Problem 22 - Advanced Engineering Mathematics

Question

(a) \(u=e^{x^{2}-y^{2}} \cos 2 x y, \quad v=e^{x^{2}-y^{2}} \sin 2 x y ; \quad \frac{\partial u}{\partial x}=-2 y e^{x^{2}-y^{2}} \sin 2 x y+2 x e^{x^{2}-y^{2}} \cos 2 x y=\frac{\partial v}{\partial y}\) \(\frac{\partial u}{\partial y}=-2 x e^{x^{2}-y^{2}} \sin 2 x y-2 y e^{x^{2}-y^{2}} \cos 2 x y=-\frac{\partial v}{\partial x}\) Since \(u, v,\) and their first partial derivatives are continuous, and \(u\) and \(v\) satisfy the Cauchy-Riemann equations everywhere, the function \(f\) is differentiable everywhere. Hence \(f\) is entire. (b) \(\frac{\partial^{2} u}{\partial x^{2}}=-4 y^{2} e^{x^{2}-y^{2}} \cos 2 x y-4 x y e^{x^{2}-y^{2}} \sin 2 x y-4 x y e^{x^{2}-y^{2}} \sin 2 x y+\cos 2 x y\left[4 x^{2} e^{x^{2}-y^{2}}+2 e^{x^{2}-y^{2}}\right]\) \(\frac{\partial^{2} u}{\partial y^{2}}=-4 x^{2} e^{x^{2}-y^{2}} \cos 2 x y+4 x y e^{x^{2}-y^{2}} \sin 2 x y+4 x y e^{x^{2}-y^{2}} \sin 2 x y+\cos 2 x y\left[4 y^{2} e^{x^{2}-y^{2}}-2 e^{x^{2}-y^{2}}\right]\) since \(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\) the function \(u\) is harmonic throughout the \(z\) -plane.

Step-by-Step Solution

Answer

(a) Function \(f\) is entire. (b) Function \(u\) is harmonic.

Step 1: Verify Cauchy-Riemann Equations for Part (a)

Let's look at the given Cauchy-Riemann equations: \[ \frac{\partial u}{\partial x} = -2y e^{x^{2}-y^{2}} \sin 2xy + 2x e^{x^{2}-y^{2}} \cos 2xy = \frac{\partial v}{\partial y} \]and\[ \frac{\partial u}{\partial y} = -2x e^{x^{2}-y^{2}} \sin 2xy - 2y e^{x^{2}-y^{2}} \cos 2xy = -\frac{\partial v}{\partial x} \]These equations must be satisfied for \(u\) and \(v\) to be the real and imaginary parts of an entire function \(f\). Since both pairs of equations are explicitly stated to hold, \(f = u + iv\) is entire.

Step 2: Differentiate u and Verify Laplace's Equation for Part (b)

Given:\[ \frac{\partial^{2} u}{\partial x^{2}} = -4 y^{2} e^{x^{2}-y^{2}} \cos 2xy - 4xy e^{x^{2}-y^{2}} \sin 2xy - 4xy e^{x^{2}-y^{2}} \sin 2xy + \cos 2xy[4x^{2} e^{x^{2}-y^{2}} + 2e^{x^{2}-y^{2}}] \]and\[ \frac{\partial^{2} u}{\partial y^{2}} = -4x^{2} e^{x^{2}-y^{2}} \cos 2xy + 4xy e^{x^{2}-y^{2}} \sin 2xy + 4xy e^{x^{2}-y^{2}} \sin 2xy + \cos 2xy[4y^{2} e^{x^{2}-y^{2}} - 2e^{x^{2}-y^{2}}] \]Adding these gives:\[ \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0 \]This equation verifies that \(u\) is harmonic in the complex plane, which is another requirement for \(f\) to be differentiable everywhere.

Key Concepts

Cauchy-Riemann Equations

The Cauchy-Riemann equations play a crucial role in determining whether a complex function is differentiable, which in turn helps establish if it is entire. In complex analysis, a function of a complex variable, say \( f(z) = u(x, y) + iv(x, y) \), must satisfy specific partial differential equations to be considered differentiable everywhere in the complex plane. These are the Cauchy-Riemann equations.

They are given as follows:
\[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]
Here, \( u \) is the real part and \( v \) is the imaginary part of the complex function \( f(z) \).

By checking the exercise's functions \( u \) and \( v \) against these equations, we determine they satisfy the Cauchy-Riemann equations everywhere. Therefore, the function \( f(z) = u(x, y) + iv(x, y) \) is entire, meaning it is differentiable across the whole complex plane.

Harmonic Functions

Harmonic functions are a significant concept in mathematics, particularly when analyzing the behavior of complex functions. A function is called harmonic if it satisfies Laplace's equation, which means it has a second partial derivative (the Laplacian) summing to zero. In mathematical terms, a function \( u(x, y) \) is harmonic if:

\[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]

This condition ensures that \( u \) is smooth and continuous. In the context of complex analysis, verifying harmonicity is essential for determining the analyticity and differentiability of complex functions.

For example, in the given exercise, it is shown that \( u(x, y) \) satisfies Laplace's equation, thus confirming its harmonic nature. By being harmonic, the real component \( u(x, y) \) fulfills another necessary condition for the complex function \( f = u + iv \) to be entire across the complex plane.

Partial Differentiation

Partial differentiation is a foundational tool in mathematics, especially in understanding how functions change with respect to multiple variables. When dealing with functions of two or more variables, like those found in complex analysis, partial differentiation allows us to isolate and analyze the change of a function with respect to one of those variables, treating all others as constants.

In the context of the exercise, the partial derivatives \( \frac{\partial u}{\partial x} \) and \( \frac{\partial u}{\partial y} \) are critical calculations. They help verify whether a function satisfies the Cauchy-Riemann equations. Furthermore, calculating second-order partial derivatives, such as \( \frac{\partial^2 u}{\partial x^2} \) and \( \frac{\partial^2 u}{\partial y^2} \), is used to test if a function is harmonic.

Understanding how to compute and interpret these derivatives is key. They allow analysts to conclude important properties like continuity, differentiability, and smoothness, ensuring the complex function in question behaves predictably across its domain.