Problem 26 - Advanced Engineering Mathematics

Question

Verify that the given function \(u\) is harmonic. Find \(v\), the harmonic conjugate function of \(u\). Form the corresponding analytic function \(f(z)=u+i v .\) \(u(x, y)=4 x y^{3}-4 x^{3} y+x\)

Step-by-Step Solution

Answer

The function \( u(x, y) \) is harmonic and the analytic function is \( f(z) = (4xy^3 - 4x^3y + x) + i(-6xy^2 + x^4 + y^4 + y) \).

Step 1: Verify that u is Harmonic

To verify that the function \( u(x, y) = 4xy^3 - 4x^3y + x \) is harmonic, we must show that it satisfies the Laplace equation, which states that \( abla^2 u = 0 \), where \( abla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \). First, calculate \( \frac{\partial u}{\partial x} = 4y^3 - 12x^2y + 1 \).Next, calculate \( \frac{\partial^2 u}{\partial x^2} = -24xy \).Now, calculate \( \frac{\partial u}{\partial y} = 12xy^2 - 4x^3 \).Then, calculate \( \frac{\partial^2 u}{\partial y^2} = 24xy \).Add the second partial derivatives: \( abla^2 u = -24xy + 24xy = 0 \). Thus, \( u(x, y) \) is harmonic.

Step 2: Find v, Harmonic Conjugate of u

To find the harmonic conjugate \( v(x, y) \), we use the Cauchy-Riemann equations:\( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \) and \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \).From \( \frac{\partial u}{\partial x} = 4y^3 - 12x^2y + 1 \), we set \( \frac{\partial v}{\partial y} = 4y^3 - 12x^2y + 1 \).Integrating with respect to \( y \), we get:\( v(x, y) = \int (4y^3 - 12x^2y + 1) \, dy = y^4 - 6x^2y^2 + y + g(x) \), where \( g(x) \) is an arbitrary function of \( x \).Now, from \( \frac{\partial u}{\partial y} = 12xy^2 - 4x^3 \), we set \( -\frac{\partial v}{\partial x} = 12xy^2 - 4x^3 \).Thus, \( \frac{\partial v}{\partial x} = -12xy^2 + 4x^3 \).Integrating with respect to \( x \), we find:\( v(x,y) = -6xy^2 + x^4 + h(y) \), where \( h(y) \) is an arbitrary function of \( y \).Comparing terms with earlier results, we find:\( v(x, y) = -6xy^2 + x^4 + y^4 + y \).

Step 3: Form the Analytic Function f(z)

The analytic function \( f(z) = u(x, y) + iv(x, y) \) is formed by combining the harmonic function \( u \) and its harmonic conjugate \( v \):\( f(z) = (4xy^3 - 4x^3y + x) + i(-6xy^2 + x^4 + y^4 + y) \).This is the required analytic function.

Key Concepts

Laplace Equation

The Laplace equation finds its essential role in verifying whether a function is harmonic. A function is called harmonic if it satisfies the Laplace equation, written in mathematical terms as:

• \( abla^2 u = 0 \)
• Where \( abla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \)

Understanding this means comprehending how the second partial derivatives of a function add up to zero, indicating zero net curvature. Typically, harmonic functions depict scenarios that are usually smooth and exhibit stability, such as potential flow in physics.

For the function \( u(x, y) = 4xy^3 - 4x^3y + x \), calculating the second derivatives with respect to both \( x \) and \( y \) and showing that they sum to zero signifies that the function is harmonic. Harmonic functions like this one are crucial in mathematics, particularly because they serve as foundations for complex analyses in various scientific fields.

Cauchy-Riemann Equations

Cauchy-Riemann equations are the cornerstone for complex analysis when working with functions of complex variables. They provide the criteria that determine if two real functions combine to form an analytic function. The equations are:

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

Integrating these equations helps us find the harmonic conjugate \( v(x, y) \) of \( u(x, y) \), allowing us to construct a complex function \( f(z) \) such that \( f(z) = u(x, y) + iv(x, y) \).

The use of these equations helps to build the crucial understanding that, by finding \( v \), the imaginary counterpart to \( u \), each function can be represented in a complex plane. This enables deeper explorations into complex numbers and their applications in fields like fluid dynamics and electromagnetism.

Analytic Function

An analytic function is a vital concept in complex analysis. It refers to a function of a complex variable that is differentiable at every point in its domain, a trait implying the utmost smoothness and continuous differentiability across the domain.

When forming an analytic function from given real functions \( u(x, y) \) and \( v(x, y) \), it's critical that they both satisfy the conditions of harmonicity and the Cauchy-Riemann equations. For example, if \( u \) and \( v \) are derived correctly as in this problem, the function \( f(z) = u(x, y) + iv(x, y) \) can indeed be considered analytic.

Analytic functions are significant because they preserve the properties of smoothness and are crucial in representing complex phenomena in both theoretical and applied mathematics. They facilitate solving integral and differential equations and play a part in predicting fluid and electromagnetic behavior.