Problem 7 - Advanced Engineering Mathematics

Question

In Problems 3-8, show that the given function is not analytic at any point. $$ f(z)=x^{2}+y^{2} $$

Step-by-Step Solution

Answer

The function is not analytic at any point except possibly the origin.

Step 1: Recall Cauchy-Riemann Equations

In complex analysis, to determine whether a complex function \( f(z) = u(x, y) + iv(x, y) \) is analytic at a point, we use the Cauchy-Riemann equations. These equations are:\[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \] \[ \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]The function must satisfy both these conditions at a point for it to be analytic there.

Step 2: Identify Real and Imaginary Parts

For the given function \( f(z) = x^2 + y^2 \), where \( z = x + iy \), the real part \( u(x, y) = x^2 + y^2 \) and there is no imaginary part, so \( v(x, y) = 0 \).

Step 3: Calculate Partial Derivatives of \( u \)

Compute the partial derivatives of \( u(x, y) = x^2 + y^2 \):- \( \frac{\partial u}{\partial x} = 2x \)- \( \frac{\partial u}{\partial y} = 2y \)

Step 4: Calculate Partial Derivatives of \( v \)

Compute the partial derivatives of the imaginary part \( v(x, y) = 0 \):- \( \frac{\partial v}{\partial x} = 0 \)- \( \frac{\partial v}{\partial y} = 0 \)

Step 5: Apply Cauchy-Riemann Equations

Substitute the derivatives into the Cauchy-Riemann equations:- For \( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \): \( 2x = 0 \)- For \( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \): \( 2y = 0 \)These equations imply that both \( x \) and \( y \) must be zero, which is only true at the origin \( (0, 0) \).

Step 6: Conclusion on Analyticity

Since the Cauchy-Riemann equations are not satisfied at any point except the origin, the function \( f(z) = x^2 + y^2 \) is not analytic at any point in the complex plane.

Key Concepts

Cauchy-Riemann equations

The Cauchy-Riemann equations are fundamental in determining whether a complex function is analytic at a point. Analytic functions, which we'll discuss later, are central concepts in complex analysis, defined by their differentiability over a region. To verify if a function is analytic, we use these powerful equations. Given a complex function \( f(z) = u(x, y) + iv(x, y) \), the Cauchy-Riemann equations state:

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

Checking these requires calculating partial derivatives of the real part \( u \) and the imaginary part \( v \). If a function satisfies both equations everywhere in a region, it is termed analytic there. For instance, given \( f(z) = x^2 + y^2 \), the real part \( u(x, y) = x^2 + y^2 \) has derivatives \( \frac{\partial u}{\partial x} = 2x \) and \( \frac{\partial u}{\partial y} = 2y \), while for the imaginary part \( v = 0 \), both derivatives are zero. Substituting into the Cauchy-Riemann equations, you obtain equations only true when \( x = y = 0 \), showing the function is not analytic beyond the origin.

analytic functions

Analytic functions in complex analysis are like the perfect all-rounders of the function world. They are differentiable, not just at isolated points but throughout a region in the complex plane. This differentiability needs to be complex differentiability, stronger than just real differentiability, and is ensured when the Cauchy-Riemann equations we discussed earlier are satisfied.
To conceptualize, an analytic function can be expanded into a power series, meaning local behavior is entirely captured and described by its derivatives at a point. Such functions exhibit a smooth, continuous nature over their domain.

• They behave predictably; no surprises across their path.
• Solutions to various challenges in physics or engineering often rely on these stable properties.

Returning to the exercise, \( f(z) = x^2 + y^2 \) is non-analytic outside the origin because it doesn't meet the rigorous demands of the Cauchy-Riemann conditions except at that single point.

partial derivatives

In simple terms, partial derivatives help us understand how a function changes as we tweak one variable while keeping others fixed. They are like taking a car for a spin to see how acceleration responds to steering while speed remains steady. For the function \( f(z) = u(x, y) + iv(x, y) \), the process involves differentiating \( u \) and \( v \) with respect to \( x \) and \( y \), providing insights into its local behavior.

• The notation \( \frac{\partial u}{\partial x} \) refers to the rate of change of \( u \) with respect to \( x \), assuming \( y \) is constant.
• Similarly, \( \frac{\partial v}{\partial y} \) indicates how \( v \) varies as \( y \) changes, keeping \( x \) fixed.

In evaluating whether \( f(z) \) is analytic, you compute these derivatives. As we saw, \( f(z) = x^2 + y^2 \) has partial derivatives \( 2x \) and \( 2y \) for \( u \), while \( v = 0 \) leads to derivatives \( 0 \). When matched against the Cauchy-Riemann equations, these derivatives rule out the function's analyticity almost entirely, illustrating the importance of finely detailed calculations.