Function $F$
Values for calculation
Calculation
Function $ F $
$$F=1167.0521452767\cdot β^2+12020.824702470\cdot β-4823.2657361591$$
| 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 |
| 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 |
| AVEDEV | Returns the average of the absolute deviations of data points from their mean. STANDARD DEVIATION is a measure of the variability of a data set. | AVEDEV(10, 30, 2, 3)=9.375 |
| 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 |
| COUNT | Returns the number of items in the argument list. | COUNT(1, 2)=2 |
| 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(9.5)=362880 |
| 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 |
| MODE | Returns the most frequently occurring or recurring value in a range of data. | MODE(20, 3, 20, 3, 26, 3)=3 |
| MROUND | Returns a number rounded to the desired multiple. | MROUND(20, 2)=20 |
| MULTINOMIAL | Returns the factorial of the sum of values divided by the product of their factorials. | MULTINOMIAL(10, 20, 30)=3.553261127085E+24 |
| PERMUT | Returns the number of permutations for a given number of objects that can be selected from number objects. A permutation is any set or subset of objects or events where internal order is significant. Permutations are different from combinations, for which the internal order is not significant. Use this function for lottery-style probability calculations. | PERMUT(10, 2)=90 |
| PERMUTATIONA | Returns the number of permutations for a given number of objects (with repetitions) that can be selected from the total objects. | PERMUTATIONA(10, 2)=100 |
| PHI | Returns the value of the density function for the standard deviation. | PHI(2)=0.053990966513188 |
| PI | Returns the number 3.1415926535898, the mathematical constant pi. | PI()=3.1415926535898 |
| POWER | Returns the power of a number. | POWER(3, 2)=9 |
| QUOTIENT | Returns the whole part of a division. This function is used to remove the remainder after division. | QUOTIENT(3, 2)=1 |
| RADIANS | Converts degrees to radians. | RADIANS(180)=3.1415926535898 |
| RAND | RAND returns uniformly distributed random real numbers greater than or equal to 0 and less than 1. A new random real number is returned each time the worksheet is recalculated. | RAND()=0.49304542899739 |
| RANDBETWEEN | Returns a random integer from the specified interval. A new random integer is returned each time the worksheet is recalculated. | RANDBETWEEN(180, 220)=207 |
| ROUNDDOWN | Rounds a number down towards zero. | ROUNDDOWN(180.66669, 3)=180.666 |
| ROUNDUP | Rounds a number up, away from zero. | ROUNDUP(180.66669, 3)=180.667 |
| SEC | Returns the secant of an angle. | SEC(0.5)=1.1394939273245 |
| SECH | Returns the hyperbolic secant of an angle. | SECH(1)=0.64805427366389 |
| 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 |
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F=1167.0521452767\cdot β^2+12020.824702470\cdot β-4823.2657361591