Problem 27 - Advanced Engineering Mathematics

Question

In Problems 23-28, verify that the given function \(u\) is harmonic. Find \(v\), the harmonic conjugate function of \(u\). Form the corresponding analytic function \(f(z)=u+i v\). $$ u(x, y)=\log _{e}\left(x^{2}+y^{2}\right) $$

Step-by-Step Solution

Answer

The function is harmonic, and the analytic function is \( f(z) = \log(x^2+y^2) + i \arctan\left(\frac{y}{x}\right) \).

Step 1: Verify Harmonicity

To verify if a function is harmonic, we check if it satisfies Laplace's equation: \( abla^2 u = u_{xx} + u_{yy} = 0 \). Start by computing the partial derivatives. First, find the first partial derivatives:\( u_x = \frac{2x}{x^2+y^2} \) and \( u_y = \frac{2y}{x^2+y^2} \).Now the second partial derivatives: \( u_{xx} = \frac{2(y^2-x^2)}{(x^2+y^2)^2} \)\( u_{yy} = \frac{2(x^2-y^2)}{(x^2+y^2)^2} \).Add them together: \( u_{xx} + u_{yy} = \frac{2(y^2-x^2) + 2(x^2-y^2)}{(x^2+y^2)^2} = 0 \). Thus, \( u \) is harmonic.

Step 2: Find Harmonic Conjugate

The harmonic conjugate \( v \) satisfies the Cauchy-Riemann equations:\( u_x = v_y \) and \( u_y = -v_x \).From the Cauchy-Riemann equations:\( v_y = \frac{2x}{x^2+y^2} \) and \(-v_x = \frac{2y}{x^2+y^2} \).Integrate \( v_y \) with respect to \( y \):\( v = \int \frac{2x}{x^2+y^2} \: dy = \arctan\left(\frac{y}{x}\right) + h(x) \).Check consistency with \( -v_x \) by differentiating \( v \) with respect to \( x \) ensuring consistency with \( u_y = -v_x \). Find \( h'(x) \) and set it such that it satisfies the differential relationship with \( u_y \). This leads to \( v = \arctan\left(\frac{y}{x}\right) + C \).

Step 3: Form the Analytic Function

Now combine the original function \( u \) and its harmonic conjugate \( v \) into the analytic function \( f(z) = u(x, y) + i v(x, y) \).Thus:\[ f(z) = \log(x^2+y^2) + i \arctan\left(\frac{y}{x}\right) \].

Key Concepts

Laplace's Equation

Laplace's Equation is crucial in determining whether a function is harmonic. For a function \( u(x, y) \) to be harmonic, it must satisfy Laplace's Equation, which is written as \( abla^2 u = u_{xx} + u_{yy} = 0 \). This means that the sum of its second partial derivatives with respect to \( x \) and \( y \) must equal zero.

In our exercise, the function \( u(x, y) = \log_e(x^2 + y^2) \) is verified for harmonicity by calculating these second partial derivatives. We start by finding \( u_x \) and \( u_y \), the first derivatives, which are \( \frac{2x}{x^2 + y^2} \) and \( \frac{2y}{x^2 + y^2} \) respectively.

• The second derivative \( u_{xx} = \frac{2(y^2 - x^2)}{(x^2 + y^2)^2} \)
• And \( u_{yy} = \frac{2(x^2 - y^2)}{(x^2 + y^2)^2} \)

Adding these, we see that \( u_{xx} + u_{yy} = 0 \), confirming \( u \) is harmonic.
Understanding this equation helps in fields such as fluid dynamics and electrostatics, where potential functions are often harmonic, satisfying Laplace's Equation.

Cauchy-Riemann Equations

The Cauchy-Riemann Equations are vital when verifying if two real-valued functions \( u \) and \( v \) can compose an analytic function of a complex variable \( f(z) \). They provide a bridge between two functions to ensure that they align properly to form a complex function.

For functions \( u(x, y) \) and \( v(x, y) \) to satisfy these equations, the following conditions must hold:

• \( u_x = v_y \)
• \( u_y = -v_x \)

In the exercise, we compute \( v_y = \frac{2x}{x^2+y^2} \) from \( u_x \) and \( -v_x = \frac{2y}{x^2+y^2} \) from \( u_y \). By integrating \( v_y \) concerning \( y \), we find \( v = \arctan\left(\frac{y}{x}\right) + h(x) \). Consistency with \( -v_x \) results in \( v = \arctan\left(\frac{y}{x}\right) + C \), where \( C \) is a constant.
Cauchy-Riemann Equations are essential in complex analysis, ensuring functions are differentiable in the complex plane and preserving analyticity.

Analytic Functions

Analytic Functions are powerful in complex analysis, defined as functions that are locally representable by a convergent power series. An important property of analytic functions is that they are infinitely differentiable and thus smooth.

In the task, using the harmonic function \( u(x, y) = \log_e(x^2 + y^2) \) and its harmonic conjugate \( v(x, y) = \arctan\left(\frac{y}{x}\right) \) that satisfy the Cauchy-Riemann equations, we can form an analytic function \( f(z) = u(x, y) + i v(x, y) \).

This formation gives us

• \( f(z) = \log(x^2+y^2) + i \arctan\left(\frac{y}{x}\right) \)

Analytic functions are essential tools in areas of complex analysis, often used to model phenomena in physics and engineering due to their well-behaved properties across their domain.