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
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

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

Circular rounded contraction

Values for calculation

Diameter $ d_1 $

$ d_1 $ $ \mathrm{mm} $

Diameter $ d_2 $

$ d_2 $ $ \mathrm{mm} $

Radius

$ r $ $ \mathrm{mm} $

Calculation

Diameter ratio

$$β=\cfrac{d_2}{d_1}$$

Jet contraction ratio

$$λ=1+0.622\cdot\left(1-0.3\cdot\sqrt{\cfrac{r}{d_2}}-0.7\cdot\cfrac{r}{d_2}\right)^4\cdot\left(1-0.215\cdot β^2-0.785\cdot β^5\right)$$

Loss coefficient

$\text{if }\ r/d_2\geq 1$

$$ζ=0.03\cdot\left(1-β\right)\cdot\left(1-β^4\right)$$

$\text{else}$

$$ζ=0.0696\cdot\left(1-0.569\cdot\cfrac{r}{d_2}\right)\cdot\left(1-\sqrt{\cfrac{r}{d_2}}\cdot β\right)\cdot\left(1-β^5\right)\cdot λ^2+\left(λ-1\right)^2$$

Discharge coefficient

$$μ=\cfrac{1}{\sqrt{ζ+1}}$$

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