Therefore, the tangent mapping is a linear fraction transformation, pre
ceded by an exponential mapping and followed by a clockwise rotation
by 90
◦
.
With
Z
=
e

2
y
e
2
ix
, we see that the strip
S
:=
{
z
∈
C

x
∈
(

π/
4
, π/
4)
}
in the
z
plane is mapped onto the right half of the
Z
plane,
X >
0.
The linear fractional transformation maps the right half of the
Z
plane
onto the interior of the unit disk in the
W
plane. A clockwise rotation
by
π/
2 follows, so that
w
= tan
z
maps the strip
S
in the
z
plane onto
the interior of the unit disk in the
w
plane.
18 Complex Analysis and Potential Theory
The theory of solutions of Laplace’s equation (cf. Sec. 12.10) is called
poten
tial theory.
Laplace’s equations occurs in many fields such as gravitation,
electrostatics, heat conduction, fluid flow etc. In two space dimensions, and
in cartesian coordinates, we are interested in solutions of the secondorder
PDE
∆Φ = Φ
xx
+ Φ
yy
= 0
,
in Ω
⊆
R
2
,
(1011)
for the (realvalued)
potential
Φ : Ω
→
R
. We know from Section 13.4 that
solutions of (1011) with continuous second derivatives (harmonic functions)
are closely related to holomorphic functions Φ +
i
Ψ where Ψ is a harmonic
conjugate of Φ.
The restriction to two space dimensions is often justified when the poten
tial Φ is independent of one of the space coordinates. In this last chapter of
the lecture, we shall consider this connection and its consequences in detail
and illustrate it by typical boundary value problems from electrostatics, heat
conduction, and hydrodynamics.
18.1 Electrostatic Fields
The electric force
F
[
N
] between charged particles is given by Coulomb’s
law. This force is the gradient of a function Φ:
F
=
q
E
,
E
=
∇
Φ, with
the electric field
E
[Vm

1
] = [NC

1
] and the
electrostatic potential
Φ [V]
(cf. Sec. 12.10; there we looked at Newton’s law of universal gravitation, but
Coulomb’s law has a similar form).
Equipotential surfaces
are level sets of Φ
(Φ
≡
const
.
; lines in 2D). The electric field is perpendicular to these surfaces.
183
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Examples:
1. Two parallel plates kept at potentials Φ
1
, Φ
2
, respectively, extending
to infinity. Choose a cartesian coordinate system (
x, y, z
) such that the
plates are parallel to the
yz
plane, and located at
x
=

1 and
x
= 1.
Due to translation invariance of the problem in both
y
 and
z
direction,
we may restrict our consideration to the
x
direction, where we have a
secondorder ODE with Dirichlet boundary conditions:
Φ
′′
= 0
,
in (

1
,
1)
,
Φ(

1) = Φ
1
,
Φ(1) = Φ
2
.
(1012)
The general solution is given by Φ(
x
) =
ax
+
b
, and the constants
a, b
are determined from the boundary conditions:
Φ(

1)
=

a
+
b
= Φ
1
Φ(1)
=
a
+
b
= Φ
2
⇒
a
=
1
2
(Φ
2

Φ
1
)
b
=
1
2
(Φ
1
+ Φ
2
)
(1013)
Therefore, the potential between two parallel plates is given by
Φ(
x
) =
1
2
(Φ
2

Φ
1
)
x
+
1
2
(Φ
1
+ Φ
2
)
.
(1014)
The equipotential surfaces are parallel planes,
x
= const
.
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 Spring '08
 Staff
 Math, Cartesian Coordinate System, Complex number, Holomorphic function, harmonic function, z direction

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