Deep-Gauss constant:
The deep-gauss constant (denoted `g_d`) is a transcendental mathematical constant representing the value of the standard normal deviate with an equivalent standard deviation in a standard normal distribution. Only one such value exists.
Infinite nesting of Gaussian functions.
`PDF(x) = x`
Solution to `sqrt(-2ln(x)) = e^(-x^2/2) = x`
`sqrt(Omega)` where `Omega` is the omega constant
Mathematica: N[Sqrt[ProductLog[1]], 100]
May be calculated efficiently using Newton iteration.
0.753089164979674815796583362806350249743131750732524340813685844647225055609954728947071378202743825

Deeper-Gauss constant:
The only value such that the probability according to the standard normal function is equal to the standard deviation.
`CDF(x) = x`
`1/2 erf(x / sqrt(2)) + 1/2 = x`
0.783263929995929012648770847268103403231048133095485687445057379819632222390583200277636422370964984

Area under odd tetration of Gaussian functions:
Indefinite integral of `g(x)^{g(x)^{g(x)...}}` where `g(x) = e^(-x^2/2)` an odd number of times
Mathematica:
	PowerTower[a_, k_Integer] := Nest[Power[a, #] &, 1, k]
	NIntegrate[PowerTower[E^(-x^2/2), 10001], {x, -8, 8}]
3.23738

Elevation of area bifurcation of the Gaussian function:
The y-coordinate of the horizontal line on the standard Gaussian function which divides the area into two equal halves.
Value of a such that the `int_-oo^oo max(e^(-x^2/2) - a, 0) dx = sqrt(pi/2)`
Mathematica: Sqrt[Pi/2]-NIntegrate[Max[Exp[-x^2/2]-v,0] == 0
0.30636228415302713294

Balance point of the Gaussian function:
The height of the center of gravity of the standard Gaussian function.
Value of h such that `int_0^1 (x - h)*e^(-x^2/2)=0`
`1 / (2 sqrt(2))`
Mathematica: 1/(2*Sqrt[2])
OEIS A020765
0.353553390593273762200422181052424519642417968844237018294169934497683119615526759712596883581910393

Solution to `sin(x)=x-1`:
Mathematica: FindRoot[Sin[x]+1==x,{x,1.5}]
1.934563210752024267563261453768850027623302849353341290328231795541351269445927283100778722332410744

Solution to `cos(x)=x-1`:
Mathematica: FindRoot[Cos[x]==x-1,{x,1.5}]
1.283428741745765316797130600633044743705435627207677051364715283693170638284330854619597152925448185

The Dottie number:
Solution to `cos(x) = x`
Solution to arccos(x) = cos(x)
Mathematica: FindRoot[Cos[x]==x,{x,1.5}]
OEIS A003957
0.739085133215160641655312087673873404013411758900757464965680635773284654883547594599376106931766531

Solution to `tan(x) = x+1`:
Mathematica: FindRoot[Tan[x]==x+1,{x,1.0}]
1.132267725272885131625420696936001741528834429928502705956972230605059852849012697659932903448039734

Solution to `csc(x) = x+1`:
Mathematica: FindRoot[Csc[x]==x+1,{x,0.6}]
0.650751677964222006596404299979381935013853770031835224755443352442640151577649697854940226946445635

Solution to `cos(x) = tan(x)` near zero:
Solution to `(cos(x))^2 = sin(x)`
Mathematica: FindRoot[Cos[x]==Tan[x],{x,0.6}]
OEIS A175288
0.6662394324925152551040048959777927206674901387259478428314738428039789893790592817079068311695811353

Lower bound of radius of convergence of `x!!!!!...` near `-0.56`:
Range of convergence is approximately `(-0.5571, 2]`
Equal to the inverse factorial of 2 near `x=-0.56`
Mathematica: FindRoot[(x!)==2,{x,-0.56}]
-0.5571226035152725625479674834793943282896354619336335970087692801041475177158259195922990622701551779

Statistical deep-gauss constant (denoted `k`):
Solution to `e^(-x^2/2)/sqrt(2pi) = x`
Infinite nesting of Gaussian functions normalized to an area of 1.
`sqrt(LambertW(1/(2pi)))`
Mathematica: N[Sqrt[ProductLog[1/(2*Pi)]], 100]
May be calculated efficiently using Newton iteration.
0.3722388980356186389429106469095330269078295289174196538739184756542884705209392543696106849303749431

Classical deep-gauss constant:
Solution to `e^(-x^2)/sqrt(pi) = x`
Infinite nesting of Gaussian functions normalized to an area of 1, with variation = 1/2.
`sqrt(1/(2pi))`
Mathematica: N[Sqrt[ProductLog[2/Pi]/2], 100]
May be calculated efficiently using Newton iteration.
0.4575990762801477482110264370524857730600883392737928050972231875124445284808864507994380937381909696

Elevation such that the area of the Gaussian function under the elevation equals 1 - elevation:
Value of h such that `sqrt(2pi) - int_-sqrt(-2ln(1-h))^sqrt(-2ln(1-h)) e^(-x^2/2)-(1-h) dx = h`
Mathematica:
	g[a_]:=Sqrt[2*Pi]-NIntegrate[E^(-x^2/2)-a,{x,-Sqrt[-2*Log[a]],Sqrt[-2*Log[a]]},WorkingPrecision->100];
	N[b/. FindRoot[g[1-b]==b,{b,0.5},WorkingPrecision->100],100]
0.8215310678569499228996492062017322084477204695805055009194784666597653442831791413630644985491548807

Global maximum of first derivative of Gaussian function:
`1/sqrt(e)`
Mathematica: 1/Sqrt[E]
OEIS A092605
0.6065306597126334236037995349911804534419181354871869556828921587350565194137484239986476115079894560

Global maximum of second derivative of Gaussian function:
`2/e^(2/3)`
Mathematica: 2/E^(3/2)
0.4462603202968596578665609415280250426843432587221586574876706375206503332626288823551274606651551816

Global maximum of third derivative of Gaussian function:
`e^(1/2(-3 + sqrt(6)))sqrt(6(3 - sqrt(6)))`
Mathematica: Sqrt[6 (3-Sqrt[6])] E^(1/2 (-3+Sqrt[6]))
1.380119046160749111716612383227940318094682174207461803870163787656101914967089235003328880724784629

X coordinate of global maximum of third derivative of Gaussian function:
`sqrt(3 - sqrt(6))`
Mathematica: Sqrt[3-Sqrt[6]]
0.7419637843027258576485135967263602248295201475089189536114738789949997546500052952140537804671914252

Infinite summation of `1/(n^n)`:
Also known as "Sophomore's Dream"
`sum_{n=1}^oo 1/(n^n)`
Mathematica: Sum[1/(n^n),{n,1,Infinity}]
OEIS A073009
1.291285997062663540407282590595600541498619368274522317310002445136944538765234455558817041129429709

X coordinate of `tan(x) = cos(x)` near 0.6
Value of x such that `tan(x) = cos(x)` near 0.6
`arcsin((sqrt(5)-1)/2) = arcsin(phi-1) = arcsin(1/phi)` where `phi` is the golden ratio
Mathematica: ArcSin[GoldenRatio-1]
OEIS A175288
0.6662394324925152551040048959777927206674901387259478428314738428039789893790592817079068311695811353

Y coordinate of `tan(x) = cos(x)` near 0.6
Cosine or tangent of x such that `tan(x) = cos(x)` near 0.6
`tan(arcsin((sqrt(5)-1)/2)) = tan(arcsin(phi-1)) = cos(arcsin(phi-1) = sqrt(1-(phi-1)^2)` where `phi` is the golden ratio
Mathematica:
	Tan[ArcSin[GoldenRatio-1]]
	OR
	Cos[ArcSin[GoldenRatio-1]]
	OR
	Sqrt[1-(GoldenRatio-1)^2]
OEIS A197762
0.7861513777574232860695585858429589295231220578377232376649019701011820476223109137119128891585081356

First ten positive solutions to `x = csc(x)`
First ten positive X and Y coordinates of intersection between `y=x` and `y=csc(x)`
Mathematica: N[x/.FindRoot[x==Csc[x],{x,#},WorkingPrecision->100],100]&/@{1.1,2.8,6.4,9.3,12.6,15.6,18.9,21.9,25.2,28.2}
OEIS A133866 (for the first solution only)
1.114157140871930087300525178169203903954101376049375595337370555351019135450088826340464542817468949
2.772604708265991233953569721499279279322291225726785124329373158751894369005509293834845203192641825
6.439117238417246461724514840310879478656965054672609119153022903155477936959362644122838634851535248
9.317242941414809618601288513569511562449802182373635824335817177562093145056972653015421016315005368
12.64553257878910795240798538668067580398251798321028736603345183871121377161491841458998521879709135
15.64399737477314310180879264382812549863234072011157728642015901473289124731152616052576798512861511
18.90248373034244521760822967518209890277839177145978851539504303116477143307622122042002496265168765
21.94556549881968413352034035301547665667286719107257346703874804220094173195602758881297955883643044
25.17247761173437938733130203570685958875462290028172162507415108847817178157524858477515029963796502
28.23891435063900153941345335992077895624552020488688575034400582097258728370022198431175041116816101

First ten positive solutions to `x = tan(x)`:
First ten positive roots of the tanc function (see MathWorld, Tanc Function)
X and Y coordinates of intersection between `y=x` and `y=tan(x)`
Mathematica: N[x/.FindRoot[x==Tan[x],{x,#},WorkingPrecision->100],100]&/@{4.5,7.7,10.9,14.0,17.22,20.4,23.5,26.7,29.8,32.9}
OEIS A115365 (first solution)
OEIS A255272 (second solution)
4.493409457909064175307880927280322082215583872290040802895823961926950314597104098729057809455879692
7.725251836937707164195068933062986626378159304610791186649328216729645001682688816184504845740695787
10.90412165942889982714870279018868387209485885781875858875838037854152403129767924534795971449299529
14.06619391283147347997896560060156499694237831509622273835055936239790678068662399708995660145597824
17.22075527193076873957371892506096229893332971829167936478403467046043142257400443881724600687007969
20.37130295928756284509100440543504575049474077627828310900472481472406399769752471312034311206721095
23.51945249868900654645110203911830424635003114823871066842675410735279392086189788831563784209112032
26.66605425881267352839414354427355510035094967713474434380833716343936396570339416811983469926519861
29.81159879089295883681058812719248542381985582067960421020287818369901454045657431457801550155387214
32.95638903982247672529905648511000477277068028983182458330416774436300013420899180523883684352187886

First ten positive solutions to `tan(x) = arctan(x)`:
X coordinates of intersection between `y=tan(x)` and `y=arctan(x)`
Mathematica: Do[Print[N[x/.FindRoot[ArcTan[x]==Tan[x],{x,i},WorkingPrecision->100],100]],{i,4,Pi*11,Pi}]
4.067588865765862790917085025312411319068300674493957922637263436551465862660547101559028237704400117
7.244916979143423162128851608909041331842892378156524425323576343777403305384206366724156490826807976
10.39976929174371930405907564111851697598172656638934331811382119198533728423681943572174118400577866
13.54827694405954058495330601924441706870943461727181539607718809690811484000127196361383133772963852
16.69411428341000770067177353892660776364922715630448263835203166738729287107339853979265293824247756
19.83857763666212886481840336509319406712959936294659743758139982258975909320943020972277444943680702
22.98224115529880631878536875493410196445318251794518737318720553673262176981144465603465527703838756
26.12539822043640316400897413208107378179301792451908868463822083429805684715838262660376286494809566
29.26821436776660446929644858615114305285095418763229607502500523035319097842376205260885502218317458
32.41079008240094902391080855928580456854838496665698904277246524869031674587421418059832814570681321

Area under `tan(tan(x))` between 0 and 1:
Mathematica: N[NIntegrate[Tan[Tan[x]],{x,0,1},WorkingPrecision->100],100]
1.650310610726968108967907066271364240999490509346991959943053356050282006630973331053353071780822272

Infinite product of `1+1/n^2`:
`prod_{n=1}^oo (1+1/n^2)`
Limit of recursive function `f_{n+1} = f_n + f_n / (n+1)^2` where `f_0=1` as n approaches infinity
`sinh(pi)/pi = e^pi/(2pi) - e^(-pi)/(2pi) = 1/(picsch(pi)) = int_0^1 cosh(pix) dx`
Mathematica: Sinh[Pi]/Pi
3.676077910374977720695697492028260666507156346827630277478003593557447324111022073213255926590323023

Infinite product of `1+1/2^n`:
`prod_{n=1}^oo (1+1/2^n)`
`1 / (1 - P)` where `P` is the Pell constant
Mathematica: N[Product[1+1/2^n,{n,1,Infinity}],100]
2.384231029031371724149899288678397238771619516508433457692101507989181293036037255186535210365680520

Infinite product of `1+1/e^n`
`prod_{n=1}^oo (1+1/e^n)`
Mathematica: N[Product[1+1/E^n,{n,1,Infinity}],100]
1.677928684989354197439072369836846846820834870876556493041873514464050933830679696121607677171839395

Infinite product of `1+1/n!`
`prod_{n=1}^oo (1+1/n!)`
Mathematica: N[Product[1+1/n!,{n,1,90}],100]
3.682154136183628628186386254815526547827128418034453830396277476848119082203811990990702528185740895

Infinite sum of `n^-n`
`sum_{n=1}^oo (n^-n)`
Mathematica: N[Sum[n^-n,{n,1,100}],100]
1.291285997062663540407282590595600541498619368274522317310002445136944538765234455558817041129429709