es
In this section, we focus on the real parts of holomorphic/analytic functions and their connection with real harmonic functions. Be aware that there is no universal notation for imaginary unit vector, mathematicians use i or ⅈ, while engineerers and physists prefer to use j or ⅉ. We follow the latter.

Harmonic functions

A real-valued function u defined on an open subset Ω of the complex plane is called harmonic if it has continuous partial derivatives of first and second order in Ω and satisfies Laplace’s equation written in a Cartesian coordinate system
\begin{equation} \label{EqHarmonic.1} \Delta u \equiv \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \end{equation}
Here Δ = ∇ • ∇ = ∂xx + ∂yy (also informally written as ∇²) is the Laplace operator or Laplacian, where ∇ is the gradient operator (also symbolized "grad").
The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who in 1786 applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. However, many years before Laplace, L. Euler used this equation in his study of hydrodynamics.
Theorem 1: If f = u + ⅉ v is a holomorphic function in Ω, then u and v are harmonic functions in Ω.
Both functions, u and v have continuous partial derivatives of all orders since f is analytic. By the Cauchy-Riemann equations \[ u_x = v_y \qquad\mbox{and} \qquad u_y = - v_x . \] Upon their differentiation, we get \[ u_{xx} = v_{yx} = v_{xy} = - u_{yy} \] Hence, u is harmonic. By the same argument, v is harmonic since it is the real part of the holomorphic function −ⅉ f.
   
Example 1: The converse of Theorem 1 is not true. For example, \[ u(x,y) = \ln \left( x^2 + y^2 \right) \] is harmonic in punctured plane, but it is not the real part of a holomorphic function    ■
End of Example 1
A simply connected domain in the complex plane (ℂ) is a connected, open set with no "holes," meaning any simple closed curve within the domain can be continuously shrunk to a point without leaving the domain. It is a domain whose complement in the Riemann sphere (ℂ &cup> {∞}) is connected.
For instance, an annulus \( \displaystyle \quad a < |z| < b \quad \) is not simply connected. But the unit disk minus the real axis is simply connected.
Theorem 2: Let u be harmonic in domain Ω. Then
  1. partial derivative ux is a real part of a holomorphic function in Ω;
  2. if Ω is simply conencted, u is a real part of a holomorphic function.
  1. Let f = ux − ⅉuy. Since u ∈ ℭ², f has continuous first-order partial derivatives. Moreover, by the harmonicity of u, \[ f_y = u_{xy} - {\bf j}\,u_{yy} = u_{xy} + {\bf j}\,u_{xx} = {\bf j}\,f_x . \] So f satisfies the Cauchy--Riemann equations. Hence, f is holomorphic in Ω.
  2. We need the follwoing theorem:

    Theorem: If f is a holomorphic function in simply connected domain Ω, then there exist a primitive F, holomorphic in Ω, and such that its derivarive is fF′ = f.

    If Ω is simply connected, then by Theorem, f = ux − ⅉuy is the derivative of a holomorphic function F. If F = A + ⅉB, then \[ F' (z) = A_x + {\bf j}\,B_x = A_x - {\bf j}\,A_y = u_x - {\bf j}\,u_y . \] So \[ A(x,y) = u(x,y ) + c . \] Hence, u(x, y) is the real part of the holomorphic function F(z) − c.

   
Example 2: Function u(x, y) = x − exsiny is harmonic in the whole plane ℂ. Hence, f(z) = ux − ⅉuy = 1 − exsiny + ⅉexcosy is entire function. In fact, f(z) = 1 + ⅉez and if we set \[ F(z) = \int_0^z f(\zeta )\,{\text d}\zeta = z + ⅉe^z - ⅉ, \] then \[ u(z) = \Re\,F(z) . \]    ■
End of Example 2
    Since Laplace's equation is a homogeneous equation, any linear combination
\[ \sum_{k=1}^n c_k u_k (x,y) \]
of harmonic functions uk with real coefficients ck is again a harmonic function.
Two harmonic in domain Ω functions u(x, y) and v(x, y) connected by the Cauchy--Riemann equations are called conjugate.
The word "conjugate" originates from the Latin verb coniugāre, which means "to yoke together," "to connect," or "to join together". It entered English usage in the 15th and 16th centuries, initially as an adjective meaning "joined together" (1471) and later as a verb (1530) to describe the grammatical process of inflecting a ver
Theorem 3: Any harmonic function u(x, y) is an anlytic function in an neighborhood of every point z₀ = x₀ + ⅉy₀ ∈ Ω, so it is represented by absolutely convergent power series \[ u(x, y) = \sum_{n,k=0}^{\infty} c_{kn} \left( x - x_0 \right)^k \left( y - y_0 \right)^n . \]
"Neighborhood" originates from the Middle English neighborehode (c. 15th century), combining neighbor (one who lives near) with the suffix -hood (state or condition). It stems from the Old English nēahgebūr, meaning "near dweller," composed of nēah ("nigh/near") and gebūr ("dweller/farmer"), ultimately rooted in Proto-Germanic.
   
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