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{
"Title": "Generalized log-sinh integrals",
"Definition": "The generalized log-sinh integral is $\\operatorname{Lsh}_n^{(k)}(x)=-\\int_0^x t^k\\log^{n-1-k}(2\\sinh(t/2))\\,dt$, with natural logarithms CITE{BS}, equation (33). For $x>0$, integers $n\\geq1$ and $0\\leq k<n$, this is an ordinary convergent improper integral.",
"Parameters": {
"n": {
"type": "Z",
"title": "total order",
"display": "$n$",
"constraints": "$n\\geq1$"
},
"k": {
"type": "Z",
"title": "integration-variable moment order",
"display": "$k$",
"constraints": "$0\\leq k<n$"
},
"x": {
"type": "R",
"title": "positive real endpoint",
"display": "$x$",
"constraints": "$x>0$",
"values": {
"logphi": "$a$",
"2logphi": "$2a$",
"2logsilver": "$b$"
}
}
},
"Comments": {
"endpoints": "Put $\\varphi=(1+\\sqrt5)/2$, the HREF{T32}[golden ratio], $a=\\log\\varphi$, and $b=2\\log(1+\\sqrt2)=2\\operatorname{arsinh}(1)$. The endpoint keys $\\mathtt{logphi}$, $\\mathtt{2logphi}$ and $\\mathtt{2logsilver}$ denote exactly $a$, $2a$ and $b$. All radicals and logarithms use their positive real branches.",
"conventions": "The exponent $k$ is the moment order and $n-1-k$ is the logarithmic power. Write $\\operatorname{Lsh}_n=\\operatorname{Lsh}_n^{(0)}$ for the basic integrals. The leading minus sign, the factor $2$ and the half-argument are retained; no factorial is absorbed into the stored value. The logarithm itself can be negative, and its absolute value is not taken.",
"selection": "The kernel $A(t)=\\log(2\\sinh(t/2))$ has its unique positive zero at $2a$, the alternating Apéry endpoint of CITE{BS}, Example 11. The smaller endpoint $a$ occurs in the harmonic-weighted binomial reductions of CITE{CGLXZ}, Lemma 2.2 and section 4; its basic-moment slice supplies the boundary terms of the duplication formula. The endpoint $b$ is a bounded exploratory reference: $u=-4\\sinh^2(x/2)$ gives $u=-4$ there, the negative convergence endpoint of the inverse-binomial family CITE{DK}. It also includes integration across the zero of $A$. No separate application is asserted for every higher moment at $b$.",
"convergence": "Near $t=0$, the logarithm is $\\log t+O(t^2)$, so its finite powers multiplied by $t^k$ are integrable. The elementary edge $k=n-1$ is given by a formula and is omitted from the numerical selection."
},
"Formulas": {
"elementary": "$\\operatorname{Lsh}_n^{(n-1)}(x)=-x^n/n$.",
"dilog": "$\\operatorname{Lsh}_2^{(0)}(x)=\\pi^2/6-\\operatorname{Li}_2(e^{-x})-x^2/4$. Here $\\operatorname{Li}_2(z)=\\sum_{j\\geq1}z^j/j^2$ for $|z|<1$. Differentiation and the limit at $x=0$ give this identity.",
"special": "$\\operatorname{Lsh}_2^{(0)}(2a)=\\pi^2/10$ and $\\operatorname{Lsh}_3^{(1)}(2a)=\\zeta(3)/5$, using the dilogarithm identity and evaluation in CITE{BS}, Example 11. Here $\\zeta$ is the HREF{T14}[Riemann zeta function].",
"binomial": "For integers $s\\geq3$, $\\operatorname{Lsh}_s^{(1)}(2a)=\\dfrac{(s-2)!}{(-2)^{s-2}}F_s(-1)$, where $F_s(u)=\\sum_{j\\geq1}u^j/(j^s\\binom{2j}{j})$ is the HREF{T698}[inverse binomial sum]. This follows from repeated integration of $uF_s'(u)=F_{s-1}(u)$, starting with $F_2(-4\\sinh^2(x/2))=-x^2/2$, and using $A(2a)=0$.",
"duplication": "For integers $r\\geq0$, $\\operatorname{Lsh}_{r+1}^{(0)}(2x)=2\\sum_{j=0}^r\\binom rj H_{j+1,r-j+1}(x)$, where $H_{m,n}(x)=-\\int_0^x A(t)^{m-1}C(t)^{n-1}\\,dt$ and $C(t)=\\log(2\\cosh(t/2))$ CITE{CGLXZ}, Definition 2.1. This follows by substitution from $A(2t)=A(t)+C(t)$; in particular, $H_{m,1}(x)=\\operatorname{Lsh}_m^{(0)}(x)$."
},
"Similar tables": [
"HREF{T697}[Generalized log-sine integrals] use the trigonometric kernel and real-angle endpoints; they are the trigonometric counterpart discussed in CITE{BS}.",
"HREF{T698}[Inverse binomial sums] supply the alternating first-moment relation CITE{binomial}; the higher moments here are distinct quantities.",
"HREF{T14}[Riemann zeta values] include the value entering CITE{special}.",
"HREF{T176}[Polylogarithms at rational arguments] give related special-function context. The algebraic arguments in CITE{dilog} are not its current rational-argument family."
],
"References": {
"BS": {
"bib": "J. M. Borwein and A. Straub, Special Values of Generalized Log-sine Integrals (2011), section 5.2, equation (33), Theorem 4 and Example 11.",
"arxiv": "1103.4298"
},
"CGLXZ": {
"bib": "S. Charlton, H. Gangl, L. Lai, C. Xu and J. Zhao, On two conjectures of Sun concerning Apery-like series (2022), Definition 2.1, Lemma 2.2 and section 4.",
"arxiv": "2210.14704"
},
"DK": {
"bib": "A. I. Davydychev and M. Yu. Kalmykov, Massive Feynman diagrams and inverse binomial sums (2004), equations (1.1)-(1.3) and section 3.",
"arxiv": "hep-th/0303162"
}
},
"Links": {
"DLMF": {
"url": "https://dlmf.nist.gov/4.36#E1",
"title": "DLMF 4.36.1: Infinite product for sinh"
}
},
"Keywords": [
"log-sinh",
"Lsh",
"hyperbolic logarithmic integrals",
"alternating binomial sums"
],
"Tags": [
"integral",
"moments",
"special functions",
"special values",
"number theory"
],
"Programs": {
"mpmath": {
"language": "Python (mpmath)",
"code": "import mpmath as mp\n\ndef log_sinh(n, k, endpoint, dps=140):\n with mp.workdps(dps):\n phi = (1 + mp.sqrt(5))/2\n x = {\"logphi\": mp.log(phi),\n \"2logphi\": 2*mp.log(phi),\n \"2logsilver\": 2*mp.asinh(1)}[endpoint]\n if n < 1 or not 0 <= k < n:\n raise ValueError(\"require n>=1 and 0<=k<n\")\n def f(s):\n t = x*mp.exp(-s)\n return -x**(k+1)*mp.exp(-(k+1)*s)*mp.log(2*mp.sinh(t/2))**(n-1-k)\n return mp.quad(f, [0, 1, 4, 16, 64, 256, mp.inf])\n\ndef selected_arguments():\n for n in range(2, 11):\n for k in range(n-1):\n for x in (\"logphi\", \"2logphi\", \"2logsilver\"):\n if x != \"logphi\" or (k == 0 and n <= 9):\n yield n, k, x\n\nlog_sinh(3, 1, \"2logphi\")\n# Quadrature is an independent numerical check; generate.py supplies bounds.\n"
}
},
"Data properties": {
"type": "R",
"rigour": "proven",
"complete": "no",
"complete-note": "it holds every $2\\leq n\\leq10$, $0\\leq k\\leq n-2$ at $x=2a,b$, and $k=0$, $2\\leq n\\leq9$ at $x=a$, for 98 entries. The triangle covers all non-elementary moments through a noncanonical order cutoff of ten at the intrinsic golden endpoint and the exploratory inverse-binomial boundary. The shorter basic slice at $a$ supplies the mixed-family duplication boundaries through logarithmic degree eight. These conditional selections are documented in argument-selection.md",
"rigour details": "All 98 values carry 100 significant digits from real ball arithmetic. Expanding $B(t)=\\log(2\\sinh(t/2)/t)$ by the infinite product CITE{DLMF} gives coefficients $(-1)^{j+1}\\zeta(2j)/(j(2\\pi)^{2j})$. The generator integrates a finite polynomial in $t^2$ and $\\log t$ exactly in form, using ball coefficients. With $q=(x/(2\\pi))^2<1/10$, the omitted kernel tail after $N$ terms is bounded by $2q^{N+1}/((N+1)(1-q))$; a difference-of-powers bound is integrated and added to every result. Endpoint radicals and logarithms are enclosed and their errors propagated. At 100 requested digits, $N=150$ and 614-bit working arithmetic retain at least 547 bits of relative accuracy. The derivation is attached as computation.md. Independent 140-digit mpmath quadrature checks every value, with 175-digit repeats at four difficult cases; the checker also tests the dilogarithm formula and every stored alternating first-moment identity. Known-answer and deliberately wrong controls validate the checker. The attached executable checks and results distinguish these numerical checks from the rigorous error bound."
},
"Display properties": {
"number-header": "$\\operatorname{Lsh}_n^{(k)}(x)$"
},
"Numbers": {
"2": {
"0": {
"logphi": "0.8316472421721863689616628700820596456998847230196935282056992132653494973365017092722424705626947784",
"2logphi": "0.9869604401089358618834490999876151135313699407240790626413349376220044822419205243001773403718552232",
"2logsilver": "0.6885603792371556576720292986960330623610000339861255019932610311364092573515431512581030571477967464"
}
},
"3": {
"0": {
"logphi": "-1.920558570844575980359306539335704773936491706139814055426056102065234188949998177590788614444768818",
"2logphi": "-1.991089882371392968230758181371506714228628916274806756183866731846102530925464516611467853641321672",
"2logsilver": "-2.133836291964733045126740847998276306279356383350715642254543635531027805478607913990540727369073728"
},
"1": {
"2logphi": "0.2404113806319188570799476323022899981529972584680997763584543110683676411572626180372911747218670516",
"2logsilver": "-0.2027971653867127485613947205175517647164536982627038277333890843450460671356801713472345319693650946"
}
},
"4": {
"0": {
"logphi": "5.954091854150286887022779289711324584956160165232850994974845445820501176222992800083460315506639357",
"2logphi": "5.990954427333662221085106748708180940427344354228792633526814798897901054422674888999883635498441967",
"2logsilver": "5.915236506143167754844778574177040243597185064234240308295572588017845210866340514389589325642086422"
},
"1": {
"2logphi": "-0.2450752164550237958624617052561286374048717628812191953944903337648896550319136125399626878978259695",
"2logsilver": "-0.4661207128818308941799209621854503883862611169399606846900972446182191955940328953754716214035212911"
},
"2": {
"2logphi": "0.1035880131109227799385749806249477468593384003349826370025881738402959057376265310749152711185280718",
"2logsilver": "-0.5656963978854031797569491107674444693840882395568473817233017422476954798083550983375992794764510230"
}
},
"5": {
"0": {
"logphi": "-23.96703691755914213829482919684496686791838099503370809999665649832504625142430195004920705675751147",
"2logphi": "-23.98782774996036739008086693933519050078158200014743696518691790050998809675781401313022772994955727",
"2logsilver": "-24.03037054035826138494263678343550627602321602349952895745467733775141688577711715883358932178540906"
},
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"2logsilver": "0.2510174008484959781125374477621693612970577267001533149035925103942885996256772604064229409523434008"
},
"2": {
"2logphi": "-0.07097846975780870437600884496294466581226504002466154756722903803660887715960903406575991492979022369",
"2logsilver": "-0.4169883171974070579571371643981951543515167721411589146557937445353296386785284464541442504122109294"
},
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"2logphi": "0.05635021092254895548199207679907508608731180763021678659351978144719442201928257319033614936053741190",
"2logsilver": "-0.9693157542805685947205473687493779060705150061190425803317676320613475453922113445809062700007662857"
}
},
"6": {
"0": {
"logphi": "119.9673019555723062187209955803653207843245894285143937960422482859615955727899098198845720557624957",
"2logphi": "119.9795991140767549617852804401899260295177634710678367913287476417972577767963041882919641936540323",
"2logsilver": "119.9547970146880617482244974167698416628577529306980274860720161993020688198945638720267943957886361"
},
"1": {
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"2logsilver": "-0.8148830477083279483516619888567767476038080527678457301288556971462895587945130719082892274975837033"
},
"2": {
"2logphi": "0.07217812101045059169243780659423267293022775068444960651637610957743570625749114535983681149591197398",
"2logsilver": "-0.1201610646943565173591955726309151732585703538383183285341367068264648654905515344658204817602270588"
},
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"2logphi": "-0.02913913084544511266108445805391049200990890624238116393117913847044386045958514351725272706875794984",
"2logsilver": "-0.5759880939613256056977312798893099243639111102366883214322925736233275858437998051295690792055422507"
},
"4": {
"2logphi": "0.03482823759404336066830207968506789519067689562077743363797893061348368090558626742046699928487620242",
"2logsilver": "-1.557504324358268010108296880053671317439162755702770007542561349660849001932366815291169007208363335"
}
},
"7": {
"0": {
"logphi": "-719.9515467796277676055028434632198949376790915111168512655026830972305592317266606799615606481940446",
"2logphi": "-719.9590596202139214466857093815071116187750488057438630452200425028482207018652522897532957864256776",
"2logsilver": "-719.9738965971572269405645457924176495896131583132926120837652277033395806276534146396028407706548940"
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"1": {
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"2logsilver": "1.829666179828968134965623511208500811842877846528274325115412850670517482193927752496528622538352049"
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"2": {
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"2logphi": "0.02235822126444696699747966318839605598315300776045682433901318839840920165202925036951698424154233948",
"2logsilver": "-0.2874664562894984464751896241415333896110887990336814132478143044427942749678600135783771157641024518"
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"2logsilver": "-0.8861403107811902797093595874364662248361044341066912893360610139601258530060011129872963744465188864"
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"2logsilver": "-2.476990395500770208641809410550268417182117653732718882335541556381019168951894197561859070594000664"
}
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