Chapter 4
Three-Dimensional Harmonic Function
4.1 Simplest Examples of Harmonic Functions
In the simple case, a harmonic function depends only on one variable x. Wquation
(3.7) goes to a regular homogeneous differential equation of the second order
d 2 f
dx 2 = 0.
Hence it follows that
f = ax + b, (a, b − const),
e. g. the harmonic function depending on one coordinate in the Cartesian system is
linear.
In the cylindrical system of coordinates (ρ, ϕ, z), we have
ρ =
x 2 + y 2 , ϕ = arctg
y
x
,
and Laplace’s equation (3.7) looks as follows:
∂
∂ρ
ρ
∂f
∂ρ
+
1
ρ
∂ 2 f
∂ϕ 2 + ρ
∂ 2 f
∂z 2 = 0.
(4.1)
By analyzing the last formula, we may notice that the simplest harmonic function
depending only on ϕ or only on z will be the linear function. If the harmonic function
(f 1 ) in cylindrical coordinates depends only on the variable ρ, we will obtain from
Eq. (4.1)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_4
37