Show that the following functions are harmonic, that is,...

Show that the following functions are harmonic, that is, that they satisfy Laplace's equation, and find for each a function $f(z)$ of which the given function is the real part. Show that the function $v(x, y)$ which you find also satisfies Laplace's equation.
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Key Concepts

Cauchy-Riemann Equations

The Cauchy-Riemann equations are a set of conditions u_x = v_y and u_y = -v_x that must be satisfied by the real part u(x, y) and the imaginary part v(x, y) of an analytic function. These equations ensure that the function behaves well in the complex sense and that both u and v are harmonic.

Analytic (Holomorphic) Function

An analytic, or holomorphic, function is a complex function that is differentiable at every point in its domain. Such functions can be expressed as f(z) = u(x, y) + iv(x, y), where both u and v are harmonic functions that satisfy additional conditions given by the Cauchy-Riemann equations.

Harmonic Conjugate

For a given harmonic function u(x, y), a harmonic conjugate v(x, y) is another harmonic function such that the combination f(z) = u(x, y) + iv(x, y) becomes an analytic function. The existence of a harmonic conjugate is guaranteed under certain conditions in simply connected domains, and the pair (u, v) must satisfy the Cauchy-Riemann equations.

Harmonic Function

A harmonic function is one that satisfies Laplace's equation, meaning that the sum of its second partial derivatives with respect to x and y is zero. In two-dimensional domains, this is expressed as u_xx + u_yy = 0. Such functions are important in many areas of physics and mathematics because they describe steady-state solutions to heat and potential problems.

Laplace's Equation

Laplace's equation is the partial differential equation u_xx + u_yy = 0. It is central to potential theory and appears in various physical contexts such as electrostatics, fluid dynamics, and gravitational fields, describing the behavior of scalar potentials in a region where there are no sources.

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