Certain Integrals Arising from Ramanujan's Notebooks
In his third notebook, Ramanujan claims the specific. In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if logx were replaced by log²x in the first integral and logx were inserted in the...
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irk-123456789-1471472019-02-14T01:26:29Z Certain Integrals Arising from Ramanujan's Notebooks Berndt, B.C. Straub, A. In his third notebook, Ramanujan claims the specific. In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if logx were replaced by log²x in the first integral and logx were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly relating large classes of infinite integrals and illustrate these by several examples. 2015 Article Certain Integrals Arising from Ramanujan's Notebooks / B.C. Berndt, A. Straub // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 5 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E20 DOI:10.3842/SIGMA.2015.083 http://dspace.nbuv.gov.ua/handle/123456789/147147 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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In his third notebook, Ramanujan claims the specific. In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if logx were replaced by log²x in the first integral and logx were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly relating large classes of infinite integrals and illustrate these by several examples. |
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Berndt, B.C. Straub, A. |
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Berndt, B.C. Straub, A. Certain Integrals Arising from Ramanujan's Notebooks Symmetry, Integrability and Geometry: Methods and Applications |
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Certain Integrals Arising from Ramanujan's Notebooks |
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Certain Integrals Arising from Ramanujan's Notebooks |
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Certain Integrals Arising from Ramanujan's Notebooks |
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Certain Integrals Arising from Ramanujan's Notebooks |
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Certain Integrals Arising from Ramanujan's Notebooks |
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certain integrals arising from ramanujan's notebooks |
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Інститут математики НАН України |
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Certain Integrals Arising from Ramanujan's Notebooks / B.C. Berndt, A. Straub // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 5 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT berndtbc certainintegralsarisingfromramanujansnotebooks AT strauba certainintegralsarisingfromramanujansnotebooks |
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2025-07-11T01:27:40Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 083, 11 pages
Certain Integrals Arising
from Ramanujan’s Notebooks?
Bruce C. BERNDT † and Armin STRAUB ‡
† University of Illinois at Urbana–Champaign, 1409 W Green St, Urbana, IL 61801, USA
E-mail: berndt@illinois.edu
‡ University of South Alabama, 411 University Blvd N, Mobile, AL 36688, USA
E-mail: straub@southalabama.edu
Received September 05, 2015, in final form October 11, 2015; Published online October 14, 2015
http://dx.doi.org/10.3842/SIGMA.2015.083
Abstract. In his third notebook, Ramanujan claims that∫ ∞
0
cos(nx)
x2 + 1
log xdx+
π
2
∫ ∞
0
sin(nx)
x2 + 1
dx = 0.
In a following cryptic line, which only became visible in a recent reproduction of Ramanujan’s
notebooks, Ramanujan indicates that a similar relation exists if log x were replaced by log2 x
in the first integral and log x were inserted in the integrand of the second integral. One of
the goals of the present paper is to prove this claim by contour integration. We further
establish general theorems similarly relating large classes of infinite integrals and illustrate
these by several examples.
Key words: Ramanujan’s notebooks; contour integration; trigonometric integrals
2010 Mathematics Subject Classification: 33E20
1 Introduction
If you attempt to find the values of the integrals∫ ∞
0
cos(nx)
x2 + 1
log x dx and
∫ ∞
0
sin(nx)
x2 + 1
dx, n > 0, (1.1)
by consulting tables such as those of Gradshteyn and Ryzhik [3] or by invoking a computer
algebra system such as Mathematica, you will be disappointed, if you hoped to evaluate these
integrals in closed form, that is, in terms of elementary functions. On the other hand, the latter
integral above can be expressed in terms of the exponential integral Ei(x) [3, formula (3.723),
no. 1]. Similarly, if 1/(x2 + 1) is replaced by any even rational function with the degree of the
denominator at least one greater than the degree of the numerator, it does not seem possible to
evaluate any such integral in closed form.
However, in his third notebook, on p. 391 in the pagination of the second volume of [5],
Ramanujan claims that the two integrals in (1.1) are simple multiples of each other. More
precisely,∫ ∞
0
cos(nx)
x2 + 1
log x dx+
π
2
∫ ∞
0
sin(nx)
x2 + 1
dx = 0. (1.2)
?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica-
tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html
mailto:berndt@illinois.edu
mailto:straub@southalabama.edu
http://dx.doi.org/10.3842/SIGMA.2015.083
http://www.emis.de/journals/SIGMA/OPSFA2015.html
2 B.C. Berndt and A. Straub
Moreover, to the left of this entry, Ramanujan writes, “contour integration”. We now might
recall a couple of sentences of G.H. Hardy from the introduction to Ramanujan’s Collected
papers [4, p. xxx], “. . . he had [before arriving in England] never heard of . . . Cauchy’s theorem,
and had indeed but the vaguest idea of what a function of a complex variable was”. On the
following page, Hardy further wrote, “In a few years’ time he had a very tolerable knowledge of
the theory of functions . . . ”. Generally, the entries in Ramanujan’s notebooks were recorded by
him in approximately the years 1904–1914, prior to his departure for England. However, there is
evidence that some of the entries in his third notebook were recorded while he was in England.
Indeed, in view of Hardy’s remarks above, almost certainly, (1.2) is such an entry. A proof
of (1.2) by contour integration was given by the first author in his book [2, pp. 329–330].
The identity (1.2) is interesting because it relates in a simple way two integrals that we are
unable to individually evaluate in closed form. On the other hand, the simpler integrals∫ ∞
0
cos(nx)
x2 + 1
dx =
πe−n
2
and
∫ ∞
−∞
sin(nx)
x2 + 1
dx = 0
have well-known and trivial evaluations, respectively.
With the use of the most up-to-date photographic techniques, a new edition of Ramanujan’s
Notebooks [5] was published in 2012 to help celebrate the 125th anniversary of Ramanujan’s
birth. The new reproduction is vastly clearer and easier to read than the original edition. When
the first author reexamined (1.2) in the new edition, he was surprised to see that Ramanujan
made a further claim concerning (1.2) that was not visible in the original edition of [5]. In
a cryptic one line, he indicated that a relation similar to (1.2) existed if log x were replaced by
log2 x in the first integral and log x were inserted in the integrand of the second integral of (1.2).
One of the goals of the present paper is to prove (by contour integration) this unintelligible entry
in the first edition of the notebooks [5]. Secondly, we establish general theorems relating large
classes of infinite integrals for which individual evaluations in closed form are not possible by
presently known methods. Several further examples are given.
2 Ramanujan’s extension of (1.2)
We prove the entry on p. 391 of [5] that resurfaced with the new printing of [5].
Theorem 2.1. We have∫ ∞
0
sin(nx)
x2 + 1
dx+
2
π
∫ ∞
0
cos(nx)
x2 + 1
log x dx = 0 (2.1)
and ∫ ∞
0
sin(nx) log x
x2 + 1
dx+
1
π
∫ ∞
0
cos(nx) log2 x
x2 + 1
dx =
π2e−n
8
. (2.2)
Proof. Define a branch of log z by −1
2π < θ = arg z ≤ 3
2π. We integrate
f(z) :=
einz log2 z
z2 + 1
over the positively oriented closed contour CR,ε consisting of the semi-circle CR given by
z = Reiθ, 0 ≤ θ ≤ π, the interval [−R,−ε], the semi-circle Cε given by z = εeiθ, π ≥ θ ≥ 0, and
the interval [ε,R], where 0 < ε < 1 and R > 1. On the interior of CR,ε there is a simple pole
at z = i, and so by the residue theorem,∫
CR,ε
f(z)dz = 2πi
e−n ·
(
−1
4π
2
)
2i
= −e
−nπ3
4
. (2.3)
Certain Integrals Arising from Ramanujan’s Notebooks 3
Parameterizing the respective semi-circles, we can readily show that∫
Cε
f(z)dz = o(1), (2.4)
as ε→ 0, and∫
CR
f(z)dz = o(1), (2.5)
as R→∞. Hence, letting ε→ 0 and R→∞ and combining (2.3)–(2.5), we conclude that
−e
−nπ3
4
=
∫ 0
−∞
einx(log |x|+ iπ)2
x2 + 1
dx+
∫ ∞
0
einx log2 x
x2 + 1
dx (2.6)
=
∫ ∞
0
(cos(nx)− i sin(nx))(log x+ iπ)2
x2 + 1
dx+
∫ ∞
0
(cos(nx) + i sin(nx)) log2 x
x2 + 1
dx.
If we equate real parts in (2.6), we find that
−e
−nπ3
4
=
∫ ∞
0
cos(nx)
(
2 log2 x− π2
)
+ 2π sin(nx) log x
x2 + 1
dx. (2.7)
It is easy to show, e.g., by contour integration, that∫ ∞
0
cos(nx)
x2 + 1
dx =
πe−n
2
. (2.8)
(In his quarterly reports, Ramanujan derived (2.8) by a different method [1, p. 322].) Putting
this evaluation in (2.7), we readily deduce (2.2). If we equate imaginary parts in (2.6), we deduce
that
0 =
∫ ∞
0
π2 sin(nx) + 2π cos(nx) log x
x2 + 1
dx,
from which the identity (2.1) follows. �
3 A second approach to the entry at the top of p. 391
Theorem 3.1. For s ∈ (−1, 2) and n ≥ 0,
π
2
e−n =
∫ ∞
0
cos(nx− πs/2)
x2 + 1
xsdx. (3.1)
Before indicating a proof of Theorem 3.1, let us see how the integral (3.1) implies Ramanujan’s
integral relations (2.1) and (2.2). Essentially, all we have to do is to take derivatives of (3.1) with
respect to s (and interchange the order of differentiation and integration); then, upon setting
s = 0, we deduce (2.1) and (2.2).
First, note that upon setting s = 0 in (3.1), we obtain (2.8). On the other hand, taking
a derivative of (3.1) with respect to s, and then setting s = 0, we find that
0 =
∫ ∞
0
cos(nx)
x2 + 1
log x dx+
π
2
∫ ∞
0
sin(nx)
x2 + 1
dx, (3.2)
4 B.C. Berndt and A. Straub
which is the formula (2.1) that Ramanujan recorded on p. 391. Similarly, taking two derivatives
of (3.1) and then putting s = 0, we arrive at
0 =
∫ ∞
0
cos(nx)
x2 + 1
log2 x dx+ π
∫ ∞
0
sin(nx)
x2 + 1
log x dx− π2
4
∫ ∞
0
cos(nx)
x2 + 1
dx,
which, using (2.8), simplifies to
π3
8
e−n =
∫ ∞
0
cos(nx)
x2 + 1
log2 x dx+ π
∫ ∞
0
sin(nx)
x2 + 1
log x dx. (3.3)
Note that this is Ramanujan’s previously unintelligible formula (2.2). If we likewise take m
derivatives before setting s = 0, we obtain the following general set of relations connecting the
integrals
Im :=
∫ ∞
0
cos(nx)
x2 + 1
logm x dx, Jm :=
∫ ∞
0
sin(nx)
x2 + 1
logm x dx.
Corollary 3.2. For m ≥ 1,
0 =
m∑
k=0
(
m
k
)(π
2
)k
(−1)[k/2]
{
Im−k, if k is even
Jm−k, if k is odd
}
.
We now provide a proof of Theorem 3.1.
Proof. In analogy with our previous proof, we integrate
fs(z) :=
einzzs
z2 + 1
over the contour CR,ε and let ε→ 0 and R→∞. Here, zs = es log z with −1
2π < arg z ≤ 3
2π, as
above. By the residue theorem,∫
CR,ε
fs(z)dz = 2πi
e−neπis/2
2i
= πe−neπis/2. (3.4)
Letting ε → 0 and R → ∞, and using bounds for the integrand on the semi-circles as we did
above, we deduce that
lim
R→∞
ε→0
∫
CR,ε
fs(z)dz =
∫ ∞
−∞
einxxs
x2 + 1
dx =
∫ ∞
0
e−inxxseπis
x2 + 1
dx+
∫ ∞
0
einxxs
x2 + 1
dx. (3.5)
Combining (3.4) and (3.5), we find that
πe−neπis/2 =
∫ ∞
0
(
einx + e−inxeπis
) xs
x2 + 1
dx. (3.6)
We then divide both sides of (3.6) by 2eπis/2 to obtain (3.1). Note that the integrals are
absolutely convergent for s ∈ (−1, 1). By Dirichlet’s test, (3.6) holds for s ∈ (−1, 2). �
Replacing s with s+ 1 in (3.1), we obtain the following companion integral.
Corollary 3.3. For s ∈ (−2, 1) and n ≥ 0,
π
2
e−n =
∫ ∞
0
x sin(nx− πs/2)
x2 + 1
xsdx. (3.7)
Certain Integrals Arising from Ramanujan’s Notebooks 5
Example 3.4. Setting s = 0 in (3.7), we find that
π
2
e−n =
∫ ∞
0
x sin(nx)
x2 + 1
dx, (3.8)
which is well known. After taking one derivative with respect to s in (3.7) and setting s = 0,
we similarly find that
0 =
∫ ∞
0
x sin(nx)
x2 + 1
log x dx− π
2
∫ ∞
0
x cos(nx)
x2 + 1
dx, (3.9)
which may be compared with Ramanujan’s formula (2.1). As a second example, after taking
two derivatives of (3.7) with respect to s, setting s = 0, and using (3.8), we arrive at the identity
π3
8
e−n =
∫ ∞
0
x sin(nx)
x2 + 1
log2 x dx− π
∫ ∞
0
x cos(nx)
x2 + 1
log x dx. (3.10)
We offer a few additional remarks before generalizing our ideas in the next section. Equating
real parts in the identity (3.6) from the proof of Theorem 3.1, we find that
πe−n cos(πs/2) =
∫ ∞
0
(
cos(nx)(1 + cos(πs)) + sin(nx) sin(πs)
) xs
x2 + 1
dx. (3.11)
Setting s = 0 in (3.11), we again obtain (2.8). On the other hand, note that[
d
ds
(
cos(nx)(1 + cos(πs)) + sin(nx) sin(πs)
)]
s=0
= π sin(nx).
Hence, taking a derivative of (3.11) with respect to s, and then setting s = 0, we find that
0 = π
∫ ∞
0
sin(nx)
x2 + 1
dx+ 2
∫ ∞
0
cos(nx)
x2 + 1
log x dx,
which is the formula (2.1) that Ramanujan recorded on p. 391. Similarly, taking two derivatives
of (3.11) and letting s = 0, we deduce that
−π
3
4
e−n = −π2
∫ ∞
0
cos(nx)
x2 + 1
dx+ 2π
∫ ∞
0
sin(nx)
x2 + 1
log x dx+ 2
∫ ∞
0
cos(nx)
x2 + 1
log2 x dx,
which, using (2.8), simplifies to
π3
8
e−n = π
∫ ∞
0
sin(nx)
x2 + 1
log x dx+
∫ ∞
0
cos(nx)
x2 + 1
log2 x dx
which is the formula (2.2) arising from Ramanujan’s unintelligible remark in the initial edition
of [5].
The integral (3.11) has the companion
πe−n sin(πs/2) =
∫ ∞
0
(
cos(nx) sin(πs) + sin(nx)(1− cos(πs))
) xs
x2 + 1
dx, (3.12)
which is obtained by equating imaginary parts in (3.6). However, taking derivatives of (3.12)
with respect to s, and then setting s = 0, does not generate new identities. Instead, we recover
precisely the previous results. For instance, taking a derivative of (3.12) with respect to s, and
then setting s = 0, we again deduce (2.8). Taking two derivatives of (3.12) with respect to s,
and then setting s = 0, we obtain
0 = π2
∫ ∞
0
sin(nx)
x2 + 1
dx+ 2π
∫ ∞
0
cos(nx)
x2 + 1
log x dx,
which is again Ramanujan’s formula (2.1).
6 B.C. Berndt and A. Straub
4 General theorems
The phenomenon observed by Ramanujan in (1.2) can be generalized by replacing the rational
function 1/(z2 + 1) by a general rational function f(z) in which the denominator has degree at
least one greater than the degree of the numerator. We shall also assume that f(z) does not have
any poles on the real axis. We could prove a theorem allowing for poles on the real axis, but in
such instances we would need to consider the principal values of the resulting integrals on the
real axis. In our arguments above, we used the fact that 1/(z2 + 1) is an even function. For our
general theorem, we require that f(z) be either even or odd. For brevity, we let Res(F (z); z0)
denote the residue of a function F (z) at a pole z0. As above, we define a branch of log z by
−1
2π < θ = arg z ≤ 3
2π.
For a rational function f(z) as prescribed above and each nonnegative integer m, define
Im :=
∫ ∞
0
f(x) cosx logm x dx and Jm :=
∫ ∞
0
f(x) sinx logm x dx. (4.1)
Theorem 4.1. Let f(z) denote a rational function in z, as described above, and let Im and Jm
be defined by (4.1). Let
S := 2πi
∑
U
Res(eizf(z) logm z, zj), (4.2)
where the sum is over all poles zj of eizf(z) logm z lying in the upper half-plane U . Suppose
that f(z) is even. Then
S =
m∑
k=0
(
m
k
)
(iπ)m−k(Ik − iJk) + (Im + iJm). (4.3)
Suppose that f(z) is odd. Then
S = −
m∑
k=0
(
m
k
)
(iπ)m−k(Ik − iJk) + (Im + iJm). (4.4)
Observe that (4.3) and (4.4) are recurrence relations that enable us to successively calcu-
late Im and Jm. With each succeeding value of m, we see that two previously non-appearing
integrals arise. If f(z) is even, then these integrals are Im and Jm−1, while if f(z) is odd, these
integrals are Jm and Im−1. The previously non-appearing integrals appear in either the real part
or the imaginary part of the right-hand sides of (4.3) and (4.4), but not both real and imaginary
parts. This fact therefore does not enable us to explicitly determine either of the two integrals.
We must be satisfied with obtaining recurrence relations with increasingly more terms.
Proof. We commence as in the proof of Theorem 2.1. Let CR,ε denote the positively oriented
contour consisting of the semi-circle CR given by z = Reiθ, 0 ≤ θ ≤ π, [−R,−ε], the semi-
circle Cε given by z = εeiθ, π ≥ θ ≥ 0, and [ε,R], where 0 < ε < d, where d is the smallest
modulus of the poles of f(z) in U . We also choose R larger than the moduli of all the poles
of f(z) in U . By the residue theorem,∫
CR,ε
eizf(z) logm z dz = S, (4.5)
where S is defined in (4.2).
Certain Integrals Arising from Ramanujan’s Notebooks 7
We next directly evaluate the integral on the left-hand side of (4.5). As in the proof of
Theorem 2.1, we can easily show that∫
Cε
eizf(z) logm z dz = o(1), (4.6)
as ε tends to 0. Secondly, we estimate the integral over CR. By hypothesis, there exist a positive
constant A and a positive number R0, such that for R ≥ R0, |f(Reiθ)| ≤ A/R. Hence, for
R ≥ R0,∣∣∣∣∫
CR
eizf(z) logm z dz
∣∣∣∣ =
∣∣∣∣∫ π
0
eiRe
iθ
f(Reiθ) logm(Reiθ)iReiθdθ
∣∣∣∣
≤
∫ π
0
e−R sin θ|f(Reiθ)|(logR+ π)mR dθ
≤ A(logR+ π)m
(∫ π/2
0
+
∫ π
π/2
)
e−R sin θdθ. (4.7)
Since sin θ ≥ 2θ/π, 0 ≤ θ ≤ π/2, upon replacing θ by π − θ, we find that∫ π
π/2
e−R sin θdθ =
∫ π/2
0
e−R sin θdθ ≤
∫ π/2
0
e−2Rθ/πdθ =
π
2R
(
1− e−R
)
. (4.8)
The bound (4.8) also holds for the first integral on the far right-hand side of (4.7). Hence,
from (4.7),∣∣∣∣∫
CR
eizf(z) logm z dz
∣∣∣∣ ≤ A(logR+ π)m
π
R
(
1− e−R
)
= o(1), (4.9)
as R tends to infinity.
Hence, so far, by (4.5), (4.6), and (4.9), we have shown that
S =
∫ 0
−∞
eixf(x)(log |x|+ iπ)mdx+
∫ ∞
0
eixf(x) logm x dx
=
∫ ∞
0
{
e−ixf(−x)(log x+ iπ)m + eixf(x) logm x
}
dx. (4.10)
Suppose first that f(x) is even. Then (4.10) takes the form
S =
∫ ∞
0
f(x)
{
e−ix(log x+ iπ)m + eix logm x
}
dx
=
∫ ∞
0
f(x)
{
e−ix
m∑
k=0
(
m
k
)
logk x(iπ)m−k + eix logm x
}
dx
=
m∑
k=0
(
m
k
)
(iπ)m−k(Ik − iJk) + (Im + iJm),
which establishes (4.3). Secondly, suppose that f(z) is odd. Then, (4.10) takes the form
S =
∫ ∞
0
f(x)
{
−e−ix(log x+ iπ)m + eix logm x
}
dx
=
∫ ∞
0
f(x)
{
−e−ix
m∑
k=0
(
m
k
)
logk x(iπ)m−k + eix logm x
}
dx
= −
m∑
k=0
(
m
k
)
(iπ)m−k(Ik − iJk) + (Im + iJm), (4.11)
from which (4.4) follows. �
8 B.C. Berndt and A. Straub
Example 4.2. Let f(z) = z/(z2 + 1). Then
2πiRes
(
eizz logm z
z2 + 1
, i
)
=
πi
e
(
iπ
2
)m
,
and so we are led by (4.4) to the recurrence relation
πi
e
(
iπ
2
)m
= −
m∑
k=0
(
m
k
)
(iπ)m−k(Ik − iJk) + (Im + iJm), (4.12)
where
Im :=
∫ ∞
0
x cosx logm x
x2 + 1
dx and Jm :=
∫ ∞
0
x sinx logm x
x2 + 1
dx.
(In the sequel, it is understood that we are assuming that n = 1 in Theorem 2.1 and in all our
deliberations of the two preceding sections.) If m = 0, (4.12) reduces to
J0 =
π
2e
, (4.13)
which is (2.8). After simplification, if m = 1, (4.12) yields
−π
2
2e
= −iπI0 − πJ0 + 2iJ1. (4.14)
If we equate real parts in (4.14), we once again deduce (4.13). If we equate imaginary parts
in (4.14), we find that
J1 −
π
2
I0 = 0, (4.15)
which is identical with (3.9). Setting m = 2 in (4.12), we find that
− iπ
3
4e
= π2(I0 − iJ0)− 2iπ(I1 − iJ1) + 2iJ2. (4.16)
Equating real parts on both sides of (4.16), we once again deduce (4.15). If we equate imaginary
parts in (4.16) and employ (4.13), we arrive at
J2 − πI1 =
π3
8e
, (4.17)
which is the same as (3.10). Lastly, we set m = 3 in (4.12) to find that
π4
8e
= iπ3(I0 − iJ0) + 3π2(I1 − iJ1)− 3iπ(I2 − iJ2) + 2iJ3. (4.18)
If we equate real parts on both sides of (4.18) and simplify, we deduce (4.17) once again. On
the other hand, when we equate imaginary parts on both sides of (4.18), we deduce that
2J3 − 3πI2 − 3π2J1 + π3I0 = 0. (4.19)
A slight simplification of (4.19) can be rendered with the use of (4.15).
We can replace the rational function 1/(x2 + 1) in Theorem 3.1 by other even rational func-
tions f(x) to obtain the following generalization of Theorem 3.1. Its proof is in the same spirit
as that of Theorem 4.1.
Certain Integrals Arising from Ramanujan’s Notebooks 9
Theorem 4.3. Suppose that f(z) is an even rational function with no real poles and with the
degree of the denominator exceeding the degree of the numerator by at least 2. Then,
πi
eπis/2
∑
U
Res(einzf(z)zs, zj) =
∫ ∞
0
cos(nx− πs/2)f(x)xsdx,
where the sum is over all poles zj of f(z) lying in the upper half-plane U .
Note that, as we did for (3.7), we can replace s with s+ 1 in Theorem 4.3 to obtain a corre-
sponding result for odd rational functions xf(x). This is illustrated in Example 4.7 below.
As an application, we derive from Theorem 4.3 the following explicit integral evaluation,
which reduces to Theorem 3.1 when r = 0.
Theorem 4.4. Let r ≥ 0 be an integer. For s ∈ (−1, 2(r + 1)) and n ≥ 0,
∫ ∞
0
cos(nx− πs/2)
(x2 + 1)r+1
xsdx =
π
2
e−n
r∑
k=0
1
2r+k
(
r + k
k
) r−k∑
j=0
(−1)j
(
s
j
)
nr−k−j
(r − k − j)!
.
Proof. Setting f(z) = 1/(z2 + 1)r in Theorem 4.3, we see that we need to calculate the residue
Res
(
einzzs
(z2 + 1)r+1
, i
)
= Res
(
α(z)
(z − i)r+1
, i
)
,
where
α(z) =
einzzs
(z + i)r+1
is analytic in a neighborhood of z = i. Equivalently, we calculate the coefficient of xr in the
Taylor expansion of α(x+ i) around x = 0. Using the binomial series
1
(x+ a)r+1
=
∑
k≥0
(−1)k
(
r + k
k
)
xka−r−k−1
with a = 2i, we find that
α(x+ i) = e−n
einx(x+ i)s
(x+ 2i)r+1
= e−n
∑
k≥0
(−1)k
(
r + k
k
)
xk(2i)−r−k−1
∑
j≥0
(
s
j
)
xjis−j
∑
l≥0
(inx)l
l!
.
Extracting the coefficient of xr, we conclude that
Res
(
einzzs
(z2 + 1)r+1
, i
)
=
e−n
(2i)r+1
r∑
k=0
(−1)k
(2i)k
(
r + k
k
) r−k∑
j=0
(
s
j
)
is−j
(in)r−k−j
(r − k − j)!
=
e−neπis/2
2i
r∑
k=0
1
2r+k
(
r + k
k
) r−k∑
j=0
(−1)j
(
s
j
)
nr−k−j
(r − k − j)!
.
Theorem 4.4 now follows from Theorem 4.3. �
10 B.C. Berndt and A. Straub
Example 4.5. In particular, in the case s = 0,
∫ ∞
0
cos(nx)
(x2 + 1)r+1
dx =
π
2
e−n
r∑
k=0
1
2r+k
(
r + k
k
)
nr−k
(r − k)!
. (4.20)
We note that, more generally, this integral can be expressed in terms of the modified Bessel
function Kr+1/2(z) of order r + 1/2. Namely, we have [3, formula (3.773), no. 6]
∫ ∞
0
cos(nx)
(x2 + 1)r+1
dx =
(n
2
)r+1/2
√
π
Γ(r + 1)
Kr+1/2(n). (4.21)
When r ≥ 0 is an integer, the Bessel function Kr+1/2(z) is elementary and the right-hand side
of (4.21) evaluates to the right-hand side of (4.20).
On the other hand, taking a derivative with respect to s before setting s = 0, and observing
that, for j ≥ 1,[
d
ds
(
s
j
)]
s=0
=
(−1)j−1
j
,
we arrive at the following generalization of Ramanujan’s formula (2.1).
Corollary 4.6. We have∫ ∞
0
sin(nx)
(x2 + 1)r+1
dx+
2
π
∫ ∞
0
cos(nx)
(x2 + 1)r+1
log x dx
= −e−n
r∑
k=0
1
2r+k
(
r + k
k
) r−k∑
j=1
1
j
nr−k−j
(r − k − j)!
.
We leave it to the interested reader to make explicit the corresponding generalization of (3.3).
Example 4.7. As a direct extension of (3.7), replacing s with s+ 1 in Theorem 4.4, we obtain
the following companion integral. For integers r ≥ 0, and any s ∈ (−2, 2r + 1) and n ≥ 0,
∫ ∞
0
x sin(nx− πs/2)
(x2 + 1)r+1
xsdx =
π
2
e−n
r∑
k=0
1
2r+k
(
r + k
k
) r−k∑
j=0
(−1)j
(
s+ 1
j
)
nr−k−j
(r − k − j)!
.
In particular, setting s = 0, we find that
∫ ∞
0
x sin(nx)
(x2 + 1)r+1
dx =
π
2
e−n
r∑
k=0
1
2r+k
(
r + k
k
){
nr−k
(r − k)!
− nr−k−1
(r − k − 1)!
}
, (4.22)
while taking a derivative with respect to s before setting s = 0 and observing that, for j ≥ 2,[
d
ds
(
s+ 1
j
)]
s=0
=
(−1)j
j(j − 1)
,
Certain Integrals Arising from Ramanujan’s Notebooks 11
we find that∫ ∞
0
x cos(nx)
(x2 + 1)r+1
dx− 2
π
∫ ∞
0
x sin(nx)
(x2 + 1)r+1
log x dx
= e−n
r∑
k=0
1
2r+k
(
r + k
k
) nr−k−1
(r − k − 1)!
−
r−k∑
j=2
1
j(j − 1)
nr−k−j
(r − k − j)!
=
2
π
∫ ∞
0
cos(nx)
(x2 + 1)r+1
dx− 2
π
∫ ∞
0
x sin(nx)
(x2 + 1)r+1
dx
− e−n
r∑
k=0
1
2r+k
(
r + k
k
) r−k∑
j=2
1
j(j − 1)
nr−k−j
(r − k − j)!
,
upon the employment of (4.20) and (4.22).
Acknowledgements
We wish to thank Khristo Boyadzhiev, Larry Glasser and the referees for their careful and
helpful suggestions.
References
[1] Berndt B.C., Ramanujan’s notebooks. Part I, Springer-Verlag, New York, 1985.
[2] Berndt B.C., Ramanujan’s notebooks. Part IV, Springer-Verlag, New York, 1994.
[3] Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 8th ed., Academic Press Inc., San
Diego, CA, 2014.
[4] Ramanujan S., Collected papers, Cambridge University Press, Cambridge, 1927, reprinted by Chelsea, New
York, 1962, reprinted by Amer. Math. Soc., Providence, RI, 2000.
[5] Ramanujan S., Notebooks. Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957.
http://dx.doi.org/10.1007/978-1-4612-1088-7
http://dx.doi.org/10.1007/978-1-4612-0879-2
1 Introduction
2 Ramanujan's extension of (1.2)
3 A second approach to the entry at the top of p. 391
4 General theorems
References
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