If u(x, y) is harmonic in the unit disk 𝔻={z=x+i y:|z|<1}, define f(z)=2 u((z)/(2), (z)/(2 i))-u

step 1 For this exercise we have f of z equals negative i z plus a constant and then for b we have nega

If u(x, y) is harmonic in the unit disk 𝔻={z=x+i y:|z|<1},...

If $u(x, y)$ is harmonic in the unit disk $\mathbb{D}=\{z=x+i y:|z|<1\}$, define $$ f(z)=2 u\left(\frac{z}{2}, \frac{z}{2 i}\right)-u(0,0) . $$ In other words, formally replace the real variables $x$ and $y$ in the two-variable Taylor series for $u$ by the complex variables $\alpha$ and $\beta$ then set $\alpha=z / 2$ and $\beta=z / 2 i$. Where will the resulting series converge? Prove that $f$ is well defined (i.e., that this substitution makes sense) and analytic in $\mathbb{D}$ and $\operatorname{Re} f=u$. Note that this procedure does not work for the functions $u=\log \left(x^2+y^2\right)$ and $\operatorname{Re} \frac{1}{z}$ which are harmonic in $\mathbb{D} \backslash\{0\}$. However, if $u$ is harmonic in a neighborhood of $\left(x_0, y_0\right)$ then, for $z_0=x_0+i y_0$, $$ f(z)=2 u\left(\left(z-z_0\right) / 2+x_0,\left(z-z_0\right) /(2 i)+y_0\right)-u\left(x_0, y_0\right) $$ satisfies $\operatorname{Re} f=u$. Try this procedure for a few familiar functions such as $e^x \cos y$ and $\operatorname{Re}\left(z^n\right)$

If $u(x, y)$ is harmonic in the unit disk $\mathbb{D}=\{z=x+i y:|z|<1\}$, define
$$
f(z)=2 u\left(\frac{z}{2}, \frac{z}{2 i}\right)-u(0,0) .
$$
In other words, formally replace the real variables $x$ and $y$ in the two-variable Taylor series for $u$ by the complex variables $\alpha$ and $\beta$ then set $\alpha=z / 2$ and $\beta=z / 2 i$. Where will the resulting series converge? Prove that $f$ is well defined (i.e., that this substitution makes sense) and analytic in $\mathbb{D}$ and $\operatorname{Re} f=u$. Note that this procedure does not work for the functions $u=\log \left(x^2+y^2\right)$ and $\operatorname{Re} \frac{1}{z}$ which are harmonic in $\mathbb{D} \backslash\{0\}$. However, if $u$ is harmonic in a neighborhood of $\left(x_0, y_0\right)$ then, for $z_0=x_0+i y_0$,
$$
f(z)=2 u\left(\left(z-z_0\right) / 2+x_0,\left(z-z_0\right) /(2 i)+y_0\right)-u\left(x_0, y_0\right)
$$
satisfies $\operatorname{Re} f=u$. Try this procedure for a few familiar functions such as $e^x \cos y$ and $\operatorname{Re}\left(z^n\right)$

Key Concepts

Harmonic Functions

Harmonic functions are twice continuously differentiable functions that satisfy Laplace’s equation, meaning that the sum of their second partial derivatives with respect to the independent variables is zero. These functions naturally arise in problems of physics such as heat flow and potential theory, and they possess strong regularity properties like the mean value property and the maximum principle.

Analytic Functions

An analytic (or holomorphic) function is one that is complex differentiable in some region of the complex plane. Such functions have power series representations and satisfy the Cauchy–Riemann equations. Importantly, the real and imaginary parts of analytic functions are interrelated—they are harmonic and are known as harmonic conjugates of each other, which is fundamental in linking harmonic and analytic functions.

Power Series and Convergence

Power series are infinite series of the form ? a?(z - z?)? that converge to analytic functions within a certain radius of convergence. The radius is determined by the distance from the expansion point to the nearest singularity. In the context of substitution in harmonic function expansions, one must ensure that the mapping of variables preserves the radius of convergence, thereby giving rise to a well?defined analytic function in the intended domain.

Substitution in Taylor Series

Substituting complex expressions into a Taylor series involves replacing the real variables in a two-by-two power series with specific complex functions. This technique requires careful attention to ensure that the series continues to be convergent and that the resulting function remains analytic. Typically, these substitutions are valid within a transformed domain where the original radius of convergence is respected.

Representation of Harmonic Functions as Real Parts of Analytic Functions

A fundamental result in complex analysis is that given a harmonic function defined on a simply connected domain, there exists an analytic function whose real part is the original harmonic function. This is accomplished by finding a harmonic conjugate and forming a complex function which inherently embodies the structure of both the harmonic and its conjugate parts, thereby explaining why certain substitutions in harmonic function series yield analytic functions with the prescribed real part.

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