It can be shown that, if u(x, y) is a harmonic function...

It can be shown that, if $u(x, y)$ is a harmonic function which is defined at $z_{0}=x_{0}+t y_{0}$ then an analytic function of which $u(x, y)$ is the real part is given by $$ f(z)=2 u\left(\frac{z+\bar{z}_{0}}{2}, \frac{z-\bar{z}_{0}}{2 i}\right)+\text { const }. $$ [See Struble, Quart. Appl. Math., 37 (1979), 79-81.] Use this formula to find $f(z)$ in Problems 54 to $63 .$ Hint $:$ If $u(0,0)$ is defined, take $z_{0}=0$.

It can be shown that, if $u(x, y)$ is a harmonic function which is defined at $z_{0}=x_{0}+t y_{0}$ then an analytic function of which $u(x, y)$ is the real part is given by
$$
f(z)=2 u\left(\frac{z+\bar{z}_{0}}{2}, \frac{z-\bar{z}_{0}}{2 i}\right)+\text { const }.
$$
[See Struble, Quart. Appl. Math., 37 (1979), 79-81.] Use this formula to find $f(z)$ in Problems 54 to $63 .$ Hint $:$ If $u(0,0)$ is defined, take $z_{0}=0$.

Key Concepts

Harmonic Conjugate

For a given harmonic function, a harmonic conjugate is another harmonic function such that together they form the real and imaginary parts of an analytic function. The existence of a harmonic conjugate is guaranteed in simply connected domains and is essential for the reconstruction of an analytic function from its real part. This construction is a key tool in complex analysis, particularly in solving boundary value problems and conformal mapping.

Cauchy-Riemann Equations

The Cauchy-Riemann equations provide necessary and sufficient conditions for a function to be analytic. They relate the partial derivatives of the real and imaginary parts of a complex function, ensuring that these components satisfy Laplace’s equation. This set of equations is fundamental because they establish the link between harmonic functions and analytic functions, allowing one to construct an analytic function from a given harmonic function by finding its harmonic conjugate.

Analytic Function

An analytic (or holomorphic) function is a complex function that is differentiable at every point in its domain. This differentiability implies that the function satisfies the Cauchy-Riemann equations and can be locally expressed as a convergent power series. Analytic functions have many remarkable properties such as infinite differentiability and the ability to be represented by their Taylor series, making them central objects of study in complex analysis.

Harmonic Function

A harmonic function is a twice continuously differentiable function that satisfies Laplace’s equation. These functions appear naturally in many areas such as physics and engineering, especially in contexts involving steady-state heat distributions and potential flows. In complex analysis, the real (or imaginary) part of any analytic function is harmonic, which provides a deep connection between potential theory and analytic function theory.

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