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 \(v_{x x}+v_{y y}=0 .\) Assume \(u\) and \(v\) satisfy the conditions in Theorem 15.4

Short answer

Question: Show that the given functions are analytic and that if \(f=u+i v\) is analytic, then \(u_{x x}+u_{y y}=0\) and \(v_{x x}+v_{y y}=0\). Answer: a. \(f(x, y)=\left(x^{2}-y^{2}\right)+i(2 x y)\) is analytic since it satisfies the Cauchy-Riemann equations, with \(u_x = v_y = 2x\) and \(u_y = -v_x = -2y\). b. \(f(x, y)=x\left(x^{2}-3 y^{2}\right)+i y\left(3 x^{2}-y^{2}\right)\) is analytic since it satisfies the Cauchy-Riemann equations, with \(u_x = v_y = 3x^2 - 3y^2\) and \(u_y = -v_x = -6xy\). c. If \(f=u+i v\) is analytic, then both u and v satisfy Laplace's equations, which implies that \(u_{xx} + u_{yy} = 0\) and \(v_{xx} + v_{yy} = 0\).

Step-by-step solution

Identify u(x, y) and v(x, y)

In this function, we have \(u(x, y) = x^2 - y^2\) and \(v(x, y) = 2xy\).

Find the first partial derivatives

Now, we'll find the first partial derivatives of u and v: \(u_x = \frac{\partial u}{\partial x} = 2x\) \(u_y = \frac{\partial u}{\partial y} = -2y\) \(v_x = \frac{\partial v}{\partial x} = 2y\) \(v_y = \frac{\partial v}{\partial y} = 2x\)

Check Cauchy-Riemann equations

We now check if the Cauchy-Riemann equations hold: \(u_x = v_y\) and \(u_y = -v_x\). Since \(2x = 2x\) and \(-2y = -2y\), the Cauchy-Riemann equations hold, so the function 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.

Identify u(x, y) and v(x, y)

Here, we have \(u(x, y) = x(x^2 - 3y^2)\) and \(v(x, y) = y(3x^2 - y^2)\).

Find the first partial derivatives

Let's find the first partial derivatives of u and v: \(u_x = \frac{\partial u}{\partial x} = 3x^2 - 3y^2\) \(u_y = \frac{\partial u}{\partial y} = -6xy\) \(v_x = \frac{\partial v}{\partial x} = 6xy\) \(v_y = \frac{\partial v}{\partial y} = 3x^2 - 3y^2\)

Check Cauchy-Riemann equations

Checking if the Cauchy-Riemann equations hold: \(u_x = v_y\) and \(u_y = -v_x\). Since \(3x^2 - 3y^2 = 3x^2 - 3y^2\) and \(-6xy = -6xy\), the Cauchy-Riemann equations hold, and therefore the function is analytic. c. Show that if \(f=u+i v\) is analytic, then \(u_{x x}+u_{y y}=0\) and $v_{x x}+v_{y y}=0 .$

Find the second partial derivatives

In this step, we'll find the second partial derivatives of u and v: \(u_{xx} = \frac{\partial^2 u}{\partial x^2} = \frac{\partial u_x}{\partial x}\) \(u_{yy} = \frac{\partial^2 u}{\partial y^2} = \frac{\partial u_y}{\partial y}\) \(v_{xx} = \frac{\partial^2 v}{\partial x^2} = \frac{\partial v_x}{\partial x}\) \(v_{yy} = \frac{\partial^2 v}{\partial y^2} = \frac{\partial v_y}{\partial y}\)

Use Cauchy-Riemann equations

As \(u_x = v_y\) and \(u_y = -v_x\), we differentiate these equations with respect to x and y, respectively: \(\frac{\partial u_x}{\partial x} = \frac{\partial v_y}{\partial x}\) (1) \(\frac{\partial u_y}{\partial y} = -\frac{\partial v_x}{\partial y}\) (2)

Add both equations

Now we'll add equations (1) and (2): \(\frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} = \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}\) Since the left-hand side of the equation is equal to \(u_{xx} + u_{yy}\), and the right-hand side is equal to \(v_{xx} + v_{yy}\), we get: \(u_{xx} + u_{yy} = 0\) \(v_{xx} + v_{yy} = 0\) Hence, if f = u + iv is analytic, both u and v satisfy Laplace's equation, which implies that \(u_{xx} + u_{yy} = 0\) and \(v_{xx} + v_{yy} = 0\).

Key concepts

Analytic Functions

In complex analysis, the term 'analytic function' refers to a function that is locally expandable into a convergent power series around any point within its domain. Simply put, an analytic function is a smooth and infinitely differentiable function in the complex plane. The definition might seem daunting at first, but consider the function given in the example:

\( f(x, y) = (x^2 - y^2) + i(2xy) \). When we say this function is 'analytic', it means that at every point in the complex plane, the function behaves nicely, allowing us to predict its behavior in the vicinity of any point using a power series expansion.

To prove a function is analytic, we often turn to the Cauchy-Riemann equations which provide a necessary condition for differentiability in the complex plane. If a function passes this test and also satisfies additional conditions like being defined on an open set, then it can be deemed analytic. These features are fantastic because analytic functions inherit a wealth of mathematical properties and stability, essential for applications across sciences and engineering.

Cauchy-Riemann Equations

The Cauchy-Riemann equations are at the heart of determining whether a complex function is differentiable, or, as we usually say in complex analysis, 'analytic'. These equations are a set of partial differential equations which the real and imaginary parts of any complex function must satisfy: \( u_x = v_y \) and \( u_y = -v_x \).

In our textbook example, finding the partial derivatives \( u_x, u_y, v_x, \) and \( v_y \) of the real and imaginary parts of the function, and then checking these derivatives against the Cauchy-Riemann equations, we ensure the function maintains the necessary harmony between its components to be analytic. Essentially, these equations guarantee that the function behaves uniformly in all directions around any given point.

The beauty of Cauchy-Riemann equations lies in their simplicity and power: by merely checking two equations, we can gleam insight into the complex structure and confirm differentiability within the complex plane.

Partial Derivatives

Partial derivatives are a crucial tool when dealing with functions of multiple variables, as is the case in complex variables where a function f can take the form \( f(x, y) = u(x, y) + iv(x, y) \). Here, \( u \) and \( v \) are functions of two real variables, x and y.

The concept of a partial derivative is similar to that of a regular derivative, but instead of looking at how the function changes as all variables change, we only look at how it changes with respect to one variable at a time, holding the others constant.

Why is this important? This process is necessary in the context of complex analysis. As we've seen from the exercise, to verify whether these functions adhere to the Cauchy-Riemann equations, we need to find these single-variable sensitivities—these partial derivatives. The hands-on calculation shows how the interplay of these individual changes responds to the holistic change of the complex function.

Laplace's Equation

Laplace's equation is a widely used second-order partial differential equation in many fields of physics, engineering, and mathematics, notably in potential theory and fluid dynamics. In the complex plane, Laplace's equation takes a prominent role in confirming the analyticity of functions.

The equation is given by \( u_{xx} + u_{yy} = 0 \) for a function u(x, y), and similarly \( v_{xx} + v_{yy} = 0 \) for v(x, y). A function that satisfies this equation is said to be 'harmonic', and in the context of complex analysis, when both the real part (u) and the imaginary part (v) of a function are harmonic, the full complex function is considered to be analytic.

The link between Laplace's equation and the Cauchy-Riemann equations consolidates the structure of what it means for a function to be analytic. If the function satisfies the very relational Cauchy-Riemann equations, it will automatically satisfy the more general Laplace's equation. This harmonious relationship between these equations ensures that the concepts of complex analysis are interconnected, intertwining the local behavior of functions with their global properties.