Name	Mathematical constant	Value	Notation
One	$$1$$	1	M_ONE
Two	$$2$$	2	M_TWO
One half	$$1/2$$	0.5	M_ONE_HALF
Ludolph's number	$$π$$	3.1415926535898	M_PI
Tau	$$τ=2\cdot π$$	6.2831853071796	M_TAU
Euler's number	$$e=\sum_{n=0}^{\infty}\cfrac{1}{n!}=1+\cfrac{1}{1}+\cfrac{1}{1\cdot 2}+\cfrac{1}{1\cdot 2\cdot 3}+\cdots$$	2.718281828459	M_E
Euler's constant	$$γ=\lim_{n\rightarrow\infty}\left(-\log{n}+\sum_{k=1}^{n}\right)\cfrac{1}{k}$$	0.57721566490153	M_EULER
Apéry's constant	$$ζ(3)=\sum_{n=1}^{\infty}\cfrac{1}{n^3}=1+\cfrac{1}{2^3}+\cfrac{1}{3^3}+\cfrac{1}{4^3}+\cfrac{1}{5^3}+\cdots$$	1.2020569031596	M_APERY
Catalan's constant	$$G=\sum_{n=0}^{\infty}\cfrac{\left(-1\right)^n}{\left(2n+1\right)^2}=\cfrac{1}{1^2}-\cfrac{1}{3^2}+\cfrac{1}{5^2}-\cfrac{1}{7^2}+\cfrac{1}{9^2}-\cdots$$	0.91596559417722	M_CATALAN
Feigenbaum constant α	$$α$$	2.5029078750959	M_FEIGENBAUM_ALPHA
Feigenbaum constant δ	$$δ$$	4.669201609103	M_FEIGENBAUM_DELTA
Lemniscate constant	$$ϖ=2\int_{0}^{1}\cfrac{\text{d}t}{\sqrt{1-t^4}}$$	2.6220575542921	M_LEMNISCATE
Glaisher–Kinkelin constant	$$A$$	1.2824271291006	M_GLAISHER
Khinchin's constant	$$K_0=\lim_{n\rightarrow\infty}\left(a_1a_2\ldots a_n\right)^{1/n}$$	2.6854520010653	M_KHINCHIN
Golden Ratio	$$φ=\cfrac{1+\sqrt{5}}{2}$$	1.6180339887499	M_GOLDEN_RATIO
Silver Ratio	$$δ_S=\sqrt{2}+1$$	2.4142135623731	M_SILVER_RATIO
Supergolden Ratio	$$ψ=\cfrac{1+\sqrt[3]{\cfrac{29+3\cdot\sqrt{93}}{2}}+\sqrt[3]{\cfrac{29-3\cdot\sqrt{93}}{2}}}{3}$$	1.4655712318768	M_SUPERGOLDEN_RATIO
Connective constant	$$μ=\sqrt{2+\sqrt{2}}$$	1.8477590650226	M_CONNECTIVE
Kepler–Bouwkamp constant	$$K'=\prod_{n=3}^{\infty}\cos\left(\cfrac{π}{n}\right)=\cos\left(\cfrac{π}{3}\right)\cos\left(\cfrac{π}{4}\right)\cos\left(\cfrac{π}{5}\right)\cdots$$	0.1149420448533	M_KEPLER_BOUWKAMP
Erdős–Borwein constant	$$E=\sum_{n=1}^{\infty}\cfrac{1}{2^n-1}=\cfrac{1}{1}+\cfrac{1}{3}+\cfrac{1}{7}+\cfrac{1}{15}\cdots$$	1.6066951524153	M_ERDOS_BORWEIN
Omega constant	$$Ω=\cfrac{1}{π}\int_{0}^{π}\log\left(1+\cfrac{\sin t}{t}e^{t \cot t}\right)dt$$	0.56714329040978	M_OMEGA
Gauss's constant	$$G=\cfrac{1}{\text{agm}\left(1, \sqrt{2}\right)}=\cfrac{1}{4π}\sqrt{\cfrac{2}{π}}Γ\left(\cfrac{1}{4}\right)^2=\cfrac{ϖ}{π}$$	0.83462684167407	M_GAUSS
Second Hermite constant	$$γ_2=\cfrac{2}{\sqrt{3}}$$	1.1547005383793	M_SECOND_HERMITE
Liouville's constant	$$L=\sum_{n=1}^{\infty}\cfrac{1}{10^{n!}}=\cfrac{1}{10^{1!}}+\cfrac{1}{10^{2!}}+\cfrac{1}{10^{3!}}+\cfrac{1}{10^{4!}}+\cdots$$	0.110001	M_LIOUVILLE
Ramanujan's constant	$${e}^{π\cdot\sqrt{163}}$$	2.6253741264077E+17	M_RAMANUJAN
Dottie number	$$D$$	0.73908513321516	M_DOTTIE
Meissel-Mertens constant	$$M=\lim_{n\rightarrow\infty}\left(\sum_{p\le n}\cfrac{1}{p}-\ln\left(\ln n\right)\right)=γ+\sum_{p}\left(\ln\left(1-\cfrac{1}{p}\right)+\cfrac{1}{p}\right)$$	0.26149721284764	M_MEISSEL_MERTENS
Universal parabolic constant	$$ \ln{\left(1+\sqrt{2}\right)}+\sqrt{2}$$	2.2955871493926	M_UNIVERSAL_PARABOLIC
Cahen's constant	$$C=\sum_{k=1}^{\infty}\cfrac{\left(-1\right)^k}{s_k-1}=\cfrac{1}{1}-\cfrac{1}{2}+\cfrac{1}{6}-\cfrac{1}{42}+\cfrac{1}{1806}\pm\cdots$$	0.64341054628834	M_CAHEN
Gelfond's constant	$${e}^π$$	23.140692632779	M_GELFOND
Gelfond-Schneider constant	$$2^{\sqrt{2}}$$	2.6651441426902	M_GELFOND_SCHNEIDER
Second Favard constant	$$K_2=\cfrac{π^2}{8}$$	1.2337005501362	M_SECOND_FAVARD
Golden angle	$$g=π\cdot\left(3-\sqrt{5}\right)$$	2.3999632297287	M_GOLDEN_ANGLE
Sierpiński's constant	$$K=π\left(2γ+\ln\cfrac{4π^3}{Γ\left(\cfrac{1}{4}\right)^4}\right)=π\left(2γ+4\ln Γ\left(\cfrac{3}{4}\right)-\ln π\right)=π\left(2\ln2+3\ln π+2γ-4\ln Γ\left(\cfrac{1}{4}\right)\right)$$	2.5849817595793	M_SIERPINSKI
Landau-Ramanujan constant	$$b=\cfrac{1}{\sqrt{2}}\prod_{p\equiv 3\ (\mod 4)}\left(1-\cfrac{1}{p^2}\right)^{-\cfrac{1}{2}}=\cfrac{π}{4}\prod_{p\equiv 1\ (\mod 4)}\left(1-\cfrac{1}{p^2}\right)^{\cfrac{1}{2}}$$	0.76422365358922	M_LANDAU_RAMANUJAN
First Nielsen-Ramanujan constant	$$a_1=\cfrac{π^2}{12}$$	0.82246703342411	M_FIRST_NIELSEN_RAMANUJAN
Gieseking constant	$$G=\cfrac{3\sqrt{3}}{4}\left(1-\sum_{n=0}^{\infty}\cfrac{1}{\left(3n+2\right)^2}+\sum_{n=1}^{\infty}\cfrac{1}{\left(3n+1\right)^2}\right)=\cfrac{\sqrt{3}}{4}\left(\cfrac{ψ_1\left(1/3\right)}{2}-\cfrac{π^2}{3}\right)$$	1.0149416064097	M_GIESEKING
Bernstein's constant	$$β=\lim_{n\rightarrow\infty}2nE_{2n}\left(f\right)$$	0.28016949902387	M_BERNSTEIN
Tribonacci constant	$$\cfrac{1+\sqrt[3]{19+3\cdot\sqrt{33}}+\sqrt[3]{19-3\cdot\sqrt{33}}}{3}$$	1.8392867552142	M_TRIBONACCI
Brun's constant	$$B_2=\sum_{p}\left(\cfrac{1}{p}+\cfrac{1}{p+2}\right)=\left(\cfrac{1}{3}+\cfrac{1}{5}\right)+\left(\cfrac{1}{5}+\cfrac{1}{7}\right)+\left(\cfrac{1}{11}+\cfrac{1}{13}\right)\cdots$$	1.902160583104	M_BRUN
Twin primes constant	$$C_2=\prod_{p\ \text{prime,}\ p\geq3}\left(1-\cfrac{1}{\left(p-1\right)^2}\right)$$	0.66016181584687	M_TWIN_PRIMES
Plastic Ratio	$$ρ=\sqrt[3]{\cfrac{1}{2}+\cfrac{\sqrt{69}}{18}}+\sqrt[3]{\cfrac{1}{2}-\cfrac{\sqrt{69}}{18}}$$	1.3247179572447	M_PLASTIC_RATIO
Prouhet-Thue-Morse constant	$$τ=\sum_{n=0}^{\infty}\cfrac{t_n}{2^{n+1}}=\cfrac{1}{4}\left[2-\prod_{n=0}^{\infty}\left(1-\cfrac{1}{2^{2^n}}\right)\right]$$	0.41245403364011	M_PROUHET_THUE_MORSE
Golomb-Dickman constant	$$λ=\int_0^1e^{Li\left(t\right)}dt=\int_0^{\infty}\cfrac{ρ\left(t\right)}{t+2}dt$$	0.62432998854355	M_GOLOMB_DICKMAN
Lebesgue constant	$$c=\lim_{n\rightarrow\infty}\left(L_n-\cfrac{4}{π^2}\ln\left(2n+1\right)\right)$$	0.98943127383115	M_LEBESGUE
Feller-Tornier constant	$$C_{FT}=\cfrac{1}{2}\prod_{p\ \text{prime}}\left(1-\cfrac{2}{p^2}\right)+\cfrac{1}{2}$$	0.66131704946962	M_FELLER_TORNIER
Champernowne constant	$$C_{10}=0.1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13\ 14\cdots$$	0.12345678910111	M_CHAMPERNOWNE
Salem constant	$$σ_{10}=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$$	1.1762808182599	M_SALEM
Lévy's constant	$$β=\cfrac{π^2}{12\cdot\ln{2}}$$	1.1865691104156	M_LEVY
Copeland-Erdős constant	$$C_{CE}=0.2\ 3\ 5\ 7\ 11\ 13\ 17\ 19\ 23\ 29\ 31\ 37\cdots$$	0.23571113171923	M_COPELAND_ERDOS
Mills' constant	$$\left\lfloor A^{3^n}\right\rfloor$$	1.3063778838631	M_MILLS
Gompertz constant	$$δ=\int_0^{\infty}\cfrac{e^{-x}}{1+x}dx$$	0.59634736232319	M_GOMPERTZ
Van der Pauw constant	$$\cfrac{π}{\ln{2}}$$	4.5323601418272	M_VAN_DER_PAUW
Magic angle	$$θ_m=\tan^{-1}{\sqrt{2}}$$	0.95531661812451	M_MAGIC_ANGLE
Artin's constant	$$C_{Artin}=\prod_{p\ \text{prime}}\left(1-\cfrac{1}{p\left(p-1\right)}\right)$$	0.3739558136192	M_ARTIN
Porter's constant	$$C=\cfrac{6\ln 2}{π^2}\left(3\ln 2+4γ-\cfrac{24}{π^2}ζ'\left(2\right)-2\right)-\cfrac{1}{2}$$	1.467078079434	M_PORTER
Lochs constant	$$L=\cfrac{6\ln 2\ln 10}{π^2}$$	0.97027011439203	M_LOCHS
Lieb's square ice constant	$$\left(\cfrac{4}{3}\right)^{\cfrac{3}{2}}$$	1.539600717839	M_LIEB_SQUARE_ICE
Niven's constant	$$C=1+\sum_{n=2}^{\infty}\left(1-\cfrac{1}{ζ\left(n\right)}\right)$$	1.7052111401054	M_NIVEN
Stephens' constant	$$C_S=\prod_{p\ \text{prime}}\left(1-\cfrac{p}{p^3-1}\right)$$	0.57595996889295	M_STEPHENS
Zero	$$0$$	0	M_ZERO
Negative one	$$-1$$	-1	M_NEGATIVE_ONE
Square Root of 2	$$\sqrt{2}$$	1.4142135623731	M_SQRT2
Square Root of 3	$$\sqrt{3}$$	1.7320508075689	M_SQRT3
Square Root of 5	$$\sqrt{5}$$	2.2360679774998	M_SQRT5
Cube Root of 2	$$\sqrt[3]{2}$$	1.2599210498949	M_CURT2
Cube Root of 3	$$\sqrt[3]{3}$$	1.4422495703074	M_CURT3
Twelfth Root of 2	$$\sqrt[12]{2}$$	1.0594630943593	M_TWRT2
Natural Log of 2	$$\ln(2)$$	0.69314718055995	M_LN2
Natural Log of 10	$$\ln(10)$$	2.302585092994	M_LN10
Natural Log of Pi	$$\ln(π)$$	1.1447298858494	M_LNPI
Base 10 Log of e	$$\log10(e)$$	0.43429448190325	M_LOG10E
Base 2 Log of e	$$\log2(e)$$	1.442695040889	M_LOG2E
Half of Pi	$$π/2$$	1.5707963267949	M_PI_2
Quarter of Pi	$$π/4$$	0.78539816339745	M_PI_4
Inverse of Pi	$$1/π$$	0.31830988618379	M_1_PI
Two over Pi	$$2/π$$	0.63661977236758	M_2_PI
Square Root of Pi	$$\sqrt{π}$$	1.7724538509055	M_SQRTPI
Two over Square Root of Pi	$$2/\sqrt{π}$$	1.1283791670955	M_2_SQRTPI
Inverse of Square Root of 2	$$1/\sqrt{2}$$	0.70710678118655	M_SQRT1_2

Name

Mathematical constant

Value

Notation

One

$$1$$

M_ONE

Two

$$2$$

M_TWO

One half

$$1/2$$

0.5

M_ONE_HALF

Ludolph's number

$$π$$

3.1415926535898

M_PI

Tau

$$τ=2\cdot π$$

6.2831853071796

M_TAU

Euler's number

$$e=\sum_{n=0}^{\infty}\cfrac{1}{n!}=1+\cfrac{1}{1}+\cfrac{1}{1\cdot 2}+\cfrac{1}{1\cdot 2\cdot 3}+\cdots$$

2.718281828459

M_E

Euler's constant

$$γ=\lim_{n\rightarrow\infty}\left(-\log{n}+\sum_{k=1}^{n}\right)\cfrac{1}{k}$$

0.57721566490153

M_EULER

Apéry's constant

$$ζ(3)=\sum_{n=1}^{\infty}\cfrac{1}{n^3}=1+\cfrac{1}{2^3}+\cfrac{1}{3^3}+\cfrac{1}{4^3}+\cfrac{1}{5^3}+\cdots$$

1.2020569031596

M_APERY

Catalan's constant

$$G=\sum_{n=0}^{\infty}\cfrac{\left(-1\right)^n}{\left(2n+1\right)^2}=\cfrac{1}{1^2}-\cfrac{1}{3^2}+\cfrac{1}{5^2}-\cfrac{1}{7^2}+\cfrac{1}{9^2}-\cdots$$

0.91596559417722

M_CATALAN

Feigenbaum constant α

$$α$$

2.5029078750959

M_FEIGENBAUM_ALPHA

Feigenbaum constant δ

$$δ$$

4.669201609103

M_FEIGENBAUM_DELTA

Lemniscate constant

$$ϖ=2\int_{0}^{1}\cfrac{\text{d}t}{\sqrt{1-t^4}}$$

2.6220575542921

M_LEMNISCATE

Glaisher–Kinkelin constant

$$A$$

1.2824271291006

M_GLAISHER

Khinchin's constant

$$K_0=\lim_{n\rightarrow\infty}\left(a_1a_2\ldots a_n\right)^{1/n}$$

2.6854520010653

M_KHINCHIN

Golden Ratio

$$φ=\cfrac{1+\sqrt{5}}{2}$$

1.6180339887499

M_GOLDEN_RATIO

Silver Ratio

$$δ_S=\sqrt{2}+1$$

2.4142135623731

M_SILVER_RATIO

Supergolden Ratio

$$ψ=\cfrac{1+\sqrt[3]{\cfrac{29+3\cdot\sqrt{93}}{2}}+\sqrt[3]{\cfrac{29-3\cdot\sqrt{93}}{2}}}{3}$$

1.4655712318768

M_SUPERGOLDEN_RATIO

Connective constant

$$μ=\sqrt{2+\sqrt{2}}$$

1.8477590650226

M_CONNECTIVE

Kepler–Bouwkamp constant

$$K'=\prod_{n=3}^{\infty}\cos\left(\cfrac{π}{n}\right)=\cos\left(\cfrac{π}{3}\right)\cos\left(\cfrac{π}{4}\right)\cos\left(\cfrac{π}{5}\right)\cdots$$

0.1149420448533

M_KEPLER_BOUWKAMP

Erdős–Borwein constant

$$E=\sum_{n=1}^{\infty}\cfrac{1}{2^n-1}=\cfrac{1}{1}+\cfrac{1}{3}+\cfrac{1}{7}+\cfrac{1}{15}\cdots$$

1.6066951524153

M_ERDOS_BORWEIN

Omega constant

$$Ω=\cfrac{1}{π}\int_{0}^{π}\log\left(1+\cfrac{\sin t}{t}e^{t \cot t}\right)dt$$

0.56714329040978

M_OMEGA

Gauss's constant

$$G=\cfrac{1}{\text{agm}\left(1, \sqrt{2}\right)}=\cfrac{1}{4π}\sqrt{\cfrac{2}{π}}Γ\left(\cfrac{1}{4}\right)^2=\cfrac{ϖ}{π}$$

0.83462684167407

M_GAUSS

Second Hermite constant

$$γ_2=\cfrac{2}{\sqrt{3}}$$

1.1547005383793

M_SECOND_HERMITE

Liouville's constant

$$L=\sum_{n=1}^{\infty}\cfrac{1}{10^{n!}}=\cfrac{1}{10^{1!}}+\cfrac{1}{10^{2!}}+\cfrac{1}{10^{3!}}+\cfrac{1}{10^{4!}}+\cdots$$

0.110001

M_LIOUVILLE

Ramanujan's constant

$${e}^{π\cdot\sqrt{163}}$$

2.6253741264077E+17

M_RAMANUJAN

Dottie number

$$D$$

0.73908513321516

M_DOTTIE

Meissel-Mertens constant

$$M=\lim_{n\rightarrow\infty}\left(\sum_{p\le n}\cfrac{1}{p}-\ln\left(\ln n\right)\right)=γ+\sum_{p}\left(\ln\left(1-\cfrac{1}{p}\right)+\cfrac{1}{p}\right)$$

0.26149721284764

M_MEISSEL_MERTENS

Universal parabolic constant

$$ \ln{\left(1+\sqrt{2}\right)}+\sqrt{2}$$

2.2955871493926

M_UNIVERSAL_PARABOLIC

Cahen's constant

$$C=\sum_{k=1}^{\infty}\cfrac{\left(-1\right)^k}{s_k-1}=\cfrac{1}{1}-\cfrac{1}{2}+\cfrac{1}{6}-\cfrac{1}{42}+\cfrac{1}{1806}\pm\cdots$$

0.64341054628834

M_CAHEN

Gelfond's constant

$${e}^π$$

23.140692632779

M_GELFOND

Gelfond-Schneider constant

$$2^{\sqrt{2}}$$

2.6651441426902

M_GELFOND_SCHNEIDER

Second Favard constant

$$K_2=\cfrac{π^2}{8}$$

1.2337005501362

M_SECOND_FAVARD

Golden angle

$$g=π\cdot\left(3-\sqrt{5}\right)$$

2.3999632297287

M_GOLDEN_ANGLE

Sierpiński's constant

$$K=π\left(2γ+\ln\cfrac{4π^3}{Γ\left(\cfrac{1}{4}\right)^4}\right)=π\left(2γ+4\ln Γ\left(\cfrac{3}{4}\right)-\ln π\right)=π\left(2\ln2+3\ln π+2γ-4\ln Γ\left(\cfrac{1}{4}\right)\right)$$

2.5849817595793

M_SIERPINSKI

Landau-Ramanujan constant

$$b=\cfrac{1}{\sqrt{2}}\prod_{p\equiv 3\ (\mod 4)}\left(1-\cfrac{1}{p^2}\right)^{-\cfrac{1}{2}}=\cfrac{π}{4}\prod_{p\equiv 1\ (\mod 4)}\left(1-\cfrac{1}{p^2}\right)^{\cfrac{1}{2}}$$

0.76422365358922

M_LANDAU_RAMANUJAN

First Nielsen-Ramanujan constant

$$a_1=\cfrac{π^2}{12}$$

0.82246703342411

M_FIRST_NIELSEN_RAMANUJAN

Gieseking constant

$$G=\cfrac{3\sqrt{3}}{4}\left(1-\sum_{n=0}^{\infty}\cfrac{1}{\left(3n+2\right)^2}+\sum_{n=1}^{\infty}\cfrac{1}{\left(3n+1\right)^2}\right)=\cfrac{\sqrt{3}}{4}\left(\cfrac{ψ_1\left(1/3\right)}{2}-\cfrac{π^2}{3}\right)$$

1.0149416064097

M_GIESEKING

Bernstein's constant

$$β=\lim_{n\rightarrow\infty}2nE_{2n}\left(f\right)$$

0.28016949902387

M_BERNSTEIN

Tribonacci constant

$$\cfrac{1+\sqrt[3]{19+3\cdot\sqrt{33}}+\sqrt[3]{19-3\cdot\sqrt{33}}}{3}$$

1.8392867552142

M_TRIBONACCI

Brun's constant

$$B_2=\sum_{p}\left(\cfrac{1}{p}+\cfrac{1}{p+2}\right)=\left(\cfrac{1}{3}+\cfrac{1}{5}\right)+\left(\cfrac{1}{5}+\cfrac{1}{7}\right)+\left(\cfrac{1}{11}+\cfrac{1}{13}\right)\cdots$$

1.902160583104

M_BRUN

Twin primes constant

$$C_2=\prod_{p\ \text{prime,}\ p\geq3}\left(1-\cfrac{1}{\left(p-1\right)^2}\right)$$

0.66016181584687

M_TWIN_PRIMES

Plastic Ratio

$$ρ=\sqrt[3]{\cfrac{1}{2}+\cfrac{\sqrt{69}}{18}}+\sqrt[3]{\cfrac{1}{2}-\cfrac{\sqrt{69}}{18}}$$

1.3247179572447

M_PLASTIC_RATIO

Prouhet-Thue-Morse constant

$$τ=\sum_{n=0}^{\infty}\cfrac{t_n}{2^{n+1}}=\cfrac{1}{4}\left[2-\prod_{n=0}^{\infty}\left(1-\cfrac{1}{2^{2^n}}\right)\right]$$

0.41245403364011

M_PROUHET_THUE_MORSE

Golomb-Dickman constant

$$λ=\int_0^1e^{Li\left(t\right)}dt=\int_0^{\infty}\cfrac{ρ\left(t\right)}{t+2}dt$$

0.62432998854355

M_GOLOMB_DICKMAN

Lebesgue constant

$$c=\lim_{n\rightarrow\infty}\left(L_n-\cfrac{4}{π^2}\ln\left(2n+1\right)\right)$$

0.98943127383115

M_LEBESGUE

Feller-Tornier constant

$$C_{FT}=\cfrac{1}{2}\prod_{p\ \text{prime}}\left(1-\cfrac{2}{p^2}\right)+\cfrac{1}{2}$$

0.66131704946962

M_FELLER_TORNIER

Champernowne constant

$$C_{10}=0.1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12\ 13\ 14\cdots$$

0.12345678910111

M_CHAMPERNOWNE

Salem constant

$$σ_{10}=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1$$

1.1762808182599

M_SALEM

Lévy's constant

$$β=\cfrac{π^2}{12\cdot\ln{2}}$$

1.1865691104156

M_LEVY

Copeland-Erdős constant

$$C_{CE}=0.2\ 3\ 5\ 7\ 11\ 13\ 17\ 19\ 23\ 29\ 31\ 37\cdots$$

0.23571113171923

M_COPELAND_ERDOS

Mills' constant

$$\left\lfloor A^{3^n}\right\rfloor$$

1.3063778838631

M_MILLS

Gompertz constant

$$δ=\int_0^{\infty}\cfrac{e^{-x}}{1+x}dx$$

0.59634736232319

M_GOMPERTZ

Van der Pauw constant

$$\cfrac{π}{\ln{2}}$$

4.5323601418272

M_VAN_DER_PAUW

Magic angle

$$θ_m=\tan^{-1}{\sqrt{2}}$$

0.95531661812451

M_MAGIC_ANGLE

Artin's constant

$$C_{Artin}=\prod_{p\ \text{prime}}\left(1-\cfrac{1}{p\left(p-1\right)}\right)$$

0.3739558136192

M_ARTIN

Porter's constant

$$C=\cfrac{6\ln 2}{π^2}\left(3\ln 2+4γ-\cfrac{24}{π^2}ζ'\left(2\right)-2\right)-\cfrac{1}{2}$$

1.467078079434

M_PORTER

Lochs constant

$$L=\cfrac{6\ln 2\ln 10}{π^2}$$

0.97027011439203

M_LOCHS

Lieb's square ice constant

$$\left(\cfrac{4}{3}\right)^{\cfrac{3}{2}}$$

1.539600717839

M_LIEB_SQUARE_ICE

Niven's constant

$$C=1+\sum_{n=2}^{\infty}\left(1-\cfrac{1}{ζ\left(n\right)}\right)$$

1.7052111401054

M_NIVEN

Stephens' constant

$$C_S=\prod_{p\ \text{prime}}\left(1-\cfrac{p}{p^3-1}\right)$$

0.57595996889295

M_STEPHENS

Zero

$$0$$

M_ZERO

Negative one

$$-1$$

-1

M_NEGATIVE_ONE

Square Root of 2

$$\sqrt{2}$$

1.4142135623731

M_SQRT2

Square Root of 3

$$\sqrt{3}$$

1.7320508075689

M_SQRT3

Square Root of 5

$$\sqrt{5}$$

2.2360679774998

M_SQRT5

Cube Root of 2

$$\sqrt[3]{2}$$

1.2599210498949

M_CURT2

Cube Root of 3

$$\sqrt[3]{3}$$

1.4422495703074

M_CURT3

Twelfth Root of 2

$$\sqrt[12]{2}$$

1.0594630943593

M_TWRT2

Natural Log of 2

$$\ln(2)$$

0.69314718055995

M_LN2

Natural Log of 10

$$\ln(10)$$

2.302585092994

M_LN10

Natural Log of Pi

$$\ln(π)$$

1.1447298858494

M_LNPI

Base 10 Log of e

$$\log10(e)$$

0.43429448190325

M_LOG10E

Base 2 Log of e

$$\log2(e)$$

1.442695040889

M_LOG2E

Half of Pi

$$π/2$$

1.5707963267949

M_PI_2

Quarter of Pi

$$π/4$$

0.78539816339745

M_PI_4

Inverse of Pi

$$1/π$$

0.31830988618379

M_1_PI

Two over Pi

$$2/π$$

0.63661977236758

M_2_PI

Square Root of Pi

$$\sqrt{π}$$

1.7724538509055

M_SQRTPI

Two over Square Root of Pi

$$2/\sqrt{π}$$

1.1283791670955

M_2_SQRTPI

Inverse of Square Root of 2

$$1/\sqrt{2}$$

0.70710678118655

M_SQRT1_2

Function	Description	Sample example
ABS	Returns the absolute value of a number. The absolute value of a number is the same number without a sign.	ABS(-9)=9
ACOT	Returns the value of the arccotangent (inverse cotangent) of the specified number.	ACOT(3.14)=0.30831566219543
ACOTH	Returns the inverse hyperbolic cotangent of the specified number.	ACOTH(3.14)=0.32994497940173
ARCCOS	Returns the arccosine (the inverse of the cosine function) of the specified number. The arccosine is the angle whose cosine is the specified number. The resulting angle is given in radians in the range zero to pi.	ARCCOS(0.5)=1.0471975511966
ARCCOSH	Returns the inverse hyperbolic cosine of a number. The number must be greater than or equal to 1. The inverse hyperbolic cosine is a value whose hyperbolic cosine is a number, so ACOSH(COSH(number)) equals the number.	ARCCOSH(2)=1.3169578969248
ARCSIN	Returns the arcsine (the inverse of the sine) of the specified number. The arcsine is the angle whose sine is the specified number. The resulting angle is given in radians in the interval from pi/2 to pi/2.	ARCSIN(1)=1.5707963267949
ARCSINH	Returns the hyperbolic arcsine of the specified number. The hyperbolic arcsine is a value whose hyperbolic sine is the specified number, so ARCSINH(SINH(number)) is equal to the specified number.	ARCSINH(1)=0.88137358701954
ARCTG	Returns the arctangent (the inverse of the tangent function) of the specified number. The arctangent is the angle whose tangent is the specified number. The resulting angle is given in radians in the range pi/2 to pi/2.	ARCTG(2)=1.1071487177941
ARCTG2	Returns the arctangent (the inverse of tangent) of the specified x and y coordinates. The arctangent is the angle between the x axis and the line containing the origin (0;0) and the point at coordinates (x_number, y_number). This angle is given in radians in the interval -pi to pi, except for the value -pi.	ARCTG2(2, 3)=0.98279372324733
ARCTGH	Returns the arctangent (the inverse of the tangent function) of the specified number. The arctangent is the angle whose tangent is the specified number. The resulting angle is given in radians in the range pi/2 to pi/2.	ARCTGH(0.5)=0.54930614433405
AVERAGEA	Calculates the average (arithmetic mean) of the values in the argument list.	AVERAGEA(10, 20, 30)=20
CEILING	Rounds a number up to the nearest integer or to the nearest multiple of significance.	CEILING(1.2)=2
COS	Returns the cosine of the specified angle.	COS(1)=0.54030230586814
COSH	Returns the hyperbolic cosine of the specified number.	COSH(1)=1.5430806348152
COT	Returns the cotangent of an angle specified in radians.	COT(1)=0.64209261593433
COTH	Returns the hyperbolic cotangent of a hyperbolic angle.	COTH(1)=1.3130352854993
CSC	Returns the cosecant of an angle specified in radians.	CSC(1)=1.1883951057781
CSCH	Returns the hyperbolic cosecant of an angle specified in radians.	CSCH(1)=0.85091812823932
DEGREES	Converts radians to degrees.	DEGREES(PI())=180
EXP	Returns e raised to the power of the argument number. The constant e is equal to 2.718281828459, the base of natural logarithms.	EXP(1)=2.718281828459
FACT	Returns the factorial of a number. The factorial of a number is equal to 123... number.	FACT(12)=479001600
FACTDOUBLE	Returns the double factorial of the specified number.	FACTDOUBLE(12)=46080
FISHER	Returns the Fisher transformation value of x. This transformation produces a function with a normal distribution rather than a skewed distribution. You can use this function to test the hypothesis about the correlation coefficient.	FISHER(0.9)=1.4722194895832
FISHERINV	Returns the inverse of the Fisher transform. You can use this transform to analyze correlations between regions or matrices of data. If y = FISHER(x), FISHERINV(y) = x.	FISHERINV(10)=0.99999999587769
FLOOR	Rounds a number down to the nearest whole number or to the nearest multiple of the specified value.	FLOOR(1.2)=1
LCM	Returns the least common multiple of integers. The least common multiple is the smallest positive integer that is a multiple of all integer arguments number1, number2, and so on. Use LCM to add fractions with different denominators.	LCM(10, 6)=30
LN	Returns the natural logarithm of a number. The base of the natural logarithm is the constant e (2.71828182845904).	LN(10)=2.302585092994
LOG	Returns the base 10 logarithm of a number.	LOG(10)=1
LOGZ	Returns the logarithm of a number to a given base.	LOGZ(10, 3)=2.0959032742894
MAX	Returns the maximum value in a set of values.	MAX(10, -10)=10
MEDIAN	Returns the median of the specified numbers. The median is the number that lies in the middle of a set of numbers.	MEDIAN(10, 20, 15, 1)=12.5
MIN	Returns the minimum value in a set of values.	MIN(10, -10)=-10
MOD	Returns the remainder after dividing a number by a divisor.	MOD(20, 3)=2
PI	Returns the number 3.1415926535898, the mathematical constant pi.	PI()=3.1415926535898
POWER	Returns the power of a number.	POWER(3, 2)=9
RADIANS	Converts degrees to radians.	RADIANS(180)=3.1415926535898
SEC	Returns the secant of an angle.	SEC(0.5)=1.1394939273245
SIN	Returns the sine of a given angle.	SIN(PI()/6)=0.5
SINH	Returns the hyperbolic sine of a number.	SINH(PI()/6)=0.54785347388804
SQRT	Returns the positive square root.	SQRT(100)=10
TG	Returns the tangent of the specified angle.	TG(1)=1.5574077246549
TGH	Returns the hyperbolic tangent of the argument.	TGH(1)=0.76159415595576

Function

Description

Sample example

ABS

Returns the absolute value of a number. The absolute value of a number is the same number without a sign.

ABS(-9)=9

ACOT

Returns the value of the arccotangent (inverse cotangent) of the specified number.

ACOT(3.14)=0.30831566219543

ACOTH

Returns the inverse hyperbolic cotangent of the specified number.

ACOTH(3.14)=0.32994497940173

ARCCOS

Returns the arccosine (the inverse of the cosine function) of the specified number. The arccosine is the angle whose cosine is the specified number. The resulting angle is given in radians in the range zero to pi.

ARCCOS(0.5)=1.0471975511966

ARCCOSH

Returns the inverse hyperbolic cosine of a number. The number must be greater than or equal to 1. The inverse hyperbolic cosine is a value whose hyperbolic cosine is a number, so ACOSH(COSH(number)) equals the number.

ARCCOSH(2)=1.3169578969248

ARCSIN

Returns the arcsine (the inverse of the sine) of the specified number. The arcsine is the angle whose sine is the specified number. The resulting angle is given in radians in the interval from pi/2 to pi/2.

ARCSIN(1)=1.5707963267949

ARCSINH

Returns the hyperbolic arcsine of the specified number. The hyperbolic arcsine is a value whose hyperbolic sine is the specified number, so ARCSINH(SINH(number)) is equal to the specified number.

ARCSINH(1)=0.88137358701954

ARCTG

Returns the arctangent (the inverse of the tangent function) of the specified number. The arctangent is the angle whose tangent is the specified number. The resulting angle is given in radians in the range pi/2 to pi/2.

ARCTG(2)=1.1071487177941

ARCTG2

Returns the arctangent (the inverse of tangent) of the specified x and y coordinates. The arctangent is the angle between the x axis and the line containing the origin (0;0) and the point at coordinates (x_number, y_number). This angle is given in radians in the interval -pi to pi, except for the value -pi.

ARCTG2(2, 3)=0.98279372324733

ARCTGH

ARCTGH(0.5)=0.54930614433405

AVERAGEA

Calculates the average (arithmetic mean) of the values in the argument list.

AVERAGEA(10, 20, 30)=20

CEILING

Rounds a number up to the nearest integer or to the nearest multiple of significance.

CEILING(1.2)=2

COS

Returns the cosine of the specified angle.

COS(1)=0.54030230586814

COSH

Returns the hyperbolic cosine of the specified number.

COSH(1)=1.5430806348152

COT

Returns the cotangent of an angle specified in radians.

COT(1)=0.64209261593433

COTH

Returns the hyperbolic cotangent of a hyperbolic angle.

COTH(1)=1.3130352854993

CSC

Returns the cosecant of an angle specified in radians.

CSC(1)=1.1883951057781

CSCH

Returns the hyperbolic cosecant of an angle specified in radians.

CSCH(1)=0.85091812823932

DEGREES

Converts radians to degrees.

DEGREES(PI())=180

EXP

Returns e raised to the power of the argument number. The constant e is equal to 2.718281828459, the base of natural logarithms.

EXP(1)=2.718281828459

FACT

Returns the factorial of a number. The factorial of a number is equal to 1*2*3*...* number.

FACT(12)=479001600

FACTDOUBLE

Returns the double factorial of the specified number.

FACTDOUBLE(12)=46080

FISHER

Returns the Fisher transformation value of x. This transformation produces a function with a normal distribution rather than a skewed distribution. You can use this function to test the hypothesis about the correlation coefficient.

FISHER(0.9)=1.4722194895832

FISHERINV

Returns the inverse of the Fisher transform. You can use this transform to analyze correlations between regions or matrices of data. If y = FISHER(x), FISHERINV(y) = x.

FISHERINV(10)=0.99999999587769

FLOOR

Rounds a number down to the nearest whole number or to the nearest multiple of the specified value.

FLOOR(1.2)=1

LCM

Returns the least common multiple of integers. The least common multiple is the smallest positive integer that is a multiple of all integer arguments number1, number2, and so on. Use LCM to add fractions with different denominators.

LCM(10, 6)=30

Returns the natural logarithm of a number. The base of the natural logarithm is the constant e (2.71828182845904).

LN(10)=2.302585092994

LOG

Returns the base 10 logarithm of a number.

LOG(10)=1

LOGZ

Returns the logarithm of a number to a given base.

LOGZ(10, 3)=2.0959032742894

MAX

Returns the maximum value in a set of values.

MAX(10, -10)=10

MEDIAN

Returns the median of the specified numbers. The median is the number that lies in the middle of a set of numbers.

MEDIAN(10, 20, 15, 1)=12.5

MIN

Returns the minimum value in a set of values.

MIN(10, -10)=-10

MOD

Returns the remainder after dividing a number by a divisor.

MOD(20, 3)=2

Returns the number 3.1415926535898, the mathematical constant pi.

PI()=3.1415926535898

POWER

Returns the power of a number.

POWER(3, 2)=9

RADIANS

Converts degrees to radians.

RADIANS(180)=3.1415926535898

SEC

Returns the secant of an angle.

SEC(0.5)=1.1394939273245

SIN

Returns the sine of a given angle.

SIN(PI()/6)=0.5

SINH

Returns the hyperbolic sine of a number.

SINH(PI()/6)=0.54785347388804

SQRT

Returns the positive square root.

SQRT(100)=10

Returns the tangent of the specified angle.

TG(1)=1.5574077246549

TGH

Returns the hyperbolic tangent of the argument.

TGH(1)=0.76159415595576

Contact stress of sphere on flat surface

Values for calculation

Total force

$ F $ $ \mathrm{N} $

Young's modulus for $ E_1 $

$ E_1 $ $ \mathrm{MPa} $

Poisson's ratio for $ ν_1 $

$ ν_1 $

Hardened material 1

Hardness for $ HB_1 $

$ HB_1 $ $ \mathrm{HB} $

The minimum yield strength for $ S_{y-1} $

$ S_{y-1} $ $ \mathrm{MPa} $

Sphere diameter $ D_2 $

$ D_2 $ $ \mathrm{mm} $

Young's modulus for $ E_2 $

$ E_2 $ $ \mathrm{MPa} $

Poisson's ratio for $ ν_2 $

$ ν_2 $

Hardened material 2

Hardness for $ HB_2 $

$ HB_2 $ $ \mathrm{HB} $

The minimum yield strength for $ S_{y-2} $

$ S_{y-2} $ $ \mathrm{MPa} $

Coefficient according to load

Load	Value
Static load	1
Unidirectional load, non-impact load	0.8
Unidirectional load, with a small impact load	0.7
Unidirectional load, with a big impact load	0.6
Alternating load, with a small impact load	0.45
Alternating load, with a big impact load	0.25

$ C_c $

Safety factor

$ S_F $

Calculation

Allowable Hertz pressure

$\text{if }\ \text{Hardened }$$\text{material }$$\text{1}= \text{yes}\wedge\text{Hardened }$$\text{material }$$\text{2}= \text{yes}$

$$σ_H=\min\left(\cfrac{4.2\cdot S_{y-1}}{S_F}\cdot C_c, \cfrac{4.2\cdot S_{y-2}}{S_F}\cdot C_c\right)$$

$\text{else if }\ \text{Hardened }$$\text{material }$$\text{1}= \text{yes}\wedge\text{Hardened }$$\text{material }$$\text{2}= \text{no}$

$$σ_H=\min\left(\cfrac{4.2\cdot S_{y-1}}{S_F}\cdot C_c, \cfrac{7\cdot HB_2}{S_F}\cdot C_c\right)$$

$\text{else if }\ \text{Hardened }$$\text{material }$$\text{1}= \text{no}\wedge\text{Hardened }$$\text{material }$$\text{2}= \text{no}$

$$σ_H=\min\left(\cfrac{7\cdot HB_1}{S_F}\cdot C_c, \cfrac{7\cdot HB_2}{S_F}\cdot C_c\right)$$

$\text{else}$

$$σ_H=\min\left(\cfrac{7\cdot HB_1}{S_F}\cdot C_c, \cfrac{4.2\cdot S_{y-2}}{S_F}\cdot C_c\right)$$

Material coefficient

$$C_E=\cfrac{1-ν_1^2}{E_1}+\cfrac{1-ν_2^2}{E_2}$$

Dimensional coefficient

$$K_D=D_2$$

Diameter of sphere contact area

$$b=1.442\cdot\sqrt[3]{F\cdot K_D \cdot C_E}$$

Contact stress

$$σ_c=0.918\cdot\sqrt[3]{\cfrac{F}{K_D^2\cdot C_E^2}}$$

$$σ_c\le σ_H$$

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