Complex potentials

25
Applications of complex variables
In chapter 24, we developed the basic theory of the functions of a complex
variable, z = x + iy, studied their analyticity (differentiability) properties and
derived a number of results concerned with values of contour integrals in the
complex plane. In this current chapter we will show how some of those results
and properties can be exploited to tackle problems arising directly from physical
situations or from apparently unrelated parts of mathematics.
In the former category will be the use of the differential properties of the real
and imaginary parts of a function of a complex variable to solve problems involving Laplace’s equation in two dimensions, whilst an example of the latter might
be the summation of certain types of infinite series. Other applications, such as
the Bromwich inversion formula for Laplace transforms, appear as mathematical
problems that have their origins in physical applications; the Bromwich inversion
enables us to extract the spatial or temporal response of a system to an initial
input from the representation of that response in ‘frequency space’ – or, more
correctly, imaginary frequency space.
Other topics that will be considered are the location of the (complex) zeros of
a polynomial, the approximate evaluation of certain types of contour integrals
using the methods of steepest descent and stationary phase, and the so-called
‘phase-integral’ solutions to some differential equations. For each of these a brief
introduction is given at the start of the relevant section and to repeat them here
would be pointless. We will therefore move on to our first topic of complex
potentials.
25.1 Complex potentials
Towards the end of section 24.2 of the previous chapter it was shown that the real
and the imaginary parts of an analytic function of z are separately solutions of
Laplace’s equation in two dimensions. Analytic functions thus offer a possible way
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APPLICATIONS OF COMPLEX VARIABLES
y
x
Figure 25.1 The equipotentials (dashed circles) and field lines (solid lines)
for a line charge perpendicular to the z-plane.
of solving some two-dimensional physical problems describable by a potential
satisfying ∇2 φ = 0. The general method is known as that of complex potentials.
We also found that if f = u + iv is an analytic function of z then any curve
u = constant intersects any curve v = constant at right angles. In the context of
solutions of Laplace’s equation, this result implies that the real and imaginary
parts of f(z) have an additional connection between them, for if the set of
contours on which one of them is a constant represents the equipotentials of a
system then the contours on which the other is constant, being orthogonal to
each of the first set, must represent the corresponding field lines or stream lines,
depending on the context. The analytic function f is the complex potential. It
is conventional to use φ and ψ (rather than u and v) to denote the real and
imaginary parts of a complex potential, so that f = φ + iψ.
As an example, consider the function
f(z) =
−q
ln z
2π0
(25.1)
in connection with the physical situation of a line charge of strength q per unit
length passing through the origin, perpendicular to the z-plane (figure 25.1). Its
real and imaginary parts are
φ=
−q
ln |z|,
2π0
ψ=
−q
arg z.
2π0
(25.2)
The contours in the z-plane of φ = constant are concentric circles and those of
ψ = constant are radial lines. As expected these are orthogonal sets, but in addition they are, respectively, the equipotentials and electric field lines appropriate to
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25.1 COMPLEX POTENTIALS
the field produced by the line charge. The minus sign is needed in (25.1) because
the value of φ must decrease with increasing distance from the origin.
Suppose we make the choice that the real part φ of the analytic function f
gives the conventional potential function; ψ could equally well be selected. Then
we may consider how the direction and magnitude of the field are related to f.
Show that for any complex (electrostatic) potential f(z) the strength of the electric field
is given by E = |f (z)| and that its direction makes an angle of π − arg[ f (z)] with the
x-axis.
Because φ = constant is an equipotential, the field has components
Ex = −
∂φ
∂x
and
Ey = −
∂φ
.
∂y
(25.3)
Since f is analytic, (i) we may use the Cauchy–Riemann relations (24.5) to change the
second of these, obtaining
∂φ
∂ψ
and
Ey =
;
∂x
∂x
(ii) the direction of differentiation at a point is immaterial and so
Ex = −
(25.4)
∂f
∂φ
∂ψ
df
=
=
+i
= −Ex + iEy .
dz
∂x
∂x
∂x
(25.5)
From these it can be seen that the field at a point is given in magnitude by E = |f (z)|
and that it makes an angle with the x-axis given by π − arg[ f (z)]. It will be apparent from the above that much of physical interest can be
calculated by working directly in terms of f and z. In particular, the electric field
vector E may be represented, using (25.5) above, by the quantity
E = Ex + iEy = −[ f (z)]∗ .
Complex potentials can be used in two-dimensional fluid mechanics problems
in a similar way. If the flow is stationary (i.e. the velocity of the fluid does not
depend on time) and irrotational, and the fluid is both incompressible and nonviscous, then the velocity of the fluid can be described by V = ∇φ, where φ is the
velocity potential and satisfies ∇2 φ = 0. If, for a complex potential f = φ + iψ,
the real part φ is taken to represent the velocity potential then the curves ψ =
constant will be the streamlines of the flow. In a direct parallel with the electric
field, the velocity may be represented in terms of the complex potential by
V = Vx + iVy = [ f (z)]∗ ,
the difference of a minus sign reflecting the same difference between the definitions
of E and V. The speed of the flow is equal to |f (z)|. Points where f (z) = 0, and
thus the velocity is zero, are called stagnation points of the flow.
Analogously to the electrostatic case, a line source of fluid at z = z0 , perpendicular to the z-plane (i.e. a point from which fluid is emerging at a constant rate),
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APPLICATIONS OF COMPLEX VARIABLES
is described by the complex potential
f(z) = k ln(z − z0 ),
where k is the strength of the source. A sink is similarly represented, but with k
replaced by −k. Other simple examples are as follows.
(i) The flow of a fluid at a constant speed V0 and at an angle α to the x-axis
is described by f(z) = V0 (exp iα)z.
(ii) Vortex flow, in which fluid flows azimuthally in an anticlockwise direction
around some point z0 , the speed of the flow being inversely proportional
to the distance from z0 , is described by f(z) = −ik ln(z − z0 ), where k is
the strength of the vortex. For a clockwise vortex k is replaced by −k.
Verify that the complex potential
a2
f(z) = V0 z +
z
is appropriate to a circular cylinder of radius a placed so that it is perpendicular to a
uniform fluid flow of speed V0 parallel to the x-axis.
Firstly, since f(z) is analytic except at z = 0, both its real and imaginary parts satisfy
Laplace’s equation in the region exterior to the cylinder. Also f(z) → V0 z as z → ∞, so
that Re f(z) → V0 x, which is appropriate to a uniform flow of speed V0 in the x-direction
far from the cylinder.
Writing z = r exp iθ and using de Moivre’s theorem we have
a2
f(z) = V0 r exp iθ +
exp(−iθ)
r
a2
a2
cos θ + iV0 r −
sin θ.
= V0 r +
r
r
Thus we see that the streamlines of the flow described by f(z) are given by
a2
sin θ = constant.
ψ = V0 r −
r
In particular, ψ = 0 on r = a, independently of the value of θ, and so r = a must be a
streamline. Since there can be no flow of fluid across streamlines, r = a must correspond
to a boundary along which the fluid flows tangentially. Thus f(z) is a solution of Laplace’s
equation that satisfies all the physical boundary conditions of the problem, and so, by the
uniqueness theorem, it is the appropriate complex potential. By a similar argument, the complex potential f(z) = −E(z − a2 /z) (note the
minus signs) is appropriate to a conducting circular cylinder of radius a placed
perpendicular to a uniform electric field E in the x-direction.
The real and imaginary parts of a complex potential f = φ + iψ have another
interesting relationship in the context of Laplace’s equation in electrostatics or
fluid mechanics. Let us choose φ as the conventional potential, so that ψ represents
the stream function (or electric field, depending on the application), and consider
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25.1 COMPLEX POTENTIALS
y
Q
x
P
n̂
Figure 25.2 A curve joining the points P and Q. Also shown is n̂, the unit
vector normal to the curve.
the difference in the values of ψ at any two points P and Q connected by some
path C, as shown in figure 25.2. This difference is given by
Q
Q
∂ψ
∂ψ
dx +
dy ,
dψ =
ψ(Q) − ψ(P ) =
∂x
∂y
P
P
which, on using the Cauchy–Riemann relations, becomes
Q
∂φ
∂φ
ψ(Q) − ψ(P ) =
dx +
dy
−
∂y
∂x
P
Q
Q
∂φ
=
ds,
∇φ · n̂ ds =
P
P ∂n
where n̂ is the vector unit normal to the path C and s is the arc length along the
path; the last equality is written in terms of the normal derivative ∂φ/∂n ≡ ∇φ · n̂.
Now suppose that in an electrostatics application, the path C is the surface of
a conductor; then
σ
∂φ
=− ,
∂n
0
where σ is the surface charge density per unit length normal to the xy-plane.
Therefore −0 [ψ(Q) − ψ(P )] is equal to the charge per unit length normal to the
xy-plane on the surface of the conductor between the points P and Q. Similarly,
in fluid mechanics applications, if the density of the fluid is ρ and its velocity is
V then
Q
Q
∇φ · n̂ ds = ρ
V · n̂ ds
ρ[ψ(Q) − ψ(P )] = ρ
P
P
is equal to the mass flux between P and Q per unit length perpendicular to the
xy-plane.
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