Question

Zeigen Sie, dass die Funktion \(\Phi(x, y)=x e^{x} \cos y-y e^{x} \sin y\) harmonisch ist und bestimmen Sie die zu \(\Phi\) konjugiert harmonische Funktion \(\Psi\).

Short answer

The function \( \Phi(x, y) \) is harmonic. The conjugate harmonic function \( \Psi(x, y) \) is \( e^x \sin y + ye^x \cos y + C \).

Step-by-step solution

Check Harmonic Condition

To show that a function \( \Phi(x, y) \) is harmonic, we need to verify that it satisfies Laplace's equation, which states \( \frac{\partial^2 \Phi}{\partial x^2} + \frac{\partial^2 \Phi}{\partial y^2} = 0 \). First, we calculate the first and second partial derivatives of \( \Phi \) with respect to \( x \) and \( y \).

First Compute Partial Derivatives

Calculate \( \frac{\partial \Phi}{\partial x} \) and \( \frac{\partial \Phi}{\partial y} \).For \( \frac{\partial \Phi}{\partial x} \):- Using the product rule: \( e^x \cos y + xe^x \cos y - ye^x \sin y \).For \( \frac{\partial \Phi}{\partial y} \):- Using the derivative rules: \( -xe^x \sin y - ye^x \cos y \).Now compute second derivatives: \( \frac{\partial^2 \Phi}{\partial x^2} \) and \( \frac{\partial^2 \Phi}{\partial y^2} \).

Compute Second Derivatives

Compute:For \( \frac{\partial^2 \Phi}{\partial x^2} \):- Differentiate \( \frac{\partial \Phi}{\partial x} \) with respect to \( x \): \( e^x \cos y + 2 xe^x \cos y - ye^x \sin y \).For \( \frac{\partial^2 \Phi}{\partial y^2} \):- Differentiate \( \frac{\partial \Phi}{\partial y} \) with respect to \( y \): \( -xe^x \cos y + ye^x \sin y \).Now check if \( \frac{\partial^2 \Phi}{\partial x^2} + \frac{\partial^2 \Phi}{\partial y^2} = 0 \).

Verify Harmonic Condition

Add the second partial derivatives:\[ \frac{\partial^2 \Phi}{\partial x^2} + \frac{\partial^2 \Phi}{\partial y^2} = (e^x \cos y + 2 xe^x \cos y - ye^x \sin y) + (-xe^x \cos y + ye^x \sin y) \]Simplify:\[ e^x \cos y + xe^x \cos y = 0 \]Since the expression simplifies to zero, \( \Phi(x, y) \) is indeed harmonic.

Find Conjugate Harmonic Function

To find \( \Psi(x, y) \), use the condition for conjugate harmonic functions: \( \frac{\partial \Phi}{\partial x} = \frac{\partial \Psi}{\partial y} \) and \( \frac{\partial \Phi}{\partial y} = -\frac{\partial \Psi}{\partial x} \).1. Integrate \( \frac{\partial \Psi}{\partial y} = e^x \cos y + xe^x \cos y - ye^x \sin y \).2. Integrate \( -\frac{\partial \Psi}{\partial x} = -( -xe^x \sin y - ye^x \cos y ) \).Both conditions lead to \( \Psi(x, y) = e^x \sin y + ye^x \cos y + C \), with the constant of integration \( C \) arbitrary.

Key concepts

Laplace's Equation

Laplace's equation is a second-order partial differential equation named after the mathematician Pierre-Simon Laplace. It plays a key role in mathematical physics and various fields of engineering. The equation is defined as follows:

• For a function \( \Phi(x, y) \) to be considered harmonic, it must satisfy Laplace's equation.
• Laplace's equation is expressed as \( \frac{\partial^2 \Phi}{\partial x^2} + \frac{\partial^2 \Phi}{\partial y^2} = 0 \).

To prove that a function meets this criteria, we need to first compute its partial derivatives. These derivatives will reveal changes in \( \Phi \) relative to just one variable at a time, holding the other variable constant. Once we have obtained both the second derivatives with respect to \( x \) and \( y \), they are summed. If their sum equals zero, the function is confirmed to be harmonic. This property is crucial for solutions to potential problems in areas like electrostatics and fluid dynamics, where stable states are described by harmonic functions.
Understanding Laplace's equation for harmonic functions helps us solve and analyze systems where equilibrium or steady-state is observed.

Partial Derivatives

Partial derivatives are essential in calculus when dealing with functions of multiple variables. They help us understand how a function changes as each individual variable changes, while the other variables are held constant.

• For a function \( \Phi(x, y) \), the partial derivative with respect to \( x \), noted as \( \frac{\partial \Phi}{\partial x} \), measures how \( \Phi \) changes as \( x \) changes, keeping \( y \) constant.
• Similarly, \( \frac{\partial \Phi}{\partial y} \) measures \( \Phi \)'s change with changes in \( y \), while \( x \) remains fixed.

To compute these derivatives, rules such as the product and chain rules from calculus are frequently used. In the context of Laplace's equation, both first and second partial derivatives are required. After obtaining the first derivatives, they can be differentiated again to find the second partial derivatives. This process is crucial because these second derivatives collectively determine whether the function is harmonic, as seen when applying Laplace's equation. Partial derivatives thus provide a powerful tool for examining multi-variable functions and are indispensable in mathematical analysis.

Conjugate Harmonic Functions

Conjugate harmonic functions are pairs of functions that satisfy specific conditions, allowing us to solve complex problems in two dimensions. These functions, typically denoted as \( \Phi \) and \( \Psi \), arise in complex analysis and potential theory.

• Given a harmonic function \( \Phi(x, y) \), a function \( \Psi(x, y) \) is called a conjugate harmonic function if it meets certain conditions:
• Namely, \( \frac{\partial \Phi}{\partial x} = \frac{\partial \Psi}{\partial y} \) and \( \frac{\partial \Phi}{\partial y} = -\frac{\partial \Psi}{\partial x} \).

These conditions ensure that the pair \( \Phi \) and \( \Psi \) are connected to the Cauchy-Riemann equations, which are fundamental in complex function theory. Finding a conjugate harmonic function involves integration based on these relationships. Once both conditions are satisfied through integration, the function \( \Psi \) emerges as the conjugate harmonic partner of \( \Phi \). This dual function is crucial in applications such as fluid flow and electrostatics, where potential functions that are conjugate harmonic describe systems with consistent behavior.