Problem 37 - Advanced Engineering Mathematics

Question

Show that the function \(f(z)=-(2 x y+5 x)+i\left(x^{2}-5 y-y^{2}\right)\) is analytic for all \(z\). Find \(f^{\prime}(z)\).

Step-by-Step Solution

Answer

The function is analytic for all \(z\) and \(f'(z) = (-2y - 5) + i(2x)\).

Step 1: Identify Real and Imaginary Parts

The function given is \(f(z)=u(x,y) + iv(x,y)\) where \(u(x,y) = -(2xy + 5x)\) and \(v(x,y) = x^{2} - 5y - y^{2}\). These represent the real and imaginary parts of \(f\).

Step 2: Verify the Cauchy-Riemann Equations

The function \(f(z)\) is analytic if the Cauchy-Riemann equations are satisfied. The equations are \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\). Calculate these partial derivatives: \(\frac{\partial u}{\partial x} = -2y - 5\), \(\frac{\partial u}{\partial y} = -2x\), \(\frac{\partial v}{\partial x} = 2x\), \(\frac{\partial v}{\partial y} = -5 - 2y\). Compare them to check if the equations hold.

Step 3: Check the First Cauchy-Riemann Equation

Compare \(\frac{\partial u}{\partial x} = -2y - 5\) with \(\frac{\partial v}{\partial y} = -5 - 2y\). They are equal, so the first Cauchy-Riemann equation is satisfied.

Step 4: Check the Second Cauchy-Riemann Equation

Compare \(\frac{\partial u}{\partial y} = -2x\) with \(-\frac{\partial v}{\partial x} = -2x\). They are equal, so the second Cauchy-Riemann equation is satisfied.

Step 5: Conclude Analyticity

Since both Cauchy-Riemann equations are satisfied, \(f(z)\) is analytic for all \(z\).

Step 6: Compute the Derivative

The derivative \(f'(z)\) is given by \(\frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}\). From Step 2, we calculated \(\frac{\partial u}{\partial x} = -2y - 5\) and \(\frac{\partial v}{\partial x} = 2x\). Therefore, \(f'(z) = (-2y - 5) + i(2x)\).

Key Concepts

Cauchy-Riemann Equations

In complex analysis, the Cauchy-Riemann equations are vital to determine whether a complex function is analytic. A complex function \(f(z)\), where \(z = x + iy\), is expressed as \(u(x, y) + iv(x, y)\). These equations provide conditions that help confirm if the function is differentiable in a complex sense.

The Cauchy-Riemann equations state that for \(f(z)\) to be analytic, the following must hold:

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

In our exercise, these equations are satisfied as
\[\begin{align*}\frac{\partial u}{\partial x} &= -2y - 5, \\frac{\partial v}{\partial y} &= -5 - 2y \\end{align*}\]The equations are equal, thus confirming part of the function's analyticity. When \(\frac{\partial u}{\partial y} = -2x\) is compared with \(-\frac{\partial v}{\partial x} = -2x\), the continuity of satisfying the Cauchy-Riemann equations is upheld. This step is crucial; if the equations do not hold, the function is not analytic.

Analytic Functions

An analytic function is a complex function that is differentiable at every point in its domain. Analyticity is a strong condition, as differentiation implies continuity. In the context of our example, once we have confirmed the Cauchy-Riemann equations, we affirm that \(f(z)\) is analytic for all complex numbers \(z\).

If a function is analytic, it can be infinitely differentiated, and it satisfies the conditions of complex differentiability. This means there is one unique derivative at each point in the domain of \(f(z)\). The achieved derivative can be found through the formula

• \(f'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}\).

In the exercise, we found \(f'(z) = (-2y - 5) + i(2x)\). Having a derivative at every point emphasizes that the function describes a well-behaved, smooth mapping in the complex plane.

Partial Derivatives

Partial derivatives, denoted as \(\frac{\partial}{\partial x}\) and \(\frac{\partial}{\partial y}\), are derivatives taken with respect to one variable while holding other variables constant. They are essential in bridging real-valued functions and complex functions through the Cauchy-Riemann equations.

When dealing with a complex function's analytic nature,
partial derivatives describe how the function behaves along the real \((x)\) and imaginary \((y)\) axes:

• \(\frac{\partial u}{\partial x}\) indicates the variation of the real part \(u\) in the \(x\) direction.
• \(\frac{\partial u}{\partial y}\) represents the change of \(u\) in the \(y\) direction.
• \(\frac{\partial v}{\partial x}\) and \(\frac{\partial v}{\partial y}\) similarly affect the imaginary part \(v\).

These directions are crucial when assessing the Cauchy-Riemann equations, ensuring \(f(z)\) smoothly transitions through every point \(z\) in its domain. Understanding partial derivatives is fundamental to solving complex analysis exercises and demonstrating a function's differentiability.