Definite integrals using contour integration

24.13 DEFINITE INTEGRALS USING CONTOUR INTEGRATION
We note that result (24.60) is a special case of (24.63) in which θ2 is equal to
θ1 + 2π.
24.13 Definite integrals using contour integration
The remainder of this chapter is devoted to methods of applying contour integration and the residue theorem to various types of definite integral. However, three
applications of contour integration, in which obtaining a value for the integral is
not the prime purpose of the exercise, have been postponed until chapter 25. They
are the location of the zeros of a complex polynomial, the evaluation of the sums
of certain infinite series and the determination of inverse Laplace transforms.
For the integral evalations considered here, not much preamble is given since,
for this material, the simplest explanation is felt to be via a series of worked
examples that can be used as models.
24.13.1 Integrals of sinusoidal functions
Suppose that an integral of the form
2π
F(cos θ, sin θ) dθ
(24.64)
0
is to be evaluated. It can be made into a contour integral around the unit circle
C by writing z = exp iθ, and hence
cos θ = 12 (z + z −1 ),
sin θ = − 12 i(z − z −1 ),
dθ = −iz −1 dz.
(24.65)
This contour integral can then be evaluated using the residue theorem, provided
the transformed integrand has only a finite number of poles inside the unit circle
and none on it.
Evaluate
2π
I=
0
cos 2θ
dθ,
a2 + b2 − 2ab cos θ
b > a > 0.
(24.66)
By de Moivre’s theorem (section 3.4),
cos nθ = 12 (z n + z −n ).
(24.67)
Using n = 2 in (24.67) and straightforward substitution for the other functions of θ in
(24.66) gives
0
i
z4 + 1
I=
dz.
2
2ab C z (z − a/b)(z − b/a)
Thus there are two poles inside C, a double pole at z = 0 and a simple pole at z = a/b
(recall that b > a).
We could find the residue of the integrand at z = 0 by expanding the integrand as a
Laurent series in z and identifying the coefficient of z −1 . Alternatively, we may use the
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COMPLEX VARIABLES
formula (24.56) with m = 2. Choosing the latter method and denoting the integrand by
f(z), we have
d
z4 + 1
d 2
[z f(z)] =
dz
dz (z − a/b)(z − b/a)
=
(z − a/b)(z − b/a)4z 3 − (z 4 + 1)[(z − a/b) + (z − b/a)]
.
(z − a/b)2 (z − b/a)2
Now setting z = 0 and applying (24.56), we find
a b
+ .
b a
For the simple pole at z = a/b, equation (24.57) gives the residue as
R(0) =
R(a/b) = lim
z→(a/b)
=−
(z − a/b)f(z) =
(a/b)4 + 1
(a/b)2 (a/b − b/a)
a4 + b4
.
ab(b2 − a2 )
Therefore by the residue theorem
2
2πa2
i
a + b2
a4 + b4
= 2 2
I = 2πi ×
−
.
2ab
ab
ab(b2 − a2 )
b (b − a2 )
24.13.2 Some infinite integrals
We next consider the evaluation of an integral of the form
∞
f(x) dx,
−∞
where f(z) has the following properties:
(i) f(z) is analytic in the upper half-plane, Im z ≥ 0, except for a finite number
of poles, none of which is on the real axis;
(ii) on a semicircle Γ of radius R (figure 24.14), R times the maximum of |f|
on Γ tends to zero as R → ∞ (a sufficient condition is that zf(z) → 0 as
|z| → ∞);
∞
0
(iii) −∞ f(x) dx and 0 f(x) dx both exist.
Since
f(z) dz ≤ 2πR × (maximum of |f| on Γ),
Γ
condition (ii) ensures that the integral along Γ tends to zero as R → ∞, after
which it is obvious from the residue theorem that the required integral is given
by
∞
f(x) dx = 2πi × (sum of the residues at poles with Im z > 0).
−∞
(24.68)
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24.13 DEFINITE INTEGRALS USING CONTOUR INTEGRATION
y
Γ
−R
O
x
R
Figure 24.14 A semicircular contour in the upper half-plane.
Evaluate
∞
I=
0
dx
,
(x2 + a2 )4
where a is real.
The complex function (z 2 + a2 )−4 has poles of order 4 at z = ±ai, of which only z = ai
is in the upper half-plane. Conditions (ii) and (iii) are clearly satisfied. For higher-order
poles, formula (24.56) for evaluating residues can be tiresome to apply. So, instead, we put
z = ai + ξ and expand for small ξ to obtain§
1
1
1
=
=
(z 2 + a2 )4
(2aiξ + ξ 2 )4
(2aiξ)4
1−
iξ
2a
−4
.
The coefficient of ξ −1 is given by
1 (−4)(−5)(−6)
(2a)4
3!
and hence by the residue theorem
∞
−∞
−i
2a
3
=
−5i
,
32a7
10π
dx
=
,
(x2 + a2 )4
32a7
and so I = 5π/(32a7 ). Condition (i) of the previous method required there to be no poles of the
integrand on the real axis, but in fact simple poles on the real axis can be
accommodated by indenting the contour as shown in figure 24.15. The indentation
at the pole z = z0 is in the form of a semicircle γ of radius ρ in the upper halfplane, thus excluding the pole from the interior of the contour.
§
This illustrates another useful technique for determining residues.
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COMPLEX VARIABLES
y
Γ
γ
−R
O
R
x
Figure 24.15 An indented contour used when the integrand has a simple
pole on the real axis.
What is then obtained from a contour integration, apart from the contributions
for Γ and γ, is called the principal value of the integral, defined as ρ → 0 by
R
z0 −ρ
R
P
f(x) dx ≡
f(x) dx +
f(x) dx.
−R
−R
z0 +ρ
The remainder of the calculation goes through as before, but the contribution
from the semicircle, γ, must be included. Result (24.63) of section 24.12 shows
that since only a simple pole is involved its contribution is
−ia−1 π,
(24.69)
where a−1 is the residue at the pole and the minus sign arises because γ is
traversed in the clockwise (negative) sense.
We defer giving an example of an indented contour until we have established
Jordan’s lemma; we will then work through an example illustrating both. Jordan’s
lemma enables infinite integrals involving sinusoidal functions to be evaluated.
For a function f(z) of a complex variable z, if
(i) f(z) is analytic in the upper half-plane except for a finite number of poles in Im z > 0,
(ii) the maximum of |f(z)| → 0 as |z| → ∞ in the upper half-plane,
(iii) m > 0,
then
eimz f(z) dz → 0
IΓ =
as R → ∞,
(24.70)
Γ
where Γ is the same semicircular contour as in figure 24.14.
Note that this condition (ii) is less stringent than the earlier condition (ii) (see the
start of this section), since we now only require M(R) → 0 and not RM(R) → 0,
where M is the maximum§ of |f(z)| on |z| = R.
§
More strictly, the least upper bound.
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24.13 DEFINITE INTEGRALS USING CONTOUR INTEGRATION
The proof of the lemma is straightforward once it has been observed that, for
0 ≤ θ ≤ π/2,
2
sin θ
≥ .
θ
π
Then, since on Γ we have | exp(imz)| = | exp(−mR sin θ)|,
π
IΓ ≤
|eimz f(z)| |dz| ≤ MR
e−mR sin θ dθ = 2MR
1≥
0
Γ
(24.71)
π/2
e−mR sin θ dθ.
0
Thus, using (24.71),
IΓ ≤ 2MR
π/2
e−mR(2θ/π) dθ =
0
πM
πM 1 − e−mR <
;
m
m
hence, as R → ∞, IΓ tends to zero since M tends to zero.
Find the principal value of
∞
−∞
cos mx
dx,
x−a
for a real, m > 0.
Consider the function (z − a)−1 exp(imz); although it has no poles in the upper half-plane
it does have a simple pole at z = a, and further |(z − a)−1 | → 0 as |z| → ∞. We will use a
contour like that shown in figure 24.15 and apply the residue theorem. Symbolically,
a−ρ R + +
+
= 0.
(24.72)
−R
γ
a+ρ
Γ
Now as R → ∞ and ρ → 0 we have Γ → 0, by Jordan’s lemma, and from (24.68) and
(24.69) we obtain
∞ imx
e
P
(24.73)
dx − iπa−1 = 0,
−∞ x − a
where a−1 is the residue of (z − a)−1 exp(imz) at z = a, which is exp(ima). Then taking the
real and imaginary parts of (24.73) gives
∞
cos mx
P
dx = −π sin ma,
as required,
−∞ x − a
∞
sin mx
dx = π cos ma,
as a bonus. P
−∞ x − a
24.13.3 Integrals of multivalued functions
We have discussed briefly some of the properties and difficulties associated with
certain multivalued functions such as z 1/2 or Ln z. It was mentioned that one
method of managing such functions is by means of a ‘cut plane’. A similar
technique can be used with advantage to evaluate some kinds of infinite integral
involving real functions for which the corresponding complex functions are multivalued. A typical contour employed for functions with a single branch point
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COMPLEX VARIABLES
y
Γ
γ
A
B
C
D
x
Figure 24.16 A typical cut-plane contour for use with multivalued functions
that have a single branch point located at the origin.
located at the origin is shown in figure 24.16. Here Γ is a large circle of radius R
and γ is a small one of radius ρ, both centred on the origin. Eventually we will
let R → ∞ and ρ → 0.
The success of the method is due to the fact that because the integrand is
multivalued, its values along the two lines AB and CD joining z = ρ to z = R
are not equal and opposite although both are related to the corresponding real
integral. Again an example provides the best explanation.
Evaluate
∞
I=
0
dx
,
(x + a)3 x1/2
a > 0.
We consider the integrand f(z) = (z + a)−3 z −1/2 and note that |zf(z)| → 0 on the two
circles as ρ → 0 and R → ∞. Thus the two circles make no contribution to the contour
integral.
The only pole of the integrand inside the contour is at z = −a (and is of order 3).
To determine its residue we put z = −a + ξ and expand (noting that (−a)1/2 equals
a1/2 exp(iπ/2) = ia1/2 ):
1
1
= 3 1/2
(z + a)3 z 1/2
ξ ia (1 − ξ/a)1/2
1ξ
1
3 ξ2
= 3 1/2 1 +
+
·
·
·
.
+
iξ a
2a
8 a2
The residue is thus −3i/(8a5/2 ).
The residue theorem (24.61) now gives
−3i
.
+ +
+ = 2πi
5/2
8a
AB
Γ
DC
γ
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