Complex Analysis - Maths KU

φ = –2
y
D
φ = 3
Figure 4.22 Equipotential curves and
lines of force for Example 1
x
4.5 Applications 227
respectively, and so the function:
φ(x, y) =Φ(u(x, y),v(x, y)) = 5
(x − y +1)+1
2 2
= 5
2
5
x − y + 3 (9)
2
isa solution of the Dirichlet problem in D. You are encouraged to verify by
direct calculation that the function φ given in (9) satisfies Laplace’s equation
and the boundary conditions φ(x, x) = 3 and φ(x, x +2)=−2.
In Section 3.4, we saw that if φ isharmonic in a domain D and if ψ isa
harmonic conjugate of φ in D, then the complex potential function Ω(z)
given by:
Ω(z) =φ(x, y)+iψ(x, y)
isan analytic function in D. Thus, the level curves of φ and ψ are orthogonal
familiesof curvesasdefined in Section 3.4. The physical meaning of the level
curvesof φ and ψ for applications to electrostatics, fluid flow, gravitation,
and heat flow are summarized in Table 3.1 in Section 3.4. For example, if the
function φ in Example 1 represents the electrostatic potential between two
infinitely long conducting plates, then the level curves φ(x, y) =C1 represent
equipotential curves. Since φ(x, y) = 5 5
2x − 2y + 3, the equipotential curves
are given by y = x + c1, where c1 = 2
5 (3 − C1). These equipotential curves,
which are lines with slope 1, are shown in color in Figure 4.22. In order
to find a harmonic conjugate ψ of φ, we proceed asin part (b) of Example
2 in Section 3.3. Since the conjugate ψ must satisfy the Cauchy-Riemann
equations ∂ψ/∂y = ∂φ/∂x and ∂ψ/∂x = −∂φ/∂y, wemusthave:
∂ψ
∂y
= 5
2
and
∂ψ
∂x
= 5
2 .
Partial integration of the first of these equations with respect to y gives
ψ(x, y) = 5
2y + h(x). The partial derivative with respect to x of thisequation
is ∂ψ/∂x = h ′ (x). Substituting this into the second of the Cauchy-Riemann
equationsimpliesthat h ′ (x) = 5
5
2 , and so h(x) = 2x + c, where c isany real
constant. Setting c = 0, we obtain the harmonic conjugate ψ(x, y) = 5 5
2x+ 2y of φ(x, y). Therefore, a complex potential function for φ is
Ω(z) =φ(x, y)+iψ(x, y) = 5
�
5 5 5
x − y +3+i x +
2 2 2 2 y
�
.
If φ represents electrostatic potential, then the level curves ψ(x, y) =C2
represent lines of force. Since ψ(x, y) = 5 5
2x + 2y, the linesof force are given
by y = −x + c2 where c2 = 2
5C2. The linesof force are shown in black in
Figure 4.22.
The method used in Example 1 can be generalized to solve a Dirichlet
problem in any domain D bounded by two parallel lines. The key to solving
such a problem is finding an appropriate linear function that maps the
boundary linesof D onto the boundary linesof the domain shown in Figure
4.18. See Problems1–4 in Exercises4.5.