Partial derivative of $φ$ with respect to $τ$
Values for calculation
Calculation
Partial derivative of $ φ $ with respect to $ τ $
$\text{if }\ \text{Region}= 3$$$φ_τ=\sum_{i=2}^{40}{n[i]\cdot δ^{I[i]}\cdot J[i]\cdot τ^{J[i]-1}}$$
$$φ_τ=0$$
| 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 |
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\text{if }\ \text{Region}= 3φ_τ=\sum_{i=2}^{40}{n[i]\cdot δ^{I[i]}\cdot J[i]\cdot τ^{J[i]-1}}\text{else}φ_τ=0