Log-sinh-cosh integrals

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Numbers

$m$

$n$

$x$

$H_{m,n}(x)$

$a$:

-0.3381670221177184380199383200882520889341997526576539968850317444543472562155414471221538003767671668

$2a$:

-0.7034210104549882867812999565433247297039066019169092687429509689937806869534993164579608388757277190

$a$:

-0.2376789064211439807996731132196747868655108968631156209168759245866607008491598513297503913916994561

$a$:

-0.1670763276287538889417420341085095211581382866635554280468761141706826862212701441373161200227995316

$a$:

-0.1174639368483985951699266754923688151178280021640183511466217110442348410797459182607167026813119524

$a$:

-0.08259618415900924156907725181732778955486163985082364519197839456630993439847446567766931523815533268

$a$:

-0.05808736213013233265825804983837215723882048164003663784157264655820919270636970395084465266691734930

$a$:

-0.04085733103314098303728784103456826735567581096715825372171165704256686839517498284297714753222362304

$a$:

-0.02874255810141740192488107227719026013262056721168567679643787834789579140598267692929449831412384285

$a$:

0.5813462680400117385218002809348131018438440724327631491254993303644218121682128853074025395079037188

$2a$:

0.6968814588299875388807653634800273499800201206961823646643533561461358858239125808140137487744625852

$a$:

0.4064232339277725705484997135155685701401840619493960879080668320439632895911808586295227272102213093

$a$:

0.2841651041355944995807994927386306490995417595206800122666614873593642386785332515043067194544072172

$a$:

0.1987065686289248524136185376261804199334302162725663262343992870507141267149611374010752445926208498

$a$:

0.1389644296420430726190685169396115226669992537086543793124527808270615136153817082390202221964104220

$a$:

0.09719539967824520762840658502198350667179811587124821973253632253692992366438408903322083805963783291

$a$:

0.06798906950019744301271282089058461472435121924242435091080425423605318447787765926428879424302531367

$a$:

-1.336936004879339866394661007265143434668300629101029171296254142777585943854642595778256853121760923

$2a$:

-1.388916905460569920014623693065789000649320725316078219361699021092325908491255243483990845285273415

$a$:

-0.9307200639177042751336081450669331785193531235365613239893651671903517307116291794816329376924836503

$a$:

-0.6479677767576786100701315410883919969534195847024207047155843013640257733013951716788772622652471493

$a$:

-0.4511428811108196243691028765839072881194387769123680914659735069373339666255307631154837516033803509

$a$:

-0.3141244022421326724826003470662609488600442918478809339647202926217582797920560109980950593315557827

$a$:

-0.2187341623515586586923995351301686098599856801041586676719436071393576608889400407831668976956615954

$a$:

4.134561736597801172725693325529204226840842425065163965854841078204735118420195733154345160700329457

$2a$:

4.161589658683757898629756818132167115484360782980181991458588785650004356028072154914710508749486280

$a$:

2.871131635327092337665479363783528125677918709655582320906826595701942388644882219535349800619706887

$a$:

1.993822982700431685939410298166365227203147562798051383130360540116006232692475773926358140558314386

$a$:

1.384618606860613720381696740461086354135282233718026696843661115354414761288067392017332403484962422

$a$:

0.9615768711073654910822817325280864294241755372532419295462950766648861229292453659134313617529865480

$a$:

-16.62401552864273620685616980470705867338459767680481990965406308982587870740059188512620445004100516

$2a$:

-16.63921087322873574668135508686432670452608431870475327184244880633054998043083034599505020380042209

$a$:

-11.53082497238611510206306035817528774663233813821598066942992198399793660059747395904999367772068504

$a$:

-7.998116612654211590465802790102478961382089189955728235983500691561099942049834938728324557514772805

$a$:

-5.547764888515403972085190451746563617383891446426522325248134573593586849221832688256032199150416339

$a$:

83.17489598370734465946634637978794385445701575643287230928178989485187047438394489059277245052739230

$2a$:

83.18386424140858867377433655978091728125074287972887819321242084301407666755725953790471962463370418

$a$:

57.66633515043114033395165089633383490938804013651963993898206775419356995929705629831787116491984046

$a$:

39.98095461052796927783790681227653376503786441798734698851454217425171341729340769074221288503593654

$a$:

-499.0729921974372128517522364793903564153060724078459364740535082467263534971355479059015280505693979

$2a$:

-499.0784628486343392453795707596011617671982810389326309397937219689102753780348687774880661533665600

$a$:

-345.9593139540027351383440622110973978697701285259999341093371721098682228133067076887086367606811651

$a$:

3493.487646935524700256177035079039538434641831929886202166789761261641326272956062613521437460978515

$2a$:

3493.491064504809780140704631702211772417913415532784979054296415006298816700208241481857347357016397

Definition

The log-sinh-cosh integral is $H_{m,n}(x)=\operatorname{Lshch}_{m,n}(x)=-\int_0^x\log^{m-1}(2\sinh(t/2))\,\log^{n-1}(2\cosh(t/2))\,dt$, with natural logarithms [1], Definition 2.1. For positive integers $m,n$ and $x>0$, this is an ordinary convergent improper integral.

Parameters

$m$

— sinh logarithmic order ($m\geq1$)

$n$

— cosh logarithmic order ($n\geq1$)

$x$

— positive real endpoint ($x>0$; endpoint labels use $a=\log((1+\sqrt5)/2)$)

Formulas

(1)

$H_{m,1}(x)=\operatorname{Lsh}_m^{(0)}(x)$, where $\operatorname{Lsh}_m^{(0)}(x)=-\int_0^x A(t)^{m-1}\,dt$.

(2)

For integers $r\geq0$, $\operatorname{Lsh}_{r+1}^{(0)}(2x)=2\sum_{j=0}^r\binom rj H_{j+1,r-j+1}(x)$. This follows from $A(2t)=A(t)+C(t)$ and a change of variable.

(3)

$H_{1,2}(x)=-x^2/4-\operatorname{Li}_2(-e^{-x})-\pi^2/12$, where $\operatorname{Li}_2(z)=\sum_{k\geq1}z^k/k^2$ for $|z|<1$. Differentiating and taking the limit at $x=0$ gives the identity.

(4)

For integers $p\geq1$, $\displaystyle\sum_{j=1}^{\infty}\frac{(-1)^j\binom{2j}{j}h_{2j}}{4^{2j+1}(2j+1)^p}=\frac{(-1)^{p-1}}{(p-1)!}\left(\frac12H_{p,2}(2a)-\sum_{l=1}^p\binom{p-1}{l-1}H_{l,p-l+2}(a)\right)$, where $h_k=\sum_{i=1}^k1/i$. This is [1], equation (2.5), specialized at $z=1/4$; $\log(2\sinh a)=0$ leaves only its $j=p$ term.

(5)

Let $u=\tanh^2(x/2)$ and $G(\alpha,\beta;x)=2^{\alpha+\beta}B_u((\alpha+1)/2,-(\alpha+\beta)/2)$. Then $H_{m,n}(x)=-\left.\partial_\alpha^{m-1}\partial_\beta^{n-1}G(\alpha,\beta;x)\right|_{\alpha=\beta=0}$. Here $B_u(v,w)=\int_0^u q^{v-1}(1-q)^{w-1}\,dq$; since $0<u<1$, it converges for real $v>0$ and any real $w$. Substituting $q=\tanh^2(t/2)$ gives $G=\int_0^x e^{\alpha A(t)+\beta C(t)}\,dt$ for real $\alpha>-1$, which justifies differentiation near zero.

Comments

(6)

The endpoint $a$ is the logarithm of the golden ratio. Equivalently, $a=\operatorname{arsinh}(1/2)$. Square roots and logarithms use their positive real branches.

(7)

Write $A(t)=\log(2\sinh(t/2))$ and $C(t)=\log(2\cosh(t/2))$. The leading minus sign, factors $2$ and half-arguments are retained without factorial normalization. The logarithm $A(t)$ can be negative; its absolute value is not taken. The orders $(m,n)$ are ordered and have no reflection symmetry interchanging them.

(8)

Near zero, $A(t)=\log t+O(t^2)$ and $C(t)=\log2+O(t^2)$. Every finite logarithmic power is integrable there, so no regularization is used.

(9)

The harmonic-weighted binomial identity [1], Lemma 2.2, at $z=1/4$ selects $a=\operatorname{arsinh}(1/2)$ and $2a$. Its integer orders $p=1,\ldots,8$ require exactly the mixed powers selected here, including the pure-cosh boundary $m=1$. The endpoint $2a$ is also the unique positive zero of $A$. The boundary $n=1$ is a basic log-sinh integral and is supplied by the separate family Generalized log-sinh integrals.

Programs

(P1)

Python (mpmath)

import mpmath as mp

def log_sinh_cosh(m, n, endpoint, dps=140):
    with mp.workdps(dps):
        a = mp.log((1 + mp.sqrt(5))/2)
        x = {"logphi": a, "2logphi": 2*a}[endpoint]
        if m < 1 or n < 1:
            raise ValueError("require m,n>=1")
        def f(v):
            t = x*mp.exp(-v)
            return -t*mp.log(2*mp.sinh(t/2))**(m-1)*mp.log(2*mp.cosh(t/2))**(n-1)
        return mp.quad(f, [0, 1, 4, 16, 64, 256, mp.inf])

def selected_arguments():
    for m in range(1, 9):
        for n in range(2, 11-m):
            yield m, n, "logphi"
            if n == 2:
                yield m, n, "2logphi"

log_sinh_cosh(2, 2, "2logphi")
# This is numerical quadrature; generate.py supplies rigorous bounds.

References

[1]

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)

Links

[2]

DLMF 4.36.1-2: Infinite products for sinh and cosh

Similar tables

Generalized log-sinh integrals supply the pure-sinh boundary and the left-hand side of (2); the mixed powers here define a distinct quantity.

Log-sine-cosine integrals are the real-angle trigonometric counterpart in [1]. Their angle divided by $\pi$ is different from the positive real endpoint used here.

Incomplete beta values give related function values. (5) uses parameter derivatives at a second beta parameter of zero, beyond that table's positive-parameter domain; it does not rescale those stored values.

Polylogarithms at rational arguments provide related special-function context. (3) uses algebraic exponential arguments, and [1], section 4, gives higher polylogarithmic reductions.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every ordered $m\geq1$, $n\geq2$ with $m+n\leq10$ at $x=a$ (36 entries), and exactly $n=2$, $1\leq m\leq8$ at $x=2a$ (8 entries). These are the union of the terms in (4) for the noncanonical finite prefix $p=1,\ldots,8$, covering balanced and unbalanced powers and both parities. The conditional ranges and their purposes are documented in argument-selection.md)

How they were obtained:

All 44 values carry 100 significant digits. The products [2] give convergent series for $\log(2\sinh(t/2)/t)$ and $\log\cosh(t/2)$. The generator integrates a finite polynomial in $t^2$ and $\log t$ in Arb ball arithmetic. A difference-of-powers estimate propagates both kernel tails into an explicit integrated error, added to every result radius.

Exact endpoint radicals and logarithms are enclosed and their errors propagated. At 100 requested digits, 160 terms and 614 working bits retain at least 552 bits of relative accuracy. The attached computation.md derives the bounds. Each decimal interval contains the full computed enclosure. Independent 140-digit mpmath quadrature checks all 44 values, with 175-digit repeats at four extreme or balanced cases. The checker tests both dilogarithm cases, all eight harmonic-binomial identities and eight duplication identities, recomputing the pure-sinh boundary directly. Known-answer and deliberately wrong controls check the checker. Stored intervals are read back and checked again; the executable code and measured results are attached.