91Ó°ÊÓ

Laplace's equation is a crucial concept in the study of harmonic functions. A real-valued function, such as u(x, y), is considered harmonic if it satisfies this equation. Mathematically, Laplace's equation is expressed as: \[ u_{xx} + u_{yy} = 0 \] Here, \(u_{xx} \) and \(u_{yy} \) represent the second partial derivatives of the function \(u\) with respect to x and y, respectively.
A harmonic function implies there are no sources or sinks within the domain, meaning it represents a form of equilibrium. This property makes harmonic functions highly relevant in fields like physics and engineering, particularly in areas involving potential fields and steady-state heat distribution scenarios.
For example, consider the function in part (b) of the exercise: \(u(x, y) = 4 y(1+3 x) \). To determine if it is harmonic, we compute the second partial derivatives:

• \( u_x = 12 y \)
• \( u_{xx} = 0 \)
• \( u_y = 4 (1 + 3 x) \)
• \( u_{yy} = 0 \)

Adding these derivatives, we get: \( u_{xx} + u_{yy} = 0 + 0 = 0 \). Since this sums to zero, \( u(x, y) \) is indeed harmonic.
Understanding how to verify that a function satisfies Laplace's equation is the first step in identifying harmonic functions.

Partial derivatives measure how a function changes as its input variables change. In the context of multivariable functions, a partial derivative with respect to a variable shows how the function varies when all other variables are kept constant.
For a function \( u(x, y) \), the partial derivatives are denoted as \( u_x \) and \( u_y \), representing the rates of change with respect to x and y, respectively. When verifying if a function is harmonic, we need to compute the second partial derivatives, \( u_{xx} \) and \( u_{yy} \), and check if their sum is zero.
Consider part (a) from the exercise: \(u(x, y) = \sinh x \cos y \). We compute the first and second partial derivatives as follows:

• \( u_x = \cosh x \cos y \)
• \( u_{xx} = \sinh x \cos y \)
• \( u_y = -\sinh x \sin y \)
• \( u_{yy} = \sinh x \cos y \)

Summing these, \( u_{xx} + u_{yy} = \sinh x \cos y + \sinh x \cos y = 2 \sinh x \cos y e 0 \). Hence, the function isn't harmonic.
Understanding how to compute and interpret partial derivatives is essential for verifying harmonicity and further analysis in multi-variable calculus.

In complex analysis, a harmonic conjugate is a function paired with a given harmonic function to construct an analytic function. For a harmonic function \(u(x, y)\), a harmonic conjugate \(v(x, y)\) satisfies the Cauchy-Riemann equations:
\[ u_x = v_y \]
\[ u_y = -v_x \]
By solving these, we can find \(v(x, y)\).
In part (b) of the exercise, we sought the harmonic conjugate for \(u(x, y) = 4 y(1 + 3 x)\). From the partial derivatives, we have:

• \( u_x = 12 y \)
• \( u_y = 4 (1 + 3 x) \)

Using the Cauchy-Riemann equations, we find:

\( v_y = 12 y \Rightarrow v = 6 y^2 + g(x) \)

\( -v_x = 4(1 + 3 x) \Rightarrow v_x = -4 - 12 x \Rightarrow v = -4 x - 6 x^2 + h(y) \)

Combining these results, we resolve matching terms, giving us the harmonic conjugate function:
\( v(x, y) = -4 x - 6 x^2 + 6 y^2 \).
Examining harmonic conjugates helps in constructing complex functions and has applications in solving boundary value problems.