Exercises

APPLICATIONS OF COMPLEX VARIABLES
A
V0 eiωt
C
L
4
IR
D
E
R
L
C
B
Figure 25.15 The inductor–capacitor–resistor network for exercise 25.1.
which can also be simplified, and gives
1
π
2
+ √
(−x)3/2 −
.
exp i
1/4
3
4
2 π(−x)
Adding the two contributions and taking the real part of the sum, though this is not
necessary here because the sum is real anyway, we obtain
2
π
2
F(x) = √
cos
(−x)3/2 −
1/4
3
4
2 π(−x)
1
π
2
3/2
,
= √
sin
+
(−x)
3
4
π(−x)1/4
in agreement with the asymptotic form given in (25.53). 25.9 Exercises
25.1
In the method of complex impedances for a.c. circuits, an inductance L is
represented by a complex impedance ZL = iωL and a capacitance C by ZC =
1/(iωC). Kirchhoff’s circuit laws,
Ii = 0 at a node and
Zi Ii =
Vj around any closed loop,
i
25.2
i
j
are then applied as if the circuit were a d.c. one.
Apply this method to the a.c. bridge connected as in figure 25.15 to show
that if the resistance R is chosen as R = (L/C)1/2 then the amplitude of the
current, IR , through it is independent of the angular frequency ω of the applied
a.c. voltage V0 eiωt .
Determine how the phase of IR , relative to that of the voltage source, varies
with the angular frequency ω.
A long straight fence made of conducting wire mesh separates two fields and
stands one metre high. Sometimes, on fine days, there is a vertical electric field
over flat open countryside. Well away from the fence the strength of the field is
E0 . By considering the effect of the transformation w = (1 − z 2 )1/2 on the real and
920
25.9 EXERCISES
25.3
25.4
25.5
imaginary z-axes, find the strengths of the field (a) at a point one metre directly
above the fence, (b) at ground level one metre to the side of the fence, and (c)
at a point that is level with the top of the fence but one metre to the side of it.
What is the direction of the field in case (c)?
For the function
z+c
,
f(z) = ln
z−c
where c is real, show that the real part u of f is constant on a circle of radius
c cosech u centred on the point z = c coth u. Use this result to show that the
electrical capacitance per unit length of two parallel cylinders of radii a, placed
with their axes 2d apart, is proportional to [cosh−1 (d/a)]−1 .
Find a complex potential in the z-plane appropriate to a physical situation in
which the half-plane x > 0, y = 0 has zero potential and the half-plane x < 0,
y = 0 has potential V .
By making the transformation w = a(z + z −1 )/2, with a real and positive, find
the electrostatic potential associated with the half-plane r > a, s = 0 and the
half-plane r < −a, s = 0 at potentials 0 and V , respectively.
By considering in turn the transformations
z = 12 c(w + w −1 )
and w = exp ζ,
where z = x + iy, w = r exp iθ, ζ = ξ + iη and c is a real positive constant, show
that z = c cosh ζ maps the strip ξ ≥ 0, 0 ≤ η ≤ 2π, onto the whole z-plane. Which
curves in the z-plane correspond to the lines ξ = constant and η = constant?
Identify those corresponding to ξ = 0, η = 0 and η = 2π.
The electric potential φ of a charged conducting strip −c ≤ x ≤ c, y = 0,
satisfies
φ ∼ −k ln(x2 + y 2 )1/2 for large values of (x2 + y 2 )1/2 ,
25.6
with φ constant on the strip. Show that φ = Re[−k cosh−1 (z/c)] and that the
magnitude of the electric field near the strip is k(c2 − x2 )−1/2 .
For the equation 8z 3 + z + 1 = 0:
(a) show that all three roots lie between the circles |z| = 3/8 and |z| = 5/8;
(b) find the approximate location of the real root, and hence deduce that the
complex ones lie in the first and fourth quadrants and have moduli greater
than 0.5.
25.7
Use contour integration to answer the following questions about the complex
zeros of a polynomial equation.
(a) Prove that z 8 + 3z 3 + 7z + 5 has two zeros in the first quadrant.
(b) Find in which quadrants the zeros of 2z 3 + 7z 2 + 10z + 6 lie. Try to locate
them.
25.8
The following is a method of determining the number of zeros of an nth-degree
polynomial f(z) inside the contour C given by |z| = R:
(a) put z = R(1 + it)/(1 − it), with t = tan(θ/2), in the range −∞ ≤ t ≤ ∞;
(b) obtain f(z) as
A(t) + iB(t) (1 + it)n
;
(1 − it)n (1 + it)n
(c) it follows that arg f(z) = tan−1 (B/A) + n tan−1 t;
(d) and that ∆C [arg f(z)] = ∆C [tan−1 (B/A)] + nπ;
(e) determine ∆C [tan−1 (B/A)] by evaluating tan−1 (B/A) at t = ±∞ and finding
the discontinuities in B/A by inspection or using a sketch graph.
921
APPLICATIONS OF COMPLEX VARIABLES
25.9
25.10
Then, by the principle of the argument, the number of zeros inside C is given by
the integer (2π)−1 ∆C [arg f(z)].
It can be shown that the zeros of z 4 + z + 1 lie one in each quadrant. Use the
above method to show that the zeros in the second and third quadrants have
|z| < 1.
Prove that
∞
1
= 4π.
n2 + 34 n + 18
−∞
Carry out the summation numerically, say between −4 and 4, and note how
much of the sum comes from values near the poles of the contour integration.
This exercise illustrates a method of summing some infinite series.
(a) Determine the residues at all the poles of the function
f(z) =
π cot πz
,
a2 + z 2
where a is a positive real constant.
(b) By evaluating, in two different ways, the integral I of f(z) along the straight
line joining −∞ − ia/2 and +∞ − ia/2, show that
∞
n=1
(c) Deduce the value of
25.11
∞
1
π coth πa
1
1
=
− 2.
a2 + n2
2a
2a
n−2 .
By considering the integral of
2
π
sin αz
,
αz
sin πz
α<
π
,
2
around a circle of large radius, prove that
∞
(−1)m−1
m=1
25.12
1
sin2 mα
= .
(mα)2
2
Use the Bromwich inversion, and contours similar to that shown in figure 25.7(a),
to find the functions of which the following are the Laplace transforms:
(a) s(s2 + b2 )−1 ;
(b) n!(s − a)−(n+1) , with n a positive integer and s > a;
(c) a(s2 − a2 )−1 , with s > |a|.
25.13
Compare your answers with those given in a table of standard Laplace transforms.
Find the function f(t) whose Laplace transform is
e−s − 1 + s
.
s2
A function f(t) has the Laplace transform
1
s+i
,
F(s) = ln
2i
s−i
f̄(s) =
25.14
the complex logarithm being defined by a finite branch cut running along the
imaginary axis from −i to i.
(a) Convince yourself that, for t > 0, f(t) can be expressed as a closed contour
integral that encloses only the branch cut.
922
25.9 EXERCISES
(b) Calculate F(s) on either side of the branch cut, evaluate the integral and
hence determine f(t).
(c) Confirm that the derivative with respect to s of the Laplace transform
integral of your answer is the same as that given by dF/ds.
25.15
25.16
Use the contour in figure 25.7(c) to show that the function with Laplace transform
s−1/2 is (πx)−1/2 .
[ For an integrand of the form r −1/2 exp(−rx) change variable to t = r 1/2 . ]
Transverse vibrations of angular frequency ω on a string stretched with constant
tension T are described by u(x, t) = y(x) e−iωt , where
ω 2 m(x)
d2 y
+
y(x) = 0.
2
dx
T
Here, m(x) = m0 f(x) is the mass per unit length of the string and, in the general
case, is a function of x. Find the first-order W.K.B. solution for y(x).
Due to imperfections in its manufacturing process, a particular string has
a small periodic variation in its linear density of the form m(x) = m0 [ 1 +
sin(2πx/L) ], where 1. A progressive wave (i.e. one in which no energy is
lost) travels in the positive x-direction along the string. Show that its amplitude
fluctuates by ± 14 of its value A0 at x = 0 and that, to first order in , the phase
of the wave is
ω L m0
πx
sin2
2π
T
L
25.17
25.18
ahead of what it would be if the string were uniform, with m(x) = m0 .
The equation
1 1
d2 y
+ ν + − z 2 y = 0,
2
dz
2 4
sometimes called the Weber–Hermite equation, has solutions known as parabolic
cylinder functions. Find, to within (possibly complex) multiplicative constants,
the two W.K.B. solutions of this equation that are valid for large |z|. In each case,
determine the leading term and show that the multiplicative correction factor is
of the form 1 + O(ν 2 /z 2 ).
Identify the Stokes and anti-Stokes lines for the equation. On which of the
Stokes lines is the W.K.B. solution that tends to zero for z large, real and negative,
the dominant solution?
A W.K.B. solution of Bessel’s equation of order zero,
d2 y
1 dy
+
+ y = 0,
dz 2
z dz
(∗)
valid for large |z| and −π/2 < arg z < 3π/2, is y(z) = Az −1/2 eiz . Obtain an
improvement on this by finding a multiplier of y(z) in the form of an asymptotic
expansion in inverse powers of z as follows.
(a) Substitute for y(z) in (∗) and show that the equation is satisfied to O(z −5/2 ).
(b) Now replace the constant A by A(z) and find the equation that
be
must −n
satisfied by A(z). Look for a solution of the form A(z) = z σ ∞
n=0 an z ,
where a0 = 1. Show that σ = 0 is the only acceptable solution to the indicial
equation and obtain a recurrence relation for the an .
(c) To within a (complex) constant, the expression y(z) = A(z)z −1/2 eiz is the
asymptotic expansion of the Hankel function H0(1) (z). Show that it is a
divergent expansion
values of z and estimate, in terms of z, the value
for all
−n−1/2 iz
of N such that N
e gives the best estimate of H0(1) (z).
n=0 an z
923
APPLICATIONS OF COMPLEX VARIABLES
25.19
The function h(z) of the complex variable z is defined by the integral
i∞
exp(t2 − 2zt) dt.
h(z) =
−i∞
(a) Make a change of integration variable, t = iu, and evaluate h(z) using a
standard integral. Is your answer valid for all finite z?
(b) Evaluate the integral using the method of steepest descents, considering in
particular the cases (i) z is real and positive, (ii) z is real and negative and
(iii) z is purely imaginary and equal to iβ, where β is real. In each case sketch
the corresponding contour in the complex t-plane.
(c) Evaluate the integral for the same three cases as specified in part (b) using
the method of stationary phases. To determine an appropriate contour that
passes through a saddle point t = t0 , write t = t0 + (u + iv) and apply the
criterion for determining a level line. Sketch the relevant contour in each
case, indicating what freedom there is to distort it.
25.20
Comment on the accuracy of the results obtained using the approximate methods
adopted in (b) and (c).
Use the method of steepest descents to show that an approximate value for the
integral
∞
F(z) =
exp[ iz( 15 t5 + t) ] dt,
−∞
25.21
where z is real and positive, is
1/2
2π
exp(−βz) cos(βz − 18 π),
z
√
where β = 4/(5 2).
The stationary phase approximation to an integral of the form
b
g(t)eiνf(t) dt,
|ν| 1,
F(ν) =
a
where f(t) is a real function of t and g(t) is a slowly varying function (when
compared with the argument of the exponential), can be written as
1/2 N
! "
π
g(tn )
2π
√ exp i νf(tn ) + sgn νf (tn )
,
F(ν) ∼
|ν|
4
An
n=1
where the tn are the N stationary points of f(t) that lie in a < t1 < t2 < · · · <
tN < b, An = | f (tn ) |, and sgn(x) is the sign of x.
Use this result to find an approximation, valid for large positive values of ν,
to the integral
∞
1
cos[ (2t3 − 3zt2 − 12z 2 t)ν ] dt,
F(ν, z) =
2
−∞ 1 + t
25.22
where z is a real positive parameter.
The Bessel function Jν (z) is given for | arg z| < 12 π by the integral around a
contour C of the function
1 −(ν+1)
1
z
g(z) =
t−
.
exp
t
2πi
2
t
The contour starts and ends along the negative real t-axis and encircles the origin
in the positive sense. It can be considered to be made up of two contours. One
of them, C2 , starts at t = −∞, runs through the third quadrant to the point
924