Log-sine-cosine integrals

edit · history · discussion · files · short url · integral moments special functions special values number theory

Numbers

$m$, $n$

$t$

$\operatorname{Lsc}_{m,n}(\pi t)$

2, 2

1/2:

0.6459640975062462536557565638979457333796935117892730769613445438292998941138688757826396048458934624

2, 2

1.291928195012492507311513127795891466759387023578546153922689087658599788227737751565279209691786925

2, 3

1/6:

0.4071654302075600522622385021390438884381151321990349671853719636956150694610319130295240719138146297

2, 3

1/2:

0.4488992266595933490141778692690939754758911205246708215137019612505852657860167213674134178681530167

2, 3

-0.9440932840407697318008689483131357053753075931991633243957383107211386637566250829464196056664896766

3, 2

1/6:

-1.343663692080670999988482067539734772786387571810508537526114579436225170969158282596757207669683651

3, 2

1/3:

-1.383185793649199027941821340809526542863044633928931937402731701246476925008758902003009963268553632

3, 2

1/2:

-1.392992510700363080815046817582229680851198713723834145909440271971723929542641804313833023534642693

3, 2

5/6:

-1.351258714248329784063107450452179593813422725398198291581110274416753733217656995975943677580304306

2, 4

1/2:

0.3094532610636345985431577342989541707074918055106980657983198372731452709663432286282748414311017298

2, 4

4.462787069785354526748310771466769261171051971659426945571764803757593635281871190417676048571949651

3, 3

1/2:

-0.9563115149540045414460665938857362702509397082127343454796638865194843504175438265180734389797034966

3, 3

-1.912623029908009082892133187771472540501879416425468690959327773038968700835087653036146877959406993

4, 2

1/2:

4.153333808721719928205153037167815090463560166148728879773444966484448364315527961789401207140847921

2, 5

1/2:

0.2123858970851502510850349838202142960875729071297820929983175669547706144231784326434982690963659918

2, 5

-16.41994431020785549621955381577904636433644627391327600970764455956421757456162441026155289729364062

3, 4

1/2:

-0.6574272641589800368846611725131782438078219179550570496903586550021145595025466530956852374556132052

3, 4

2.215710152207518264424941814943079790016711535219640827143291384598011797643579409404061231915099650

4, 3

1/2:

2.873137416366498301309602987456258033824533453174697876833650039600126357146126062499746469370712855

5, 2

1/2:

-16.63233020729300574730458879959926066042401918104305810270596212651898818898480284290505116639000661

2, 6

1/2:

0.1454224334868578185163278238580277588116676950610518835863250198599898130770708199812601480303615808

2, 6

83.31644629572990674413424646336549307183028742195268945137517449701857395535700772340217082130500823

3, 5

1/2:

-0.4523572962273990580947316631956870388754942439018276865952412345485146576888287440811921760790308082

3, 5

-11.97219676375738921977214928348669741702365500749854377642313023995883650162200143404895117163726252

4, 4

1/2:

1.987423491593333109615080579439442024759351092088248122756811465489763853500263241653775349706692573

4, 4

3.974846983186666219230161158878884049518702184176496245513622930979527707000526483307550699413385146

5, 3

1/2:

-11.51983946752999016167741762029101037814816076359671608982788900541032184393317268996775899555823172

6, 2

1/2:

83.17102386224304892561791863950746531301861972689163756778884947715858414227993690342091067327464665

2, 7

1/2:

0.09945468991573744850470943477443167855636530921135086075246260081081746852053211044604129865083090060

2, 7

-498.9538062272859448836867350819642185805176928045798172348330363009863632398056897643485954618895749

3, 6

1/2:

-0.3114413240786725167565070294823147176976186045577582689114018086280609891915023989685153802326035003

3, 6

57.32407362073383691988089238653292971407845518746488652927286395586151008535002748724682080818817569

4, 5

1/2:

1.374732676926273784507668543056139791901443686143832017819609045263886052971827460096504364179391281

4, 5

-6.604252628626615522985780391746816388134619718384139927770518815624507675499079427463434039723895541

5, 4

1/2:

-7.978985305552889307493448934802956180036063404527971945590127860888393728470906887559938403903286822

6, 3

1/2:

57.63551494481250943663739941601524443177607379202264479818426576448957107454152988621533618842077919

7, 2

1/2:

-499.0532609172016823321914445167386502590740581137911680955854989017971807083262218747946367605404058

2, 8

1/2:

0.06798389217992952826164416429019072602473664315173480945788000827032378818437675046282065747072937437

2, 8

3493.500093627774683026044995779173461558911260208262441465453796917327400129234445377257760264462014

3, 7

1/2:

-0.2145146673086564435999909959803352287099147393783727636849243536195492504348099358642472712341659823

3, 7

-346.1032771766757368440682899377719330919805617884265430105649200189192566494098104990002491763919487

4, 6

1/2:

0.9509340341561699112175776611792751799258445581491813301716779688382230693635647448859548108313124181

4, 6

40.89102061081904514324030356257071726812911413758718346603653291737732116747152423575574355166543408

5, 5

1/2:

-5.526563083997761455159417664836640636367245979608267007619555802772998940669883927261189652235076309

5, 5

-11.05312616799552291031883532967328127273449195921653401523911160554599788133976785452237930447015262

6, 4

1/2:

39.94008657666287523202272590139144208820326957943800213586485494853909809810795949086978874083412167

7, 3

1/2:

-345.8887625093670804004682989417915978632706470490481702468799956652997073989750005631360019051577827

8, 2

1/2:

3493.432109735594753497783351614883270832886523565110706655995916909057076341050068626794939606991285

Definition

The log-sine-cosine integral is $\operatorname{Lsc}_{m,n}(\pi t)=-\int_0^{\pi t}\log^{m-1}|2\sin(v/2)|\,\log^{n-1}|2\cos(v/2)|\,dv$, with natural logarithms and radian angles [1] [2]. Endpoint singularities are interpreted as ordinary convergent improper integrals.

Parameters

$m$

— sine logarithmic order ($m\geq1$)

$n$

— cosine logarithmic order ($n\geq1$)

$t$

— angle divided by $\pi$ ($0<t\leq1$)

Formulas

(1)

$\operatorname{Lsc}_{m,n}(\theta)+\operatorname{Lsc}_{n,m}(\pi-\theta)=\operatorname{Lsc}_{m,n}(\pi)=\operatorname{Lsc}_{n,m}(\pi)$ for $0\leq\theta\leq\pi$ [2], section 3, Step 1. In particular, $\operatorname{Lsc}_{m,n}(\pi/2)+\operatorname{Lsc}_{n,m}(\pi/2)=\operatorname{Lsc}_{m,n}(\pi)$.

(2)

$\operatorname{Lsc}_{m,1}(\theta)=\operatorname{Ls}_m(\theta)$ and $\operatorname{Lsc}_{1,n}(\theta)=\operatorname{Ls}_n(\pi)-\operatorname{Ls}_n(\pi-\theta)$, where $\operatorname{Ls}_r=\operatorname{Ls}_r^{(0)}$ is the basic log-sine integral.

(3)

$-\sum_{r,s\geq0}\operatorname{Lsc}_{r+1,s+1}(\theta)\dfrac{x^r y^s}{r!s!}=2^{x+y}B(\sin^2(\theta/2);(1+x)/2,(1+y)/2)$ for $|x|,|y|<1$ and $0<\theta\leq\pi$. Here $B(z;a,b)=\int_0^z u^{a-1}(1-u)^{b-1}\,du$ is the unregularized incomplete beta function; the substitution is $u=\sin^2(v/2)$.

(4)

$-\sum_{r,s\geq0}\operatorname{Lsc}_{r+1,s+1}(\pi)\dfrac{x^r y^s}{r!s!}=2^{x+y}B((1+x)/2,(1+y)/2)$ for $|x|,|y|<1$, with $B(a,b)$ the complete beta function [1], equation (11).

(5)

$\operatorname{Lsc}_{2,2}(\pi)=\pi^3/24$, $\operatorname{Lsc}_{2,2}(\pi/2)=\pi^3/48$, and $\operatorname{Lsc}_{2,3}(\pi)=\operatorname{Lsc}_{3,2}(\pi)=-\pi\zeta(3)/4$, where $\zeta(3)$ is the Riemann zeta value.

(6)

$\sum_{j=1}^{\infty}\dfrac{\binom{2j}{j}H_{2j}}{(2j+1)^3 16^j}=2\operatorname{Ls}_4(5\pi/6)+\operatorname{Lsc}_{3,2}(\pi/3)-2\operatorname{Lsc}_{3,2}(\pi/6)+4\operatorname{Lsc}_{3,2}(5\pi/6)-2\pi\zeta(3)$ [2], equation (3.5), where $H_k=\sum_{h=1}^k1/h$ and $\operatorname{Ls}_4(5\pi/6)$ is the basic log-sine value.

Comments

(7)

The logarithmic powers are $m-1$ and $n-1$. The absolute values enclose the trigonometric arguments, not the logarithms. No sign, factorial or power of $\pi$ is absorbed into a stored value.

(8)

Reflection about $\pi/2$ interchanges the sine and cosine logarithms. The midpoint moments retain ordered pairs $(m,n)$, whereas the complete-interval moments are symmetric in $m,n$. Only $m\leq n$ is enumerated at $t=1$. The pure-power boundaries $m=1$ or $n=1$ reduce to generalized log-sine integrals with angular moment order zero.

(9)

The exceptional angles $\pi/6$, $\pi/3$ and $5\pi/6$ occur in the harmonic-weighted central-binomial identity (6) from [2], section 3. They are included only for the four mixed moments used in that reduction.

Programs

(P1)

Python (mpmath)

import mpmath as mp
mp.mp.dps = 150

def lsc(m, n, t):
    theta = mp.pi * mp.mpf(t)
    def f(v):
        if v == 0 or v == mp.pi:
            return mp.mpf(0)
        return (-mp.log(2*mp.sin(v/2))**(m-1)
                * mp.log(abs(2*mp.cos(v/2)))**(n-1))
    cuts = sorted(set([mp.mpf(0), theta]
                      + [v for v in [mp.pi/3, mp.pi/2, 2*mp.pi/3]
                         if v < theta]))
    return mp.quad(f, cuts)

print(mp.nstr(lsc(3, 2, '1/3'), 100))

References

[1]

Jonathan M. Borwein and Armin Straub, Special Values of Generalized Log-sine Integrals, Proceedings of ISSAC 2011, 43–50. Section 3.2, equations (10) and (11). (arXiv) (doi)

[2]

Steven Charlton, Herbert Gangl, Li Lai, Ce Xu and Jianqiang Zhao, On two conjectures of Sun concerning Apéry-like series. Definition 2.1 and section 3. (arXiv)

Similar tables

Generalized log-sine integrals — The pure-power boundaries reduce to the angular moment order zero, and $\operatorname{Ls}_4(5\pi/6)$ occurs with mixed moments in the harmonic-binomial identity.

Inverse binomial sums — Unweighted inverse central-binomial sums form the series companion; the harmonic-weighted central-binomial series in this table uses a different summand.

Complete beta values — Mixed parameter derivatives of its scaled generating function give the complete-interval moments.

Incomplete beta values — Mixed parameter derivatives of its scaled generating function give the moments at incomplete angular endpoints.

Riemann zeta values — The complete-interval moments are polynomials in zeta values and pi, as follows by expanding the beta generating function.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every ordered pair $m,n\geq2$ with $m+n\leq10$ at $t=1/2$, every pair $2\leq m\leq n$ with $m+n\leq10$ at $t=1$, and exactly $(m,n,t)=(3,2,1/6),(2,3,1/6),(3,2,1/3),(3,2,5/6)$. The midpoint and complete endpoint are selected by reflection and the beta generating function; the four exceptional cases occur in the harmonic-binomial application. The noncanonical total logarithmic-degree cutoff of eight covers balanced and unbalanced moments and both parities before extending a single order. The 48-row selection and its bounds are explained in argument-selection.md)

How they were obtained:

Euler sine and cosine products give convergent power series for the logarithms after the singular $\log v$ term is separated. Polynomial products and logarithmic endpoint moments are computed in Arb ball arithmetic. A positive-coefficient majorant bounds the entire omitted tail and is added to the result radius. Reflection reduces every segment to an endpoint no larger than $\pi/2$. The attached certification.md derives the bound, including argument uncertainty and rounding. Computation at 525 working bits through series degree 557 retained at least 512 accuracy bits. Each stored 100-significant-digit decimal contains the complete computed enclosure, and every entry is recomputed at 140 requested digits.

Independent mpmath quadrature at 130 and 180 decimal digits checks all 48 values. Complete-beta coefficients check all 16 complete-interval rows, and transformed incomplete-beta moments check the remaining 32. Known nonzero controls and a deliberate wrong-answer control pass. Reflection, the low-order evaluations and the harmonic-binomial identity are checked, then repeated on values read back from the draft. The log-sine companion T697 is also a draft, so its agreement is supplementary evidence; the error bounds here are independent of it. The attached checks.py and numerical-evidence.json record the calculations.