---
title: Generalized Log-Sine Integrals
url: https://www.emergentmind.com/topics/generalized-log-sine-integrals
type: topic
---

# Generalized Log-Sine Integrals

Generalized log-sine integrals are a family of trigonometric-logarithmic integrals built from the kernel $\log\!\left|2\sin \frac{\theta}{2}\right|$ and its moments, shifted variants, normalized forms, and iterated analogues. In the modern literature they include the classical quantities $Ls_n(\sigma)$, the moment integrals $Ls_n^{(k)}(\sigma)$, log-sine-cosine integrals, shifted order-3 forms, normalized integrals $\ls_m^{(k)}(z)$, zeta-type shifted log-sine integrals $\mathrm{SLs}(s;\sigma)$, and multi-indexed iterated log-sine integrals. Across these formulations, generalized log-sine integrals function as an analytic bridge between Clausen functions, zeta values, multiple polylogarithms, multiple zeta values, Mahler measures, central binomial series, and, in recent work, poly-Bernoulli numbers with negative index [1103.4298] [2603.25392].

## 1. Classical kernel and standard generalizations

The basic log-sine integrals are
\[
Ls_n(\sigma):=-\int_0^\sigma \log^{\,n-1}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad n=1,2,\dots,
\]
and their generalized moments are
\[
Ls_n^{(k)}(\sigma):=-\int_0^\sigma \theta^k\,\log^{\,n-1-k}\left|2\sin\frac{\theta}{2}\right|\,d\theta,\qquad k\ge 0.
\]
They satisfy $Ls_1(\sigma)=-\sigma$, $Ls_n^{(0)}(\sigma)=Ls_n(\sigma)$, and
\[
Ls_2(\sigma)=Cl_2(\sigma)=\sum_{m=1}^\infty \frac{\sin(m\sigma)}{m^2},
\]
so the lowest nontrivial log-sine integral is already the Clausen function [1103.3035] [1103.4298].

Two common extensions appear early in the literature. The first is the log-sine-cosine family
\[
Lsc_{m,n}(\sigma):=-\int_0^\sigma \log^{m-1}\left|2\sin\frac{\theta}{2}\right|\, \log^{n-1}\left|2\cos\frac{\theta}{2}\right|\,d\theta,
\]
with $Lsc_{m,1}(\sigma)=Ls_m(\sigma)$. The second is the shifted order-3 integral
\[
Ls3(\theta,\omega):=-\int_0^\theta \log\left|2\sin\frac{\sigma}{2}\right| \log\left|2\sin\frac{\sigma+\omega}{2}\right|\,d\sigma,
\]
for which $Ls3(\theta,0)=Ls3(\theta)$ [1103.3035].

Classical identities already exhibit the arithmetic content of the kernel. Matsusaka recalls Euler’s identity
\[
\left(1-\frac{1}{2^3}\right)\zeta(3)=\frac{\pi^2}{4}\log 2+2\int_0^{\pi/2}\theta\log(\sin\theta)\,d\theta,
\]
and Clausen’s identity
\[
\sum_{n=1}^\infty \frac{\sin(nx)}{n^2} =-x\log 2-\int_0^x \log\!\left(\sin\frac{\theta}{2}\right)d\theta.
\]
These formulas show that log-sine integrals encode special values of zeta-related functions and motivate their later generalizations [2603.25392].

A related antiderivative viewpoint is supplied by Dowker’s
\[
I(M,x)=\int dx\,x^M \log |\sin x|,
\]
whose repeated integration-by-parts reduction and Newman’s 1847 formula recover the standard closed forms in terms of odd zeta values and Clausen functions [2403.14644].

## 2. Normalized, shifted, and deformed families

One widely used normalization factors out the endpoint value of the sine function:
\[
\ls_m^{(k)}(z)\coloneqq -\int_0^z t^k \log^{\,m-1-k}\!\Bigl(\frac{\sin t}{\sin z}\Bigr)\,dt,\qquad 0<z\le \frac{\pi}{2},
\]
with
\[
\Ls_m^{(k)}(2z) = 2^{k+1}\sum_{j=0}^{m-k-1}\binom{m-k-1}{j}\log(2\sin z)^j\,\ls_{m-j}^{(k)}(z).
\]
A second variant is the shifted normalized integral
\[
-\int_0^z (z-t)^k \log^{\,m-1-k}\!\Bigl(\frac{\sin t}{\sin z}\Bigr)\,dt,
\]
which appears naturally when powers of the logarithm are expanded in generating-function arguments [2508.05770].

A more recent zeta-type generalization is Matsusaka’s shifted log-sine integral
\[
\mathrm{SLs}(s;\sigma)\coloneqq \frac{1}{\Gamma(s-1)} \int_0^\sigma (\theta-\sigma) \left(-2\log\frac{\sin(\theta/2)}{\sin(\sigma/2)}\right)^{s-2} d\theta,
\]
defined for $0<\sigma\le \pi$ and $\Re(s)>1$. Its defining features are the shift by the ratio $\sin(\theta/2)/\sin(\sigma/2)$, the linear factor $(\theta-\sigma)$, and the normalization by $\Gamma(s-1)$, which makes it behave like a zeta-type function in the parameter $s$ [2603.25392].

The literature also contains affine and Mellin-type deformations. In the study of the incomplete Mellin transformation of the fractional part, the generalized log-sine integral
\[
\mathrm{Ls}_{n-1}^{\,n-3}(a,b,d) \coloneqq -\frac12\int_0^d \log\bigl(4\sin^2(\pi(ax+b))\bigr)x^{n-1}\,dx
\]
appears as one side of an affine complex deformation of the log-sine/fractional-part kernel. In that setting, equalities between hypergeometric series and zeta/log-sine expressions capture meromorphic extension and the functional equation of the Hurwitz zeta function [2009.07134].

These variants show that “generalized log-sine integral” is not a single rigid notation but a family of constructions sharing the same logarithmic sine kernel. The common pattern is that normalization, shifting, or deformation is introduced to expose additional analytic structure: endpoint regularization, polylogarithmic generating functions, Mellin transforms, or zeta-type continuation.

## 3. Special values, generating functions, and explicit evaluation

At $\pi$, the standard log-sine values are controlled by the exponential generating function
\[
-\frac{1}{\pi}\sum_{m=0}^{\infty} Ls_{m+1}(\pi)\,\frac{\lambda^m}{m!}
= \frac{\Gamma(1+\lambda)}{\Gamma^2\!\left(1+\frac{\lambda}{2}\right)}
= \binom{\lambda}{\lambda/2}.
\]
This implies that all $Ls_n(\pi)$ reduce to zeta values. Explicit examples include
\[
Ls_2(\pi)=0,\qquad -Ls_3(\pi)=\frac{1}{12}\pi^3,\qquad Ls_4(\pi)=\frac32\pi\zeta(3),
\]
\[
-Ls_5(\pi)=\frac{19}{240}\pi^5,\qquad Ls_6(\pi)=\frac{45}{2}\pi\zeta(5)+\frac54\pi^3\zeta(3).
\]
The corresponding generalized values $Ls_n^{(k)}(\pi)$ are extracted from a double generating function and lie in the algebra generated by $\pi$ and Nielsen polylogarithms at $\pm1$ [1103.4298] [1103.3893].

At $2\pi$ and multiples of $2\pi$, quasiperiodicity simplifies the picture further. The generating function
\[
-\sum_{n,k \ge 0} Ls_{n+k+1}^{(k)}(2\pi)\frac{\lambda^n}{n!}\frac{(i\mu)^k}{k!}
=2\pi e^{i\mu\pi}\binom{\lambda}{\lambda/2+\mu}
\]
shows that $Ls_n^{(k)}(2\pi)$ reduce to zeta values only, and a harmonic-sum formula reduces all multiples of $2\pi$ to the $2\pi$ case [1103.4298].

At $\pi/3$, the sixth-root-of-unity structure yields a different type of reduction. The values are expressed in terms of zeta values together with multiple Clausen and Glaisher values at $\pi/3$. Representative evaluations are
\[
-Ls_3\!\left(\frac{\pi}{3}\right)=\frac{7}{108}\pi^3,
\]
\[
Ls_4\!\left(\frac{\pi}{3}\right)=\frac12\pi\zeta(3)+\frac92\,Gl_4\!\left(\frac{\pi}{3}\right),
\]
and
\[
-Ls_4^{(1)}\!\left(\frac{\pi}{3}\right)=\frac{17}{6480}\pi^4.
\]
Algorithmic evaluations for general arguments are also available, and an implementation in Mathematica and SAGE was provided [1103.4298].

Two complementary closed-form techniques are notable. Dowker recovers standard formulas for $\int dx\,x^M\log|\sin x|$ by repeated integration by parts plus Newman’s 1847 antiderivative formula, avoiding Hurwitz-zeta derivatives [2403.14644]. Separately, the differentiation method based on
\[
F(n,m,z)=\int_0^z x^n \sin^{2m}(x)\,dx
\]
identifies generalized log-sine integrals with parameter derivatives at $m=0$, while complete Bell polynomials encode derivatives of central and shifted central binomial coefficients in terms of polygamma functions and harmonic numbers [1705.04723].

## 4. Polylogarithmic, Nielsen, and multiple-zeta structures

A central modern theme is the conversion of generalized log-sine integrals into polylogarithmic objects. For normalized integrals, the generating identity
\[
2i\int_0^z \left(\frac{\sin t}{\sin z}\right)^a e^{ia(z-t)}\,dt
= \sum_{n=1}^\infty \frac{1-e^{-2inz}}{n+a},
\qquad 0<z<\frac{\pi}{6},\ -1<a<1,
\]
yields, after expansion in $a$, explicit formulas for polylogarithms and log-sine integrals. In particular,
\[
(-1)^m \Li_{m+1}(1-e^{-2iz}) = \frac{2i}{m!}\int_0^z \log^m\!\Bigl(\frac{\sin t}{\sin z}+i(z-t)\Bigr)\,dt.
\]
This framework proves that all generalized log-sine integrals of weight $\le 4$ can be written using only classical polylogarithms, while higher weights require Nielsen polylogarithms [2508.05770].

The higher-weight transition is explicit. Nielsen polylogarithms are defined by
\[
S_{m,k}(x) \coloneqq \frac{(-1)^{m+k-1}}{(m-1)!\,k!} \int_0^1 \log^{m-1}(t)\,\log^k(1-xt)\,\frac{dt}{t},
\]
with $S_{m,1}(x)=\Li_{m+1}(x)$. The identity
\[
(-1)^m S_{m,k+1}(1-e^{-2iz}) = \frac{(2i)^{k+1}}{m!\,k!} \int_0^z t^k \log^m\!\Bigl(\frac{\sin t}{\sin z}+i(z-t)\Bigr)\,dt
\]
implies that every generalized log-sine integral $\ls_m^{(k)}(z)$ can be expressed in terms of Nielsen polylogarithms. The same method also produces functional equations for $S_{3,2}$ and $S_{4,2}$ and a hyperbolic analogue obtained by replacing $\sin$ with $\sinh$ [2508.05770].

The iterated theory enlarges the scope from single integrals to multi-indexed simplicial integrals. For
\[
A(\theta)=\log\left|2\sin\frac{\theta}{2}\right|,
\]
Umezawa defines
\[
\operatorname{Ls}_{\mathbf{k}^{(\mathbf{l})}}(\sigma)
= (-1)^n \int_{0<\theta_1<\cdots<\theta_n<\sigma}
\prod_{u=1}^n \theta_u^{\,l_u}\, A(\theta_u)^{\,k_u-1-l_u}\, d\theta_1\cdots d\theta_n,
\]
with the convergence condition $k_u-1-l_u\ge 0$. When $n=1$, these reduce to the generalized log-sine integrals. The main evaluation theorem states that for $\sigma\in[0,2\pi]$ every such iterated log-sine integral is a $\mathbb{Q}(i)$-linear combination of powers of $\sigma$, powers of $\pi$, multiple zeta values, and multiple polylogarithms $\operatorname{Li}_{\mathbf{k}}(e^{i\sigma})$ with admissible indices [1912.07201].

The algebraic mechanism behind this reduction already appears in Umezawa’s earlier iterated-integral formulation. There the iterated log-sine integrals satisfy a shuffle product, a reduction identity when some exponent $k_j-1-l_j$ vanishes, and a conversion formula expressing $\operatorname{Li}_{\mathbf{k}}(1-e^{i\theta})$ in terms of iterated log-sine integrals. Combined with the Borwein–Broadhurst–Kamnitzer identity, this yields a method for deriving relations among multiple zeta values; the paper records, for example, $\zeta(3)= -2\,\operatorname{Ls}_3\!\left(\frac{\pi}{3}\right)$ and recovers classical relations in weights $4$ and $5$ [1904.09717].

## 5. Mahler measures, central binomial sums, and related series

Generalized log-sine integrals became especially prominent through their role in higher and multiple Mahler measures. For the family
\[
\mu_k(1+x+y_*)=\mu(1+x+y_1,\dots,1+x+y_k),
\]
one has
\[
\mu_k(1+x+y_*) = \frac1\pi Ls_{k+1}\!\left(\frac\pi3\right) - \frac1\pi Ls_{k+1}(\pi).
\]
Consequently,
\[
\mu_1(1+x+y_*)=\frac{1}{\pi}Cl_2\!\left(\frac\pi3\right),\qquad
\mu_2(1+x+y_*)=\frac{\pi^2}{54},
\]
and higher values involve Clausen or Glaisher constants at $\pi/3$, zeta values, and sometimes central-binomial-sum constants [1103.3893].

Multiple Mahler measures of simpler polynomials also reduce directly to generalized log-sine-cosine integrals. For $P(x)=1-x$,
\[
\mu_{m,n}(1-x,1+x)= -\frac1\pi\,Lsc_{m+1,n+1}(\pi).
\]
For $1+x+y$, one has
\[
\mu_1(1+x+y)=\frac{1}{\pi}Ls_2\!\left(\frac{\pi}{3}\right),\qquad
\mu_2(1+x+y)=\frac{3}{\pi}Ls_3\!\left(\frac{2\pi}{3}\right)+\frac{\pi^2}{4},
\]
while $\mu_3(1+x+y)$ is expressed in terms of $Ls_4(2\pi/3)$, $Lsc_{2,3}(\pi/3)$, Clausen values, and $\zeta(3)$. The auxiliary quantity
\[
\rho_n(\alpha):=\frac1{2\pi}\int_{-\pi}^{\pi} \left(\Re\log\left(1-\alpha e^{i\omega}\right)\right)^n\,d\omega
\]
is shown to be a finite linear combination of multiple polylogarithms at $\alpha^2$; in low weight,
\[
\rho_3(\alpha)=-\frac32 Li_{2,1}(\alpha^2),\qquad
\rho_4(\alpha)=6Li_{2,1,1}(\alpha^2)+\frac32Li_{2,2}(\alpha^2)
\]
[1103.3035].

At $\pi/3$, generalized log-sine moments are tied to central binomial sums. The relation
\[
-Ls_{n+2}^{(1)}\!\left(\frac\pi3\right) = n!\left(-\frac12\right)^n \mathcal S_+(n+2),
\qquad
\mathcal S_\pm(n)=\sum_{k=1}^\infty \frac{(\pm1)^{k+1}}{\binom{2k}{k}k^n},
\]
links the log-sine values to inverse central binomial series, and specific evaluations such as
\[
\mathcal S_+(4)=\frac{17}{36}\zeta(4)
\]
follow from this framework [1103.3893].

A complementary development derives Apéry-like and Leshchiner-type identities from the same generating-function mechanism that evaluates normalized log-sine integrals. In that setting, specializations produce central binomial coefficient series, parametric Apéry-like formulas for $\Li_3(t)$ and $\Li_4(t)$, and identities of Koecher–Leshchiner type; these are not ancillary results but arise from the same polylogarithmic-sine integral correspondence [2508.05770].

## 6. Shifted zeta-type formulations, poly-Bernoulli numbers, and conjectural directions

Matsusaka’s shifted log-sine integral provides a zeta-type object whose negative integer values are governed by central binomial series and Lehmer polynomials. A key structural theorem expresses
\[
\mathrm{SLs}(s;\sigma)
= \zeta_{\mathrm{CB}}(s;\sin(\sigma/2))
-\frac{\sigma}{\pi}\eta_{\mathrm{CB}}(s;\sin(\sigma/2)),
\]
where $\zeta_{\mathrm{CB}}$ is a central binomial series and $\eta_{\mathrm{CB}}$ its half-integer-shift analogue. This yields analytic continuation of $\mathrm{SLs}(s;\sigma)$ to all $s\in\mathbb C$. For integers $n\ge -1$ and $x=\sin^2(\sigma/2)$,
\[
\mathrm{SLs}(-n;\sigma)=\frac{x\,p_n(x)}{2^n(1-x)^{n+1}},
\]
with $p_n(x)$ the Lehmer polynomials determined recursively by
\[
p_{-1}(x)=0,\qquad q_{-1}(x)=1,
\]
\[
\begin{split}
p_{n+1}(x)&=2(nx+1)p_n(x)+2x(1-x)p_n'(x)+q_n(x),\\
q_{n+1}(x)&=(2(n+1)x+1)q_n(x)+2x(1-x)q_n'(x).
\end{split}
\]
The cancellation of the $q_n$-term is the mechanism behind the closed formula [2603.25392].

The special point $\sigma=\pi/3$ produces the poly-Bernoulli connection. Since $\sin^2(\pi/6)=1/4$,
\[
\mathrm{SLs}(-n;\pi/3) = \frac13\left(\frac23\right)^n p_n\!\left(\frac14\right),
\]
and for $n\ge 0$ this becomes
\[
\mathrm{SLs}(-n;\pi/3) = \frac13\sum_{k=0}^n B_{n-k}^{(-k)}.
\]
This is the anti-diagonal sum of poly-Bernoulli numbers with negative index highlighted in the abstract. The identity was observed by Stephan and later proved in the cited work of Bényi–Matsusaka; within Matsusaka’s framework it arises naturally from analytic continuation of the shifted log-sine integral [2603.25392].

This formulation is explicitly compared with the Arakawa–Kaneko zeta function. Arakawa and Kaneko introduced a zeta function whose special values at negative integers yield the poly-Bernoulli numbers “by design,” whereas the shifted log-sine integral is presented as a natural analytic object coming from classical log-sine geometry whose negative values automatically encode poly-Bernoulli numbers [2603.25392].

Conjectural directions remain active in the iterated setting. Umezawa proposes that suitable families of shifted log-sine values at $\pi/3$ should span, and in stronger form should form bases of, the spaces of multiple zeta values, multiple Clausen values, multiple Glaisher values, and iterated log-sine integrals, with predicted dimensions governed by Fibonacci-type recurrences. Proven depth-one results show that every Riemann zeta value, every Clausen value $\operatorname{Cl}_k(\pi/3)$, and every Glaisher value $\operatorname{Gl}_k(\pi/3)$ can be written as a $\mathbb{Q}$-linear combination of appropriate shifted log-sine generators [1912.07201].

In aggregate, generalized log-sine integrals constitute a broad analytic framework rather than a single special function. Their unifying feature is the logarithmic sine kernel; their distinguishing feature is the range of arithmetic and polylogarithmic structures exposed by moments, shifts, iterated integration, endpoint normalization, Mellin deformation, and analytic continuation.

Source: https://www.emergentmind.com/topics/generalized-log-sine-integrals