Exercises

24.14 EXERCISES
We have seen that Γ and γ vanish, and if we denote z by x along the line AB then it
has the value z = x exp 2πi along the line DC (note that
exp 2πi must not be set equal to
1 until after the substitution for z has been made in DC ). Substituting these expressions,
∞
0
3π
dx
dx
= 5/2 .
+
3 1/2 exp( 1 2πi)
(x + a)3 x1/2
4a
0
∞ [x exp 2πi + a] x
2
Thus
1−
1
exp πi
∞
0
3π
dx
= 5/2
(x + a)3 x1/2
4a
and
I=
1
3π
.
×
2 4a5/2
Several other examples of integrals of multivalued functions around a variety
of contours are included in the exercises that follow.
24.14 Exercises
24.1
Find an analytic function of z = x + iy whose imaginary part is
24.2
Find a function f(z), analytic in a suitable part of the Argand diagram, for which
(y cos y + x sin y) exp x.
Re f =
24.3
Where are the singularities of f(z)?
Find the radii of convergence of the following Taylor series:
∞
zn
,
ln n
n=2
∞
(c)
z n nln n ,
(a)
(b)
(d)
n=1
24.4
sin 2x
.
cosh 2y − cos 2x
∞
n!z n
,
nn
n=1
n2
∞ n+p
n=1
n
Find the Taylor series expansion about the origin of the function f(z) defined by
f(z) =
∞
(−1)r+1 sin
pz r=1
24.5
r
,
where p is a constant. Hence verify that f(z) is a convergent series for all z.
Determine the types of singularities (if any) possessed by the following functions
at z = 0 and z = ∞:
(a) (z − 2)−1 ,
(d) ez /z 3 ,
24.6
z n , with p real.
(b) (1 + z 3 )/z 2 ,
(e) z 1/2 /(1 + z 2 )1/2 .
(c) sinh(1/z),
Identify the zeros, poles and essential singularities of the following functions:
(a) tan z,
(d) tan(1/z),
(b) [(z − 2)/z 2 ] sin[1/(1 − z)],
(e) z 2/3 .
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(c) exp(1/z),
COMPLEX VARIABLES
24.7
Find the real and imaginary parts of the functions (i) z 2 , (ii) ez , and (iii) cosh πz.
By considering the values taken by these parts on the boundaries of the region
0 ≤ x, y ≤ 1, determine the solution of Laplace’s equation in that region that
satisfies the boundary conditions
φ(x, 0) = 0,
φ(x, 1) = x,
24.8
φ(0, y) = 0,
φ(1, y) = y + sin πy.
Show that the transformation
z
w=
0
1
dζ
(ζ 3 − ζ)1/2
transforms the upper half-plane into the interior of a square that has one corner
at the origin of the w-plane and sides of length L, where
π/2
L=
cosec 1/2 θ dθ.
0
24.10
The fundamental theorem of algebra states that, for a complex polynomial pn (z)
of degree n, the equation pn (z) = 0 has precisely n complex roots. By applying
Liouville’s theorem (see the end of section 24.10) to f(z) = 1/pn (z), prove that
pn (z) = 0 has at least one complex root. Factor out that root to obtain pn−1 (z)
and, by repeating the process, prove the above theorem.
Show that, if a is a positive real constant, the function exp(iaz 2 ) is analytic and
→ 0 as |z| → ∞ for 0 < arg z ≤ π/4. By applying Cauchy’s theorem to a suitable
contour prove that
∞
π
cos(ax2 ) dx =
.
8a
0
24.11
The function
24.9
f(z) = (1 − z 2 )1/2
of the complex variable z is defined to be real and positive on the real axis in
the range −1 < x < 1. Using cuts running along the real axis for 1 < x < +∞
and −∞ < x < −1, show how f(z) is made single-valued and evaluate it on the
upper and lower sides of both cuts.
Use these results and a suitable contour in the complex z-plane to evaluate the
integral
∞
dx
I=
.
x(x2 − 1)1/2
1
24.12
Confirm your answer by making the substitution x = sec θ.
By considering the real part of
−iz n−1 dz
,
1 − a(z + z −1 ) + a2
where z = exp iθ and n is a non-negative integer, evaluate
π
cos nθ
dθ
2
0 1 − 2a cos θ + a
24.13
for a real and > 1.
Prove that if f(z) has a simple zero at z0 , then 1/f(z) has residue 1/f (z0 ) there.
Hence evaluate
π
sin θ
dθ,
−π a − sin θ
where a is real and > 1.
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24.14 EXERCISES
24.14
Prove that, for α > 0, the integral
∞
0
24.15
24.16
24.17
t sin αt
dt
1 + t2
has the value (π/2) exp(−α).
Prove that
∞
cos mx
π −m/2
− e−m
dx =
4e
4x4 + 5x2 + 1
6
0
for m > 0.
Show that the principal value of the integral
∞
cos(x/a)
dx
2
2
−∞ x − a
is −(π/a) sin 1.
The following is an alternative (and roundabout!) way of evaluating the Gaussian
integral.
(a) Prove that the integral of [exp(iπz 2 )]cosec πz around the parallelogram with
corners ±1/2 ± R exp(iπ/4) has the value 2i.
(b) Show that the parts of the contour parallel to the real axis do not contribute
when R → ∞.
(c) Evaluate the integrals along the other two sides by putting z = r exp(iπ/4)
and working in terms of z + 12 and z − 12 . Hence, by letting R → ∞ show
that
∞
2
e−πr dr = 1.
−∞
24.18
24.19
24.20
By applying the residue theorem around a wedge-shaped contour of angle 2π/n,
with one side along the real axis, prove that the integral
∞
dx
,
1 + xn
0
where n is real and ≥ 2, has the value (π/n)cosec (π/n).
Using a suitable cut plane, prove that if α is real and 0 < α < 1 then
∞ −α
x
dx
1+x
0
has the value π cosec πα.
Show that
∞
0
24.21
√
ln x
dx = − 2π 2 .
x3/4 (1 + x)
By integrating a suitable function around a large semicircle in the upper halfplane and a small semicircle centred on the origin, determine the value of
∞
(ln x)2
I=
dx
1 + x2
0
and deduce, as a by-product of your calculation, that
∞
ln x
dx = 0.
1 + x2
0
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