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17
Residue Theory
“Residue theory” is basically a theory for computing integrals by looking at certain terms in the
Laurent series of the integrated functions about appropriate points on the complex plane. We
will develop the basic theorem by applying the Cauchy integral theorem and the Cauchy integral
formulas along with Laurent series expansions of functions about the singular points. We will
then apply it to compute many, many integrals that cannot be easily evaluated otherwise. Most
of these integrals will be over subintervals of the real line.
17.1 Basic Residue TheoryThe Residue Theorem
Suppose f is a function that, except for isolated singularities, is single-valued and analytic on
some simply-connected region R . Our initial interest is in evaluating the integral
∮
C0
f (z) dz .
where C0 is a circle centered at a point z0 at which f may have a pole or essential singularity.
We will assume the radius of C0 is small enough that no other singularity of f is on or enclosed
by this circle. As usual, we also assume C0 is oriented counterclockwise.
In the region right around z0 , we can express f (z) as a Laurent series
f (z) =∞
∑
k=−∞
ak(z − z0)k ,
and, as noted earlier somewhere, this series converges uniformly in a region containing C0 . So
∮
C0
f (z) dz =∮
C0
∞∑
k=−∞
ak(z − z0)k dz =
∞∑
k=−∞
ak
∮
C0
(z − z0)k dz .
But we’ve seen ∮
C0
(z − z0)k dz for k = 0, ±1, ±2, ±3, . . .
3/10/2014
Chapter & Page: 17–2 Residue Theory
before. You can compute it using the Cauchy integral theorem, the Cauchy integral formulas,
or even (as you did way back in exercise 14.14 on page 14–17) by direct computation after
parameterizing C0 . However you do it, you get, for any integer k ,
∮
C0
(z − z0)k dz =
{
0 if k 6= −1
i2π if k = −1.
So, the above integral of f reduces to
∮
C0
f (z) dz =∞
∑
k=−∞
ak
{
0 if k 6= −1
i2π if k = −1
}
= i2πa−1 .
This shows that the a−1 coefficient in the Laurent series of a function f about a point z0
completely determines the value of the integral of f over a sufficiently small circle centered at
z0 . This quantity, a−1 , is called the residue of f at z0 , and is denoted by
a−1,z0or Resz0
( f ) or · · ·
In fact, it seems that every author has their own notation. We will use Resz0( f ) .
If, instead of integrating around the small circle C0 , we were computing∮
C
f (z) dz
where C is any simple, counterclockwise oriented loop in R that touched no point of singularity
of f but did enclose points z0 , z1 , z2 , … at which f could have singularities, then, a
consequence of Cauchy’s integral theorem (namely, theorem 15.5 on page 15–7) tells us that∮
C
f (z) dz =∑
k
∮
Ck
f (z) dz
where each Ck is a counterclockwise oriented circle centered at zk small enough that no other
point of singularity for f is on or enclosed by this circle. Combined with the calculations done
just above, we get the following:
Theorem 17.1 (Residue Theorem)
Let f be a single-valued function on a region R , and let C be a simple loop oriented counter-
clockwise. Assume C encircles a finite set of points {z0, z1, z2, . . .} at which f might not be
analytic. Assume, further, that f is analytic at every other point on or enclosed by C . Then∮
C
f (z) dz = i2π∑
k
Reszk( f ) . (17.1)
In practice, most people just write equation (17.1) as∮
C
f (z) dz = i2π ×[
sum of the enclosed residues]
.
The residue theorem can be viewed as a generalization of the Cauchy integral theorem and
the Cauchy integral formulas. In fact, many of the applications you see of the residue theorem
can be done nearly as easily using theorem 15.5 (the consequence of the Cauchy integral theorem
used above) along with the Cauchy integral formulas.
Basic Residue Theory Chapter & Page: 17–3
Computing Residues
Remember, what we are now calling the residue of a function f at z0 is simply the value of
a−1 in the Laurent series expansion of f ,
f (z) =∞
∑
k=−∞
ak(z − z0)k ,
right around z0 , and, for this a−1 to be nonzero, f must have either a pole or essential singularity
at z0 . So the discussion about such singularities in section 16.3 applies both for identifying where
a function may have residues and for computing the residues.
The basic approach to computing the residue at z0 is be to simply find the above Laurent
series. Then
Resz0( f ) = a−1 .
This may be necessary if f has an essential singularity at z0 .
If f has a pole of finite order, say, of order M , then we can use formula (16.8) on page
16–14 for a−1 ,
Resz0( f ) = a−1 = 1
(M − 1)!d M−1
dzM−1
[
(z − z0)M f (z)
]∣∣∣∣z=z0
. (17.2)
If the pole is simple (i.e., M = 1 ), this simplifies to
Resz0( f ) = a−1 = (z − z0) f (z)|z=z0
. (17.3)
Often, we may notice that
f (z) = g(z)
z − z0
for some function g which is analytic and nonzero at z0 . In this case, we clearly have a simple
pole, and formula (17.3), above, clearly reduces to
Resz0( f ) = g(z0) .
This will make computing residues very easy in many cases.
More generally, from our earlier discussion of poles, we know that if
f (z) = g(z)
(z − z0)M(17.4)
for some function g which is analytic and nonzero at z0 , then f has a pole of order M at z0 .
In this case, formula (17.2) reduces to
Resz0( f ) = 1
(M − 1)!g(M−1)(z0) . (17.5)
Keep in mind that, if g(z0) = 0 , then the pole of
f (z) = g(z)
(z − z0)M
has order less than M , and a little more work will be needed to determine the precise order of
the pole and the corresponding residue.
version: 3/10/2014
Chapter & Page: 17–4 Residue Theory
17.2 “Simple” Applications
The main application of the residue theorem is to compute integrals we could not compute (or
don’t want to compute) using more elementary means. We will consider some of the common
cases involving single-valued functions not having poles on the curves of integration. Later, we
will add poles and deal with multi-valued functions.
Integrals of the form
∫ 2π
0f (sin θ, cos θ) dθ
Suppose we have an integral over (0, 2π) of some formula involving sin(θ) and cos(θ) , say,∫ 2π
0
dθ
2 + cos θ.
We can convert this to an integral about the unit circle by using the substitution
z = eiθ .
Note that z does go around the unit circle in the counterclockwise direction as θ goes from 0
to 2π . Under this substitution, we have
dz = d[
eiθ]
= ieiθ dθ = i z dθ .
So,
dθ = 1
i zdz = −i z−1 dz .
For the sines and cosines, we have
cos θ = eiθ + e−iθ
2= z + 1/z
2= z + z−1
2
and
sin θ = eiθ − e−iθ
2i= z − 1/z
2i= z − z−1
2i.
These substitutions convert the original integral to an integral of some function of z over the unit
circle, which can then be evaluated by finding the enclosed residues and applying the residue
theorem.
!◮Example 17.1: Let’s evaluate∫ 2π
0
dθ
2 + cos θ.
Letting C denote the unit circle and applying the substitution z = eiθ , as described above,
we get∫ 2π
0
dθ
2 + cos θ=
∮
C
−i z−1 dz
2 + 12
[
z + z−1]
=∮
C
2z
2z· −i z−1
2 + 12
[
z + z−1] dz
=∮
C
−i2
4z +[
z2 + 1] dz = −2i
∮
C
1
z2 + 4z + 1dz .
“Simple” Applications Chapter & Page: 17–5
Letting
f (z) = 1
z2 + 4z + 1
and applying the residue theorem, the above becomes
∫ 2π
0
dθ
2 + cos θ= −2i × i2π ×
[
sum of the residues of f (z) in the unit circle]
= 4π ×[
sum of the residues of f (z) in the unit circle]
.
To find the necessary residues we must find where the denominator of f (z) vanishes,
z2 + 4z + 1 = 0 .
Using the quadratic formula, these points are found to be
z± = −4 ±√
42 − 4
2= −2 ±
√3 .
So
f (z) =1
(
z −[
− 2 +√
3])(
z −[
− 2 −√
3]) .
Now, since√
3 ≈ 1.7 ,
|z+| =∣∣∣−2 +
√3
∣∣∣ ≈ |−2 + 1.7| = 0.3
while
|z−| =∣∣∣−2 −
√3
∣∣∣ ≈ |−2 − 1.7| = 3.7 .
Clearly, −2 +√
3 is the only singular point of f (z) enclosed by the unit circle, and the
singularity there is a simple pole. We can rewrite f (z) as
f (z) = g(z)
z −[
− 2 +√
3] where g(z) = 1
z −[
− 2 −√
3] .
Thus,
Resz+[ f ] = g(z+) = g(−2 +√
3) = 1[
− 2 +√
3]
−[
− 2 −√
3] = 1
2√
3.
Plugging this back into the last formula obtained for our integral, we get
∫ 2π
0
dθ
2 + cos θ= 4π ×
[
sum of the residues of f (z) in the unit circle]
= 4π Resz+[ f ]
= 4π1
2√
3
= 2π√
3.
version: 3/10/2014
Chapter & Page: 17–6 Residue Theory
(a) (b)
R
R
−R
−R
C+R
C−R
Γ +R
Γ −R
X
X
Y Y
IRIR
Figure 17.1: Closed curves for integrating on the X–axis when (a) ΓR = Γ +R = IR + C
+R
and when (b) ΓR = Γ −R = −IR + C
−R .
Integrals of the form
∫∞
−∞
f (x) dx
Now let’s consider evaluating∫ ∞
−∞f (x) dx ,
assuming that:
1. Except for a finite number of poles and/or essential singularities, f is a single-valued
analytic function on the entire complex plane.
2. None of these singularities are on the X–axis.
3. One of the following holds:
(a) z f (z) → 0 as |z| → ∞ .
(b) f (z) = g(z)eiαz where α > 0 and
g(z) → 0 as |z| → ∞ .
(c) f (z) = g(z)e−iαz where α > 0 and
g(z) → 0 as |z| → ∞ .
Under these assumptions, we can evaluate the integral by constructing, for each R > 0 , a suitable
closed loop ΓR containing the interval (−R, R) . The integral over ΓR is computed via the
residue theorem, and R is allowed to go to ∞ . The conditions listed under the third assumption
above ensure that the integral over that part of ΓR which is not the interval (−R, R) vanishes
as R → ∞ , leaving us with the integral over (−∞,∞) .
The exact choice for the closed loop ΓR depends on which of the three conditions under
assumption 3 is known to hold. If either (3a) or (3b) holds, we take
ΓR = Γ +R = IR + C
+R
where IR is the subinterval (−R, R) of the X–axis and C+R is the semicirle in the upper
half plane of radius R and centered at 0 (see figure 17.1a). In this case, with Γ +R oriented
“Simple” Applications Chapter & Page: 17–7
counterclockwise, the direction of travel on the interval IR is from x = −R to x = R (the
normal ‘positive’ direction of travel on the X–axis) and, so,
∫
IR
f (z) dz =∫ R
−R
f (x) dx .
Since we are letting R → ∞ and there are only finitely many singularities, we can always
assume that we’ve taken R large enough for Γ +R to enclose all the singularities of f in the
upper half plane (UHP). Then
i2π[
sum of the residues of f in the UHP]
=∫
Γ +R
f (z) dz
=∫
IR
f (z) dz +∫
C+R
f (z) dz .
Thus,
∫ R
−R
f (x) dx =∫
IR
f (z) dz
= i2π[
sum of the residues of f in the UHP]
−∫
C+R
f (z) dz . (17.6)
To deal with the integral over C+R , first note that this curve is parameterized by
z = z(θ) = Reiθ where 0 ≤ θ ≤ π .
So “ dz = d[Reiθ ] = i Reiθ dθ ” and
∫
C+R
f (z) dz =∫ π
0
f(
Reiθ)
i Reiθ dθ .
Now assume assumption (3a) holds, and let
M(R) = the maximum of |z f (z)| when |z| = R .
By definition, then,∣∣Reiθ f
(
Reiθ)∣∣ ≤ M(R)
Moreover, it is easily verified that assumption (3a) (that z f (z) → 0 as |z| → ∞ ) implies that
M(R) → 0 as R → ∞ .
So
limR→∞
∣∣∣∣∣
∫
C+R
f (z) dz
∣∣∣∣∣
= limR→∞
∣∣∣∣
∫ π
0
f(
Reiθ)
i Reiθ dθ
∣∣∣∣
≤ limR→∞
∫ π
0
∣∣ f
(
Reiθ)
i Reiθ∣∣ dθ
≤ limR→∞
∫ π
0
M(R) dθ = limR→∞
M(R) π = 0 .
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Chapter & Page: 17–8 Residue Theory
Thus,
limR→∞
∫
C+R
f (z) dz = 0 . (17.7)
Under assumption (3b),∫
C+R
f (z) dz =∫
C+R
g(z)eiαz dz
for some α > 0 . Now,∣∣eiαz
∣∣ =
∣∣eiα(x+iy)
∣∣ =
∣∣eiαxe−αy
∣∣ = e−αy ,
which goes to zero very quickly as y → ∞ . So it is certainly reasonable to expect equation
(17.7) to hold under assumption (3b). And it does — but a rigorous verification requires a little
more space than is appropriate here. Anyone interested can find the details in the proof of lemma
17.2 on page 17–14.
So, after letting R → ∞ , formula (17.6) for the integral of f (x) on (−R, R) becomes∫ ∞
−∞f (x) dx = i2π
[
sum of the residues of f in the UHP]
. (17.8)
On the other hand, if it is assumption (3c) that holds, then∫
C+R
f (z) dz =∫
C+R
g(z)e−iαz dz
for some α > 0 . In this case, though,∣∣e−iαz
∣∣ =
∣∣e−iα(x+iy)
∣∣ =
∣∣e−iαxeαy
∣∣ = eαy ,
which rapidly blows up as y gets large. So it is not reasonable to expect equation (17.7) to hold
here. Instead, take
ΓR = Γ −R = −IR + C
−R
where C−R is the semicirle in the lower half plane of radius R and centered at 0 (see figure
17.1b). (Observe that, this time, the direction of travel on IR is opposite to the normal ‘positive’
direction of travel on the X–axis.) Again, since we are letting R → ∞ and there are only
finitely many singularities, we can assume that we’ve taken R large enough for Γ −R to enclose
all the singularities of f in the lower half plane (LHP). Then
i2π[
sum of the residues of f in the LHP]
=∫
Γ −R
f (z) dz
=∫
−IR
f (z) dz +∫
C−R
f (z) dz
= −∫
IR
f (z) dz +∫
C−R
f (z) dz .
Thus,
∫ R
−R
f (x) dx = −∫
−IR
f (z) dz
= −i2π[
sum of the residues of f in the UHP]
+∫
C+R
f (z) dz
“Simple” Applications Chapter & Page: 17–9
But, as before, it can be shown that | f (z)| → 0 fast enough as |z| → ∞ (on the lower half
plane) to ensure that
limR→∞
∫
C−R
f (z) dz = 0 .
So, after letting R → ∞ , the above formula for the integral of f (x) on (−R, R) becomes
∫ ∞
−∞f (x) dx = −i2π
[
sum of the residues of f in the LHP]
. (17.9)
!◮Example 17.2: Let us evaluate the “Fourier integral”
∫ ∞
−∞
ei2πx
1 + x2dx .
Here we have
f (z) = g(z)ei2π z with g(z) = 1
1 + z2.
Clearly, |g(z)| → 0 as |z| → ∞ and 2π > 0 ; so condition (3b) on page 17–6 holds. Thus,
we will be applying equation (17.8), which requires the residues of f from the upper half
plane.1 By inspection, we see that
f (z) = ei2π z
1 + z2= ei2π z
(z + i)(z − i),
which tells us that the only singularities of f (z) are at z = i and z = −i . Only i , though,
is in the upper half plane, so we are only interested in the residue at i . Since we can write
f (z) as
f (z) = h(z)
z − iwith h(z) = ei2π z
z + i
(and h(i) 6= 0 ), we know f (z) has a simple pole at i and
Resi [ f ] = h(i) = ei2π i
i + i= e−2π
2i.
Thus, applying equation (17.8)
∫ ∞
−∞
ei2πx
1 + x2dx = i2π
[
sum of the residues of f in the UHP]
= i2π[
Resi [ f ]]
= i2π
[
e−2π
2i
]
= πe−2π .
1 Rather than memorize that “condition (3b) implies that (17.8) is used”, keep in mind the derivation and the fact that
you want the integral over one of the semicircles to vanish as R → ∞ . Write out the exponential in terms of x and
y and see if this exponential is vanishing as y → +∞ or as y → −∞ . Then “rederive” the residue-based formula
for the integral of interest using the semicircle in the upper half plane if the exponential vanishes as y → +∞ and
the semicircle in the lower half plane if the exponential vanishes as y → −∞ . For our example
ei2π z = ei2π(x+iy) = ei2π e−2πy → 0 as y → +∞ .
So we are using the upper half plane.
version: 3/10/2014
Chapter & Page: 17–10 Residue Theory
Standard “Simple” TricksUsing Real and Imaginary Parts
It is often helpful to observe that one integral of interest may be the real or imaginary part of
another integral that may, possibly, be easier to evaluate. Do remember that
∫ b
a
Re[ f (x)] dx = Re
[∫ b
a
f (x) dx
]
and
∫ b
a
Im[ f (x)] dx = Im
[∫ b
a
f (x) dx
]
.
(If this isn’t obvious, spend a minute to (re)derive it.) In this regard, it is especially useful to
observe that
cos(X) = Re[
ei X]
and sin(X) = Im[
ei X]
.
!◮Example 17.3: Consider∫ ∞
−∞
cos(2πx)
1 + x2dx
Using the above observations and our answer from the previous exercise,
∫ ∞
−∞
cos(2πx)
1 + x2dx =
∫ ∞
−∞Re
[
ei2πx
1 + x2
]
dx
= Re
[∫ ∞
−∞
ei2πx
1 + x2dx
]
= Re[
πe−2π]
= πe−2π .
Clever Choice of Curve and Function
The main “trick” to applying residue theory in computing integrals of real interest (i.e., integrals
that actually do arise in applications) as well as other integrals you may encounter (e.g., other
integrals in assigned homework and tests) is to make clever choices for the functions and the
curves so that you really can extract the value of desired integral from the integral over the closed
curve used. Some suggestions, such as were given on page 17–6 for computing certain integrals
on (−∞,∞) , can be given. In general, though, choosing the right curves and functions is a
cross between an art and a skill that one just has to develop.
A good example of using both clever choices of functions and curves, and in using real and
imaginary parts, is given in computing the Fresnel integrals (which arise in optics).
!◮Example 17.4: Consider the Fresnel integrals
∫ ∞
0
cos(
x2)
dx and
∫ ∞
0
sin(
x2)
dx .
Rather than use cos(
x2)
and sin(
x2)
directly, it is clever to use ei x2and the fact that
ei x2 = cos(
x2)
+ i sin(
x2)
.
So, if we can evaluate∫ ∞
0
ei x2
dx ,
“Simple” Applications Chapter & Page: 17–11
R
Reiπ/4
AR
lRΓR
π/4
X
Y
IR0
Figure 17.2: The closed curve for computing the Fresnel integrals.
then we can get the two integrals we want via
∫ ∞
0
cos(
x2)
dx =∫ ∞
0
Re[
ei x2]
dx = Re
[ ∫ ∞
0
ei x2
dx
]
and∫ ∞
0
sin(
x2)
dx =∫ ∞
0
Im[
ei x2]
dx = Im
[∫ ∞
0
ei x2
dx
]
.
Now, to compute∫ ∞
0
ei x2
dx ,
we naturally choose the function f (z) = ei z2. The clever choice for the closed curve is
sketched in figure 17.2. It is
ΓR = IR + AR − lR
where, for any choice of R > 0 ,
IR = the straight line on the real line from z = 0 to z = R
= the interval on R from x = 0 to x = R ,
AR = the circular arc centered at 0 starting at z = R and going to z = Reiπ/4 ,
and
lR = the straight line from z = 0 to z = Reiπ/4 .
Notice that the chosen function, f (z) = ei z2, is analytic on the entire complex plane. So
there are no residues, and, for each R > 0 , we have
0 =∮
ΓR
ei z2
dz =∫
IR
ei z2
dz +∫
AR
ei z2
dz −∫
lR
ei z2
dz
=∫ R
x=0
ei x2
dx +∫
AR
ei z2
dz −∫
lR
ei z2
dz .
So,∫ R
x=0
ei x2
dx =∫
lR
ei z2
dz −∫
AR
ei z2
dz .
Letting R → ∞ , this becomes∫ ∞
0
ei x2
dx = limR→∞
∫
lR
ei z2
dz − limR→∞
∫
AR
ei z2
dz . (17.10)
version: 3/10/2014
Chapter & Page: 17–12 Residue Theory
Now, lR is parameterized by
z = z(r) = reiπ/4 with r going from 0 to R .
Using this,
i z2 = ir 2eiπ/2 = ir 2i = −r 2 ,
dz = d[
reiπ/4]
= eiπ/4 dr
and
limR→∞
∫
lR
ei z2
dz = limR→∞
∫ R
0
e−r2
eiπ/4 dr = eiπ/4
∫ ∞
0
e−r2
dr .
The value of the last integral is√
π/2 . This can be found by cheap tricks not involving
residues (see the appendix on the integral of the basic Gaussian starting on page 17–16).
Thus,
limR→∞
∫
lR
ei z2
dz = eiπ/4
√π
2. (17.11)
The curve AR for the other integral in equation (17.10) is parameterized by
z = z(θ) = Reiθ with θ going from 0 to π/4 .
Using this,
i z2 = i R2ei2θ = i R2[cos(2θ) + i sin(2θ)] = i R2 cos(2θ) − R2 sin(2θ) ,
dz = d[
Reiθ]
= i Reiθ dθ
and
limR→∞
∫
AR
ei z2
dz = limR→∞
∫ π/4
0
ei R2 cos(2θ) − R2 sin(2θ)i Reiθ dθ
= limR→∞
∫ π/4
0
ei R2 cos(2θ)e−R2 sin(2θ)i Reiθ dθ .
Note that
∣∣∣∣
∫ π/4
0
ei R2 cos(2θ)e−R2 sin(2θ)i Reiθ dθ
∣∣∣∣
≤∫ π/4
0
∣∣∣e
i R2 cos(2θ)e−R2 sin(2θ)i Reiθ∣∣∣ dθ
=∫ π/4
0
e−R2 sin(2θ) R dθ .
For each θ in (0, π/4] , the exponential in the last integral goes to zero much faster than R
increases. So this integral “clearly” vanishes as R → ∞ (see exercise 17.2 on page 17–16
for a rigorous verification). Thus,
limR→∞
∫
AR
ei z2
dz = 0 . (17.12)
“Simple” Applications Chapter & Page: 17–13
Gathering together what we’ve derived (equations (17.10), (17.11) and (17.12)), we finally
get
∫ ∞
0
ei x2
dx = limR→∞
∫
lR
ei z2
dz − limR→∞
∫
AR
ei z2
dz
= eiπ/4
√π
2+ 0
=[
cos(
π
4
)
+ i sin(
π
4
)] √π
2
=[
1√
2+ i
1√
2
] √π
2= 1
2
√
π
2+ i
1
2
√
π
2.
Consequently,
∫ ∞
0
cos(
x2)
dx = Re
[ ∫ ∞
0
ei x2
dx
]
= 1
2
√
π
2
and∫ ∞
0
sin(
x2)
dx = Im
[∫ ∞
0
ei x2
dx
]
= 1
2
√
π
2.
?◮Exercise 17.1: Consider the last example.
a: Why was the curve ΓR = IR + AR − lR , as illustrated in figure 17.2, such a clever
choice of curves? (Consider what happened to the function ei z2on IR .)
b: Why would the curve ΓR = IR + C+R illustrated in figure 17.1a not be a clever choice
for computing the Fresnel integrals?
In the last example, as in a previous example, the closed curve constructed so the residue
theorem can be applied included a piece of a circle of radius R , say, the AR in our last example.
Our formula for the desired integral then includes a term of the form
limR→∞
∫
AR
f (z) dz ,
and it is important that this limit be zero (or some other computable finite number). Otherwise,
either this general approach fall apart, or we can show that the desired integral does not converge.
Keep in mind that the length of AR increases as R increases; so the vanishing of the
integrand is not enough to ensure the vanishing of the above limit. To take into account the
increasing length of the curve, it is often a good idea to parameterize it using angular measurement,
z = z(θ) = Reiθ
with θ limited to some fixed interval (α, β) . Then, as we’ve seen in examples,
∣∣∣∣
∫
AR
f (z) dz
∣∣∣∣
=∣∣∣∣
∫ β
α
f(
Reiθ)
i Reiθ dθ
∣∣∣∣
≤∫ β
α
∣∣ f
(
Reiθ)
i Reiθ∣∣ dθ =
∫ β
α
∣∣ f
(
Reiθ)∣∣ R dθ .
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Chapter & Page: 17–14 Residue Theory
With luck, you will be able to look at the inside of the last integral and tell whether the desired
limit is zero or whether the limit does not exist.
Of course, you may ask, why not just compute the limit by bringing the limit inside the
integral,
limR→∞
∫ β
α
∣∣ f
(
Reiθ)∣∣ R dθ =
∫ β
α
limR→∞
∣∣ f
(
Reiθ)∣∣ R dθ ?
Because this last equation is not always valid. There are choices for f such that
limR→∞
∫ β
α
∣∣ f
(
Reiθ)∣∣ R dθ 6=
∫ β
α
limR→∞
∣∣ f
(
Reiθ)∣∣ R dθ !
For example, f (z) = e−z with (α, β) = (0, π/2) .
Appendices to this Section
Two issues are addressed here. One is how to rigorously verify that certain integrals over arcs of
radii R vanish as R goes to infinity. The other is how to evaluate∫ ∞
0e−s2
ds . These appendices
are included for the sake of completeness. You should be acquainted with the general results,
but don’t worry about reproducing the sort of analysis given here.
The Vanishing of Certain Integrals as R → ∞
Often, when using residues to compute integrals with exponentials, it is necessary to verify that
certain integrals over arcs of radii R vanish as R goes to infinity. Sometimes this is easy to show;
sometimes it is not. To illustrate how we might verify the cases involving complex exponentials,
we will rigorously verify the following lemma. It is the lemma needed to rigorously verify
equation (17.7) on page 17–8.
Lemma 17.2
Let α > 0 , and assume g is any “reasonably smooth” function on the complex plane2 satisfying
g(z) → 0 as |z| → ∞ .
Then
limR→∞
∫
C+R
g(z)eiαz dz = 0
where C+R is the upper half of the circle of radius R about the origin.
PROOF: Keep in mind that C+R is getting longer as R gets larger. So the simple fact that
the integrand gets smaller as R gets larger is not enough to ensure that the above limit is zero.
To help take into account the increasing length of the curve and to help convert the integral
to something a little more easily to deal with, we make use of the fact that this curve can be
parameterized by
z = z(θ) = Reiθ = R [cos(θ) + i sin(θ)] where 0 ≤ θ ≤ π .
For convenience, let
M(R) = maximum value of g(z(θ)) when 0 ≤ θ ≤ π .
2 it suffices to assume g(z) is continuous on the region where |z| > R0 for some finite real value R0 .
“Simple” Applications Chapter & Page: 17–15
ππ2
1
Θ
Y
y = 12θ
y = sin(θ)
Figure 17.3: The graphs of y = sin(θ) and y = θ/2 on the interval [0, π/2] .
It isn’t hard to show that the fact that g(z) → 0 as |z| → ∞ implies that
M(R) → 0 when R → ∞ .
Also note that that, using the above parametrization,
dz = i Reiθ dθ
and
∣∣eiαz(θ)
∣∣ =
∣∣eiαR[cos(θ)+i sin(θ)]∣∣ =
∣∣eiαR cos(θ) − αR sin(θ)
∣∣
=∣∣eiαR cos(θ)e−αR sin(θ)
∣∣ = e−αR sin(θ) .
So, ∣∣∣∣∣
∫
C+R
g(z)eiαz dz
∣∣∣∣∣
=∣∣∣∣
∫ π
0
g(z(θ))eiαz(θ)i Reiθ dθ
∣∣∣∣
≤∫ π
0
∣∣g(z(θ))eiαz(θ)i Reiθ
∣∣ dθ ≤
∫ π
0
M(R)e−αR sin(θ) R dθ .
Pulling out M(R) and using the fact that sin(θ) is symmetric about θ = π/2 , this becomes
∣∣∣∣∣
∫
C+R
g(z)eiαz dz
∣∣∣∣∣
≤ 2M(R)
∫ π/2
0
e−αR sin(θ) R dθ . (17.13)
Now, it is easy to see and easy to confirm that
sin(θ) >θ
2for 0 ≤ θ ≤ π
2(17.14)
(see figure 17.3). It then follows that
e−αR sin(θ) < e−αRθ/2 for 0 ≤ θ ≤ π
2. (17.15)
Plugging this into inequality (17.13) and computing the resulting integral yields
∣∣∣∣∣
∫
C+R
g(z)eiαz dz
∣∣∣∣∣
≤ 2M(R)
∫ π/2
0
e−αRθ/2 R dθ
= 4M(R)
α
[
1 − e−αRπ/4]
<4M(R)
α.
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Chapter & Page: 17–16 Residue Theory
Thus,
limR→∞
∣∣∣∣∣
∫
C+R
g(z)eiαz dz
∣∣∣∣∣
< limR→∞
4M(R)
α= 4 · 0
α= 0 ,
which, of course, means that
limR→∞
∫
C+R
g(z)eiαz dz = 0 ,
as claimed.
?◮Exercise 17.2: Using the ideas behind inequalities (17.14) and (17.15), show that
∫ π/4
0
e−R2 sin(2θ) R dθ ≤∫ π/4
0
e−R2θ R dθ = 1
R→ 0 as R → ∞ .
Integral of the Basic Gaussian
While there is no simple formula for the indefinite integral of e−x2, the value of the definite
integral∫ ∞
0
e−s2
ds
is easily computed via a clever trick.
To begin, observe that, by symmetry,
∫ ∞
0
e−s2
ds = 1
2I
where
I =∫ ∞
−∞e−s2
ds =∫ ∞
−∞e−x2
dx =∫ ∞
−∞e−y2
dy .
The “clever trick” is based on the observation that I 2 , the product of I with itself, can be
expressed as a double integral over the entire XY –plane,
I 2 =(∫ ∞
−∞e−x2
dx
)(∫ ∞
−∞e−y2
dy
)
=∫ ∞
−∞
(∫ ∞
−∞e−x2
dx
)
e−y2
dy
=∫ ∞
−∞
(∫ ∞
−∞e−x2
e−y2
dx
)
dy =∫ ∞
−∞
∫ ∞
−∞e−(x2+y2) dx dy .
This double integral is easily computed using polar coordinates (r, θ) where
x = r cos(θ) and y = r sin(θ) .
Recall that
x2 + y2 = r 2 and dx dy = r dr dθ .
Integrals Over Branch Cuts of Multi-Valued Functions Chapter & Page: 17–17
So, converting to polar coordinates and using elementary integration techniques,
I 2 =∫ 2π
0
∫ ∞
0
e−r2
r dr dθ =∫ 2π
0
1
2dθ = π .
Taking the square root gives
I = ±√
π .
Because e−s2is a positive function, its integral must be positive. Thus,
∫ ∞
−∞e−s2
ds = I =√
π
and ∫ ∞
0
e−s2
ds = 1
2I = 1
2
√π .
17.3 Integrals Over Branch Cuts of Multi-ValuedFunctions
If you need to compute∫ β
α
f (x) dx
when f (z) is a multi-valued function, then a clever choice for the closed curve may include
using the interval (α, β) as a branch cut for f , and letting it serve as two parts of the curve
enclosing the residues of f . Just what I mean will be a lot clearer if we do one example.
But first, go back and re-read the discussion of multi-valued functions starting on page 14–5.
Also, re-read the bit about Square Roots and Such starting on page 14–8.
!◮Example 17.5: Consider evaluating
∫ ∞
0
√x
1 + x2dx .
Naturally, we will let
f (z) = z1/2
1 + z2.
This function has singularities at z = ±i (where the denominator is zero). Also, because of
the z1/2 factor, f (z) is multi-valued with a branch point at z = 0 . We will cleverly take the
cut line to be the positive X–axis, and define z1/2 to be given by
z1/2 =
√
|z| eiθ/2
where θ is the polar angle (argument) of z satisfying
0 < θ < 2π .
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Chapter & Page: 17–18 Residue Theory
i
−i
CR
Cε
X
Y
I+εR
I−εR
RR
ΓεR
ΓεRΓεR
ε
Figure 17.4: The curves for example 17.5. The region enclosed by ΓεR is shaded, and the
arrows indicate the direction of travel along ΓεR .
This defines the branch of f that we will use. This also means that we will have to be careful
about the value of this function on the positive X–axis since, for each point x0 on this interval,
limz→x0
Im z>0
z1/2 = lim
θ→0+
√x0 eiθ/2 =
√x0 ei ·0 =
√x0
while
limz→x0
Im z<0
z1/2 = lim
θ→2π−
√x0 eiθ/2 =
√x0 eiπ = −
√x0 .
Thus,
limz→x0
Im z>0
f (z) = limz→x0
Im z>0
z1/2
1 + z2=
√x0
1 + x02
while
limz→x0
Im z<0
f (z) = limz→x0
Im z>0
z1/2
1 + z2= −
√x0
1 + x02
.
The closed curve we will use with the residue theorem is constructed from the curves in
figure 17.4. For each pair of positive values ε and R satisfying ε < 1 < R , we let ΓεR be
the closed curve given by
ΓεR = I+εR + CR + I
−εR − Cε
where, as indicated in figure 17.4,
CR = circle of radius R about 0 , oriented counterclockwise ,
Cε = circle of radius ε about 0 , oriented counterclockwise ,
I+εR = straight line on the X–axis from x = ε to x = R ,
and
I−εR = straight line on the X–axis from x = R to x = ε .
Integrals Over Branch Cuts of Multi-Valued Functions Chapter & Page: 17–19
The circle Cε isolates the branch point from the region encircled by ΓεR . This is a good
idea because “bad things” can happen near branch points. Later, we will check that we can
(or cannot) let ε → 0 .
The curves I+εR and I
−εR are each, in fact, simply the subinterval (ε, R) on the X–axis
oriented in opposing directions. They are treated, however, as two distinct pieces of ΓεR ,
with I+εR viewed as a lower boundary to the region above it, and I
−εR viewed as the upper
boundary to the region below it. Indeed, it may be best to first view them as laying a small
distance above and below the X–axis, exactly as sketched in figure 17.4, with this distance
shrunk to zero by the end of the computations. Consequently, the region encircled by ΓεR is
the shaded region in figure 17.4.
Applying the residue theorem, we have
i2π × [sum of the resides of f in the shaded region]
=∮
Γ
f (z) dz
=∫
I+εR
f (z) dz +∫
CR
f (z) dz +∫
I−εR
f (z) dz −∫
Cε
f (z) dz
where∫
I+εR
f (z) dz =∫ R
ε
[
limz→x
Im z>0
f (z)
]
dx =∫ R
ε
√x
1 + x2dx
and
∫
I−εR
f (z) dz =∫ ε
R
[
limz→x
Im z<0
f (z)
]
dx = −∫ R
ε
[
−√
x
1 + x2
]
dx =∫ R
ε
√x
1 + x2dx .
Thus,
i2π × [sum of the resides of f in the shaded region]
= 2
∫ R
ε
√x
1 + x2dx +
∫
CR
f (z) dz −∫
Cε
f (z) dz ,
and, so,
∫ R
ε
√x
1 + x2dx = iπ × [sum of the resides of f in the shaded region]
+ 1
2
∫
Cε
f (z) dz − 1
2
∫
CR
f (z) dz .
(17.16)
(Notice that, because of the multi-valueness of f ,∫
I+εR
f (z) dz and
∫
I−εR
f (z) dz
did not cancel each other out even though I+εR and I
−εR are the same curve oriented in opposite
directions. That is what will make these computations work.)
Naturally, we want ε → 0 and R → ∞ . Now, if |z| = ε and ε < 1 , then
| f (z)| =
∣∣∣∣∣
z1/2
1 + z2
∣∣∣∣∣
≤ ε1/2
1 − ε2.
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Chapter & Page: 17–20 Residue Theory
So, using the parametrization z = z(θ) = εeiθ ,
∣∣∣∣
∫
Cε
f (z) dz
∣∣∣∣
=∣∣∣∣
∫ 2π
0
f (z(θ)) iεeiθ dθ
∣∣∣∣
≤∫ 2π
0
| f (z(θ))| ε dθ
≤∫ 2π
0
ε1/2
1 − ε2ε dθ = ε
3/2
1 − ε22π → 0 as ε → 0 .
By a very similar computations, we get (since R > 1 ),
∣∣∣∣
∫
CR
f (z) dz
∣∣∣∣
≤∫ 2π
0
R1/2
R2 − 1R dθ = R
3/2
R2 − 12π → 0 as R → ∞ .
Thus, after letting ε → 0 and R → ∞ , equation (17.16) reduces to
∫ ∞
0
√x
1 + x2dx = iπ × [sum of the resides of f in the shaded region] (17.17)
Clearly, the only singularities of
f (z) = z1/2
1 + z2
are at z = ±i , and
f (z) = g(z)
z − iand f (z) = h(z)
z − (−i).
where
g(z) = z1/2
z + iand h(z) = z
1/2
z − i.
So these singularities are simple poles, and
∫ ∞
0
√x
1 + x2dx = iπ × [sum of the resides of f in the shaded region]
= iπ[
Resi ( f ) + Res−i ( f )]
= iπ[g(i) + h(−i)]
= iπ
[
i1/2
i + i+ (−i)
1/2
−i − i
]
= π
[
eiπ/4
2− ei3π/4
2
]
= · · · = π√
2.
Integrating Through Poles and the Cauchy Principal Value Chapter & Page: 17–21
17.4 Integrating Through Poles and the CauchyPrincipal Value
The Cauchy Principal Value of an Integral
The “Cauchy principal value” of an integral is actually a redefinition of the integral so that certain
types of singularities are ignored through the process of taking “symmetric limits”.
To be precise: Let f be a function on an interval (a, b) and assume f is “well behaved”
(say, is analytic) at every point in the interval except one, x0 . We then define
The Cauchy principal value of
∫ b
a
f (x) dx = CPV
∫ b
a
f (x) dx (my notation)
= limε→0+
[∫ x0−ε
a
f (x) dx +∫ b
x0+ε
f (x) dx
]
.
By the way, in Arfken, Weber and Harris you’ll find
P
∫ b
a
f (x) dx and −∫ b
a
f (x) dx
denoting the above Cauchy principal value.
If f is continuous, or even just has a jump discontinuity, at x0 , then
CPV
∫ b
a
f (x) dx = limε→0+
[∫ x0−ε
a
f (x) dx +∫ b
x0+ε
f (x) dx
]
=∫ x0
a
f (x) dx +∫ b
x0
f (x) dx
=∫ b
a
f (x) dx .
This simply means that the Cauchy principal value reduces to the integral when the integral is
well defined. However, when the singularity is a simple pole then you can easily verify that the
Cauchy principal value “removes” the effect of the singularity by “canceling out infinities”.3
!◮Example 17.6: Consider
∫ 4
−3
1
xdx and CPV
∫ 4
−3
1
xdx .
The function being integrated has a singularity at x = 0 , and
∫ 4
−3
1
xdx =
∫ 0
−3
1
xdx +
∫ 4
0
1
xdx
= ln |x |∣∣0
−3+ ln |x |
∣∣4
0
= −∞ − ln 3 + ln 4 − (−∞)
3 In practice, make sure you can justify this “canceling out of infinities”. Otherwise, your use of the Cauchy principal
value is probably fraudulent.
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Chapter & Page: 17–22 Residue Theory
= ∞ − ∞ + ln4
3,
which, for very good reasons, is considered to be “undefined”. On the other hand,
CPV
∫ 4
−3
1
xdx = lim
ε→0+
[∫ −ε
−3
1
xdx +
∫ 4
ε
1
xdx
]
= limε→0+
[ln ε − ln 3 + ln 4 − ln ε]
= limε→0+
[− ln 3 + ln 4] = ln4
3.
The Cauchy principal part should not just be thought of as a way to get around “technical
difficulties” with integrals that blow up, and its indiscriminate use can lead to dangerously
misleading results. For example, if the force on some object at position x is given by 1/x , then
one could naively argue that the work needed to move that object from x = −3 to x = 4 is just
work = CPV
∫ 4
−3
1
xdx = ln
4
3,
but just see if you really can push that object past x = 0 .
The Cauchy principal value can be useful, but its use must be justified by something more
than a desire to “cancel out infinities”. Also, sometimes those infinities just don’t cancel out.
?◮Exercise 17.3: Let n be any positive integer, and show that
CPV
∫ 2
−2
1
xndx =
{
0 if n is odd
∞ if n is even.
By the way, if the integrand has several singularities on the interval (a, b) , then the Cauchy
principal value is computed by taking the above described symmetric limit at each singularity.
Also, if (a, b) = (−∞,∞) , then we have the “improper integral version of the Cauchy
principal value” given by
CPV
∫ ∞
−∞f (x) dx = lim
R→+∞
∫ R
R
f (x) dx .
Again, its use must be justified.
Linearity of the Cauchy Principal Value
It should be briefly noted that, like the regular integral, the Cauchy principal value is linear. To
be precise, you can easily show using the linearity of integrals and limits that, if at least two of
the following exist as finite numbers,
CPV
∫ b
a
f (x) dx , CPV
∫ b
a
g(x) dx and CPV
∫ b
a
[ f (x) + g(x)] dx ,
then they all exist and, moreover,
CPV
∫ b
a
[α f (x) + βg(x)] dx = α CPV
∫ b
a
f (x) dx + β CPV
∫ b
a
g(x) dx .
Integrating Through Poles and the Cauchy Principal Value Chapter & Page: 17–23
for any pair of numbers α and β . In particular, if the principal values exist, we have
CPV
∫ b
a
[u(x) + iv(x)] dx = CPV
∫ b
a
u(x) dx + i CPV
∫ b
a
v(x) dx .
From this it almost immediately follows that
Re
[
CPV
∫ b
a
f (x) dx
]
= CPV
∫ b
a
Re[ f (x)] dx
and
Im
[
CPV
∫ b
a
f (x) dx
]
= CPV
∫ b
a
Im[ f (x)] dx .
We will be using this later.
Integrating Through Singularities
Sometimes the function you are integrating has a singularity at a point on the curve over which
you must integrate the function. When this happens, there are at least three things you can do to
deal with this integral:
1. Use the Cauchy principal value.
2. Modify the function slightly to move the singularity off the curve by a distance of ε and
then see what happens as ε → 0 .
3. Look very closely at your problem and decide if the fact that this integral doesn’t really
exist is telling you something about the physics of the problem.
If you do either the first or the second of the above, then make sure you can justify your choice
by something other than “I don’t like infinities.” This means that you often should do the third
thing in the above list to help justify the your choice.
Simple Poles and the Cauchy Principal Value
The discussion starting on page 17–6 concerning integrals of the form
∫ ∞
−∞f (x) dx ,
can be easily adapted to take into account the possibility of f having simple poles on the X–
axis. What we will discover is that we don’t actually end up with the above integral, but with the
Cauchy principal value of that integral.
So let’s consider evaluating the above integral assuming that:
1. Except for a finite number of poles and/or essential singularities, f is a single-valued
analytic function on the entire complex plane.
2. Only one of these singularities is on the X–axis, and that is a simple pole at x0
3. One of the following holds:
(a) z f (z) → 0 as |z| → ∞ .
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Chapter & Page: 17–24 Residue Theory
(a)(a) (b)
RR RR −R−R −R−R
C+RC
+R
C+ε
C−ε
x0x0 XX
YY YY
Figure 17.5: Isolating a pole on the X–axis. (Compare with figure 17.1a on page 17–6.)
(b) f (z) = g(z)eiαz where α > 0 and
g(z) → 0 as |z| → ∞ .
The computations here differ from those on page 17–6 only in that we initially isolate the pole
at x0 by a circle of radius ε centered at x0 (oriented counterclockwise) with, unsurprisingly,
C+ε and C
−ε denoting the upper and lower halves (see figures 17.5a and 17.5b). This also means
that, instead of using the entire interval (−R, R) in the closed curve for the residue theorem,
we use the subintervals (−R, x0 − ε) and (−R, x0 − ε) along with either C+ε or C
−ε (possibly
re-oriented). For these calculations, C+ε will be used, as in figure 17.5a.
Take R large enough and ε small enough that the closed curve figure 17.5a encircles all
the singularities of f in the upper half plane (UHP).
The residue theorem then tells us that
i2π[
sum of the residues of f in the UHP]
=∫ x0−ε
−R
f (x) dx −∫
C+ε
f (z) dz +∫ R
x0+ε
f (x) dx +∫
C+R
f (z) dz .
By the same arguments as given before, the integral over C+R shrinks to 0 as R → ∞ . So, after
letting R → ∞ , the last equation becomes
i2π[
sum of the residues of f in the UHP]
=∫ x0−ε
−∞f (x) dx −
∫
C+ε
f (z) dz +∫ ∞
x0+ε
f (x) dx .
Hence,∫ x0−ε
−∞f (x) dx +
∫ ∞
x0+ε
f (x) dx = i2π[
sum of the residues of f in the UHP]
+∫
C+ε
f (z) dz .
Letting ε → 0 this becomes
CPV
∫ ∞
−∞f (x) dx = i2π
[
sum of the residues of f in the UHP]
+ limε→0+
∫
C+ε
f (z) dz .
(17.18)
Integrating Through Poles and the Cauchy Principal Value Chapter & Page: 17–25
On C+ε we can use the Laurent series for f around x0 , along with the parametrization
z = z(θ) = x0 + εeiθ with 0 ≤ θ ≤ π .
Since f is assumed to have a simple pole at x0 , the Laurent series will be of the form
f (z(θ)) =∞
∑
k=−1
ckzk =∞
∑
k=−1
ckεkeikθ = c−1ε
−1e−iθ +∞
∑
k=0
ckεkeikθ
Remember c−1 = Resx0( f ) . So
limε→0+
∫
C+ε
f (z) dz = limε→0+
∫ π
0
[
c−1ε−1e−iθ +
∞∑
k=0
ckεkeikθ
]
iεeiθ dθ︸ ︷︷ ︸
dz
= limε→0+
∫ π
0
[
c−1i +∞
∑
k=0
ckiεk+1ei(k+1)θ
]
dθ
= limε→0+
[
ic−1
∫ π
0
dθ +∞
∑
k=0
ickεk+1
∫ π
0
ei(k+1)θ dθ
]
= limε→0+
[
ic−1π +∞
∑
k=0
ickεk+1
∫ π
0
ei(k+1)θ dθ
]
= limε→0+
[
ic−1π +∞
∑
k=0
ckεk+1 1
k + 1
(
ei(k+1)π − 1)
]
= c−1iπ +∞
∑
k=0
ck · 0k+1 1
k + 1
(
ei(k+1)π − 1)
= iπ Resx0( f ) .
Plugging this result back into equation 17.18 gives us
CPV
∫ ∞
−∞f (x) dx
= i2π[
sum of the residues of f in the UHP]
+ iπ Resx0( f ) .
(17.19)
This last equation was derived using the curves in figure 17.5a. It turns out that you get
exactly the same result using the curves in figure 17.5b. You should verify this yourself.
If there is more than one simple pole on the X–axis, then the residue of each contributes,
and we have the following result:
Lemma 17.3
Let f be a function which, except for a finite number of poles and/or essential singularities,
is a single-valued analytic function on the entire complex plane. Assume, further, that the
singularities on the X–axis are all simple poles, and that either
1. z f (z) → 0 as |z| → ∞ .
version: 3/10/2014
Chapter & Page: 17–26 Residue Theory
2. or f (z) = g(z)eiαz where α > 0 and
g(z) → 0 as |z| → ∞ .
Then
CPV
∫ ∞
−∞f (x) dx = i2π
[
sum of the residues of f in the UHP]
+ iπ[
sum of the residues of f in the X–axis]
.
(17.20)
Deriving the corresponding results when we use the residues in the lower half plane (LHP)
will be left to you.
?◮Exercise 17.4: Assume the following concerning some function f on C :
1. Except for a finite number of poles and/or essential singularities, f is a single-valued
analytic function on the entire complex plane.
2. All the singularities on the X–axis are simple poles.
and
3. f (z) = g(z)e−iαz where α > 0 and
g(z) → 0 as |z| → ∞ .
Then show that
CPV
∫ ∞
−∞f (x) dx = −i2π
[
sum of the residues of f in the LHP]
− iπ[
sum of the residues of f in the X–axis]
.
(17.21)
What if the singularities on the X–axis are not simple poles — say, poles of higher order or
essential singularities? The computations done above can still be attempted, but, in computing
limε→0+
∫
C+ε
f (z) dz
there will generally be terms involving 1/ε that blow up. So, in general, we cannot deal with
singularities worse than simple poles on the X–axis, at least not using the Cauchy principal
value.
Let’s do an example where the Cauchy principal value is of value.
!◮Example 17.7: Consider∫ ∞
−∞
sin(x)
xdx .
The only possible point where we may have a pole in the integrand is at x = z = 0 . In fact,
though, you can easily show that
limz→0
sin(z)
z= 1 .
Integrating Through Poles and the Cauchy Principal Value Chapter & Page: 17–27
So, in fact, sin(z)/z is analytic at z = 0 , the integrand in the above integral is continuous at
x = 0 , and we have ∫ ∞
−∞
sin(x)
xdx = CPV
∫ ∞
−∞
sin(x)
xdx .
This becomes something other than a stupid, obvious observation after also noting that, for
x ∈ R ,sin(x)
x= Im
[
ei x
x
]
and that
f (z) = ei z
z
satisfies the conditions for f in lemma 17.3. Applying that lemma (and noting that the only
residue of f is at z = 0 ), we have
CPV
∫ ∞
−∞
ei x
xdx = iπ Res0
(
ei x
x
)
= iπ Res0
(
ei ·0) = iπ · 1 = iπ .
So, ∫ ∞
−∞
sin(x)
xdx = CPV
∫ ∞
−∞
sin(x)
xdx
= CPV
∫ ∞
−∞Im
[
ei x
x
]
dx
= Im
[
CPV
∫ ∞
−∞
ei x
xdx
]
= Im[iπ] = π .
Moving Singularities
To be written, someday.
Basically, this is the idea of replacing
∫ b
a
g(x)
x − x0
dx
with either
limy0→0+
∫ b
a
g(x)
x − (x0 + y0)dx or lim
y0→0+
∫ b
a
g(x)
x − (x0 − y0)dx .
Again, there must be some justification for this. Be warned that, in general,
CPV
∫ b
a
g(x)
x − x0
dx 6= limy0→0+
∫ b
a
g(x)
x − (x0 + y0)dx
6= limy0→0+
∫ b
a
g(x)
x − (x0 − y0)dx 6= CPV
∫ b
a
g(x)
x − x0
dx .
version: 3/10/2014