Question

It can be shown that, if \(u(x, y)\) is a harmonic function which is defined at \(z_{0}=\) \(x_{0}+i y_{0},\) 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 \(z_{0}=0\)

Short answer

To find \( f(z) \), set \( z_0 = 0 \) and follow the formula: \[ f(z) = 2 u \left( \frac{z}{2}, \frac{z}{2i} \right) + \text{const.} \]

Step-by-step solution

- Identify the Harmonic Function

Identify the given harmonic function, which is the real part of the desired analytic function. This harmonic function is usually provided in the problem statement.

- Set the Given Harmonic Function

Let the harmonic function be denoted as \(u(x,y)\). For problems 54 to 63, it would be provided. Assume \(u(x,y)\) is known.

- Apply the Formula

Use the formula \[f(z)=2 u\left(\frac{z+\bar{z}_{0}}{2}, \frac{z-\bar{z}_{0}}{2 i}\right)+\text{const.}\] to determine the analytic function. For simplicity, if \(u(0,0)\) is defined, take \(z_0 = 0\).

- Simplify the Formula

Substitute \(z_0=0\) into the formula: \[f(z)=2 u\left(\frac{z}{2}, \frac{z}{2i}\right) + \text{const.}\] Simplify the arguments for \(u\).

- Evaluate the Harmonic Function

Evaluate the harmonic function \(u\) at the simplified arguments. This gives the explicit expression for \(f(z)\).

- Write the Final Expression

Include the constant term in the final expression of \(f(z)\).

Key concepts

analytic function

In the field of complex variables, an analytic (or holomorphic) function is one that is locally represented by a convergent power series. This means that within some radius around every point in its domain, the function can be written with an infinite series of powers of the variable. For instance, the function \(f(z) = z^2\) is analytic because it can be expressed as a power series. Analytic functions have derivatives of all orders and are infinitely differentiable.
To understand their importance, consider that many physical phenomena can be modeled using such functions, due to their smoothness and well-behaved nature. Examples include fluid dynamics, where the velocity field of an incompressible fluid can be described by an analytic function.

complex variables

Complex variables, essentially numbers of the form \(z = x + iy\) where \(i\) is the imaginary unit, are fundamental in many areas of mathematics and engineering. They extend the real number system and allow for the solving of equations that have no real solutions, such as \(x^2 + 1 = 0\).
One important aspect of working with complex variables is understanding how they can be visualized on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. Operations such as addition, multiplication, and taking roots are defined in ways that extend their real number counterparts. Complex variables are instrumental in the study of harmonic functions and analytic functions.

Mathematical methods

Mathematical methods encompass a wide range of techniques used to solve problems in analysis, algebra, geometry, and more. For problems involving harmonic and analytic functions, methods such as substitution and transformation are crucial. The given formula \[f(z)=2 u\frac{(z+\bar{z}_{0})}{2}, \frac{(z-\bar{z}_{0})}{2i}\right)+ \text{const.}\right)\] shows how one can derive an analytic function from a harmonic function.
Important mathematical methods include:

• Substitution: Simplifies complex equations by introducing new variables.


• Separation of Variables: Used to break down partial differential equations into simpler, solvable forms.


• Integration and Differentiation: Fundamental operations in calculus used to analyze the properties of functions.

These methods help in converting practical problems into more manageable mathematical forms, thus enabling solutions for complex physical systems.