are a set of two partial differential equations which provide a necessary and sufficient condition for a function of a complex variable to be analytic. Analytic functions are akin to differentiable functions and hold uniquely special properties. These equations are particularly useful when you want to examine the differentiability of a complex function.
To understand these, consider a complex function represented by \( f(x, y) = u(x, y) + iv(x, y) \) where \( u \) and \( v \) are real-valued functions. The Cauchy-Riemann equations are defined as:

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

These must hold true at every point in the domain of the function for it to be considered analytic. If a complex function meets these criteria, it means the function is differentiable not just at a point, but in an open neighborhood, thus offering deeper insight into its behavior.

Analytic Functions

The term "analytic" refers to a function that is more than just differentiable; it possesses derivatives of all orders, making it expressible as a power series within some radius of convergence. In the context of complex functions, being analytic typically aligns with satisfying the Cauchy-Riemann equations.
Analytic functions are central to many theorems and concepts in complex analysis. They have powerful properties such as:

• Being infinitely differentiable within their radius of convergence.
• Conforming to the maximum modulus principle, stating that their greatest absolute value within a domain lies on the boundary.
• Paving way to contour integration, a method used in evaluating complex integrals.

A common method to prove a function's analyticity is to demonstrate that the Cauchy-Riemann equations are satisfied throughout the region of interest. When a function is analytic, it can be broken down into real and imaginary components, each subject to its specific behavior governed by complex analysis principles.

Partial Derivatives

Partial derivatives are foundational in determining the behavior of functions with multiple variables, such as those defining complex variables. They reflect how a function changes as each variable shifts, providing insights into the slope of the function along different axes.
For a function \( f(x, y) = u(x, y) + iv(x, y) \), we derive the following partial derivatives:

• \( \frac{\partial u}{\partial x} \) and \( \frac{\partial u}{\partial y} \) for the real part \( u(x, y) \)
• \( \frac{\partial v}{\partial x} \) and \( \frac{\partial v}{\partial y} \) for the imaginary part \( v(x, y) \)

These derivatives are pivotal in verifying the Cauchy-Riemann equations. They are also critical in exploring more advanced concepts like Laplace's equation, where the sum of second-order partial derivatives of a function equates to zero, applicable in the context of analytic functions:

• \( u_{xx} + u_{yy} = 0 \)
• \( v_{xx} + v_{yy} = 0 \)

Thus, mastering partial derivatives enhances our understanding of the analytic landscape of complex functions, paving the way to appreciating the elegant structure underlying these mathematical constructs.