Mathematical constant
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A mathematical constant is a number, which has a special meaning for calculations. For example, the constant π (pronounced "pie") means the ratio of a circle's circumference to its diameter.[1][2] This value is always the same for any circle. A mathematical constant is often a real, non-integral number of interest.[3]
In contrast to physical constants, mathematical constants do not come from physical measurements.
Key mathematical constants
The following table contains some important mathematical constants:
| Name | Symbol | Value | Meaning |
|---|---|---|---|
| Pi, Archimedes' constant or Ludoph's number | π | ≈3.141592653589793 | A transcendental number that is the ratio of the length of a circle's circumference to its diameter. It is also the area of the unit circle. |
| e, Napier's constant or Euler's (pronounced "oilers") number | e | ≈2.718281828459045 | A transcendental number that is the base of natural logarithms. |
| Phi, golden ratio | φ | ≈1.618033988749894 | It is the value of a larger value divided by a smaller value if this is equal to the value of the sum of the values divided by the larger value. |
| Square root of 2, Pythagoras' constant | √2 | ≈1.414213562373095 | An irrational number that is the length of the diagonal of a square with sides of length 1. This number can not be written as a fraction. |
Constants and series
The following table contains a list of constants and series in mathematics, with the following columns:
- Value: Numerical value of the constant.
- LaTeX: Formula or series in TeX format.
- Formula: For use in programs such as Mathematica or Wolfram Alpha.
- OEIS: Link to On-Line Encyclopedia of Integer Sequences (OEIS), where the constants are available with more details.
- Continued fraction: In the simple form [to integer; frac1, frac2, frac3, ...] (in brackets if periodic)
- Type:
- R - Rational number
- I - Irrational number
- T - Transcendental number
- C - Complex number
Note that the list can be ordered correspondingly by clicking on the header title at the top of the table.
| Value | Name | Symbol | LaTeX | Formula | Type | OEIS | Continued fraction |
|---|---|---|---|---|---|---|---|
| 3.24697960371746706105000976800847962 | Silver, Tutte–Beraha constant | [math]\displaystyle{ \varsigma }[/math] | [math]\displaystyle{ 2+2 \cos(2\pi/7)= \textstyle 2+\frac{2+\sqrt[3]{7 + 7 \sqrt[3]{7 + 7 \sqrt[3]{\, 7 + \cdots}}}}{1+\sqrt[3]{7 + 7 \sqrt[3]{7 + 7 \sqrt[3]{\, 7 + \cdots}}}} }[/math] | 2+2 cos(2Pi/7) | I | A116425 | [3;4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,...] |
| 1.09864196439415648573466891734359621 | Paris constant | [math]\displaystyle{ C_{Pa} }[/math] | [math]\displaystyle{ \prod_{n=2}^\infty \frac{2 \varphi}{\varphi+ \varphi_n}\; , \varphi= {Fi} }[/math] | I | A105415 | [1;10,7,3,1,3,1,5,1,4,2,7,1,2,3,22,1,2,5,2,1,...] | |
| 2.74723827493230433305746518613420282 | Ramanujan nested radical R5 | [math]\displaystyle{ R_{5} }[/math] | [math]\displaystyle{ \scriptstyle \sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5-\cdots}}}}}}}\;=\textstyle\frac{2+\sqrt{5}+\sqrt{15-6\sqrt{5}}}{2} }[/math] | (2+sqrt(5)+sqrt(15-6 sqrt(5)))/2 | I | [2;1,2,1,21,1,7,2,1,1,2,1,2,1,17,4,4,1,1,4,2,...] | |
| 2.23606797749978969640917366873127624 | Square root of 5, Gauss sum | [math]\displaystyle{ \sqrt{5} }[/math] | [math]\displaystyle{ \scriptstyle \forall \, n=5, \displaystyle \sum_{k=0}^{n-1} e^{\frac{2 k^2 \pi i}{n}} = 1 + e^\frac{2 \pi i} {5} + e^\frac{8 \pi i} {5} + e^\frac{18 \pi i} {5} + e^\frac{32 \pi i} {5} }[/math] | Sum[k=0 to 4]{e^(2k^2 pi i/5)} | I | A002163 | [2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...] = [2;(4),...] |
| 3.62560990822190831193068515586767200 | Gamma(1/4) | [math]\displaystyle{ \Gamma(\tfrac14) }[/math] | [math]\displaystyle{ 4 \left(\frac{1}{4}\right)! = \left(-\frac{3}{4}\right)! }[/math] | 4(1/4)! | T | A068466 | [3;1,1,1,2,25,4,9,1,1,8,4,1,6,1,1,19,1,1,4,1,...] |
| 0.18785964246206712024851793405427323 | MRB constant, Marvin Ray Burns | [math]\displaystyle{ C_{_{MRB}} }[/math] | [math]\displaystyle{ \sum_{n=1}^{\infty} ({-}1)^n (n^{1/n}{-}1) = - \sqrt[1]{1} + \sqrt[2]{2} - \sqrt[3]{3} + \sqrt[4]{4}\,\dots }[/math] | Sum[n=1 to ∞]{(-1)^n (n^(1/n)-1)} | T | A037077 | [0;5,3,10,1,1,4,1,1,1,1,9,1,1,12,2,17,2,2,1,...] |
| 0.11494204485329620070104015746959874 | Kepler–Bouwkamp constant | [math]\displaystyle{ {\rho} }[/math] | [math]\displaystyle{ \prod_{n=3}^\infty \cos\left(\frac\pi n\right) = \cos\left(\frac\pi 3\right) \cos\left(\frac\pi 4\right) \cos\left(\frac\pi 5\right) \dots }[/math] | prod[n=3 to ∞]{cos(pi/n)} | T | A085365 | [0;8,1,2,2,1,272,2,1,41,6,1,3,1,1,26,4,1,1,...] |
| 1.78107241799019798523650410310717954 | Exp(gamma) G-Barnes function |
[math]\displaystyle{ e^{\gamma} }[/math] | [math]\displaystyle{ \prod_{n=1}^\infty \frac{e^{\frac{1}{n}}}{1+\tfrac1n} = \prod_{n=0}^\infty \left(\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}\right)^{\frac{1}{n+1}} = }[/math]
[math]\displaystyle{ \textstyle \left ( \frac{2}{1} \right )^{1/2} \left (\frac{2^2}{1 \cdot 3} \right )^{1/3} \left (\frac{2^3 \cdot 4}{1 \cdot 3^3} \right )^{1/4} \left (\frac{2^4 \cdot 4^4}{1 \cdot 3^6 \cdot 5} \right )^{1/5}\dots }[/math] |
Prod[n=1 to ∞]{e^(1/n)}/{1 + 1/n} | T | A073004 | [1;1,3,1,1,3,5,4,1,1,2,2,1,7,9,1,16,1,1,1,2,...] |
| 1.28242712910062263687534256886979172 | Glaisher–Kinkelin constant | [math]\displaystyle{ {A} }[/math] | [math]\displaystyle{ e^{\frac{1}{12}-\zeta^{\prime}(-1)} = e^{\frac{1}{8}-\frac{1}{2}\sum\limits_{n=0}^{\infty} \frac{1}{n+1} \sum\limits_{k=0}^{n} \left(-1\right)^k \binom{n}{k} \left(k+1\right)^2 \ln(k+1)} }[/math] | e^(1/2-zeta´{-1}) | T | A074962 | [1;3,1,1,5,1,1,1,3,12,4,1,271,1,1,2,7,1,35,...] |
| 7.38905609893065022723042746057500781 | Schwarzschild conic constant | [math]\displaystyle{ e^2 }[/math] | [math]\displaystyle{ \sum_{n = 0}^\infty \frac{2^n}{n!} = 1+2+\frac{2^2}{2!}+\frac{2^3}{3!}+\frac{2^4}{4!}+\frac{2^5}{5!}+\dots }[/math] | Sum[n=0 to ∞]{2^n/n!} | T | A072334 | [7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...] = [7,2,(1,1,n,4*n+6,n+2)], n = 3, 6, 9, etc. |
| 1.01494160640965362502120255427452028 | Gieseking constant | [math]\displaystyle{ {G_{Gi}} }[/math] | [math]\displaystyle{ \frac{3\sqrt{3}}{4} \left(1- \sum_{n=0}^\infty \frac{1}{(3n+2)^2}+ \sum_{n=1}^\infty\frac{1}{(3n+1)^2} \right)= }[/math] [math]\displaystyle{ \textstyle \frac{3\sqrt{3}}{4} \left( 1 - \frac{1}{2^2} + \frac{1}{4^2}-\frac{1}{5^2}+\frac{1}{7^2}-\frac{1}{8^2}+\frac{1}{10^2} \pm \dots \right) }[/math]. |
T | A143298 | [1;66,1,12,1,2,1,4,2,1,3,3,1,4,1,56,2,2,11,...] | |
| 2.62205755429211981046483958989111941 | Lemniscata constant | [math]\displaystyle{ {\varpi} }[/math] | [math]\displaystyle{ \pi \, {G} = 4 \sqrt{\tfrac2\pi} \,(\tfrac14 !)^2 }[/math] | 4 sqrt(2/pi) (1/4!)^2 | T | A062539 | [2;1,1,1,1,1,4,1,2,5,1,1,1,14,9,2,6,2,9,4,1,...] |
| 0.83462684167407318628142973279904680 | G, Gauss constant | [math]\displaystyle{ {G} }[/math] | [math]\displaystyle{ \underset{ Agm:\; Arithmetic-geometric \; mean} {\frac{1}{\mathrm{agm}(1, \sqrt{2})} = \frac{4 \sqrt{2} \,(\tfrac14 !)^2}{\pi ^{3/2}}} }[/math] | (4 sqrt(2)(1/4!)^2)/pi^(3/2) | T | A014549 | [0;1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,7,1,...] |
| 1.01734306198444913971451792979092052 | Zeta(6) | [math]\displaystyle{ \zeta(6) }[/math] | [math]\displaystyle{ \frac{\pi^6}{945} = \prod_{n=1}^\infty \underset{p_{n}: \, {primo}}\frac{1}{{1-p_n}^{-6}} = \frac{1}{1{-}2^{-6}}{\cdot}\frac{1}{1{-}3^{-6}}{\cdot}\frac{1}{1{-}5^{-6}} ... }[/math] | Prod[n=1 to ∞] {1/(1-ithprime(n)^-6)} | T | A013664 | [1;57,1,1,1,15,1,6,3,61,1,5,3,1,6,1,3,3,6,1,...] |
| 0,60792710185402662866327677925836583 | Constante de Hafner-Sarnak-McCurley | [math]\displaystyle{ \frac{1}{\zeta(2)} }[/math] | [math]\displaystyle{ \frac{6}{\pi^2} {=} \prod_{n = 0}^\infty \underset{p_{n}: \, {primo}}{\left(1- \frac{1}{{p_n}^2}\right)}{=}\textstyle \left(1{-}\frac{1}{2^2}\right)\left(1{-}\frac{1}{3^2}\right)\left(1{-}\frac{1}{5^2}\right)\dots }[/math] | Prod{n=1 to ∞} (1-1/ithprime(n)^2) | T | A059956 | [0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...] |
| 1.11072073453959156175397024751517342 | The ratio of a square and circumscribed or inscribed circles | [math]\displaystyle{ \frac{\pi}{2\sqrt 2} }[/math] | [math]\displaystyle{ \sum_{n = 1}^\infty \frac{(-1)^{\lfloor \frac{n-1}{2}\rfloor}}{2n+1} = \frac{1}{1} + \frac{1}{3} - \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \frac{1}{11} - \dots }[/math] | sum[n=1 to ∞]{(-1)^(floor((n-1)/2))/(2n-1)} | T | A093954 | [1;9,31,1,1,17,2,3,3,2,3,1,1,2,2,1,4,9,1,3,...] |
| 2.80777024202851936522150118655777293 | Fransén–Robinson constant | [math]\displaystyle{ {F} }[/math] | [math]\displaystyle{ \int_{0}^\infty \frac{1}{\Gamma(x)}\, dx. = e + \int_0^\infty \frac{e^{-x}}{\pi^2 + \ln^2 x}\, dx }[/math] | N[int[0 to ∞] {1/Gamma(x)}] | T | A058655 | [2;1,4,4,1,18,5,1,3,4,1,5,3,6,1,1,1,5,1,1,1...] |
| 1.64872127070012814684865078781416357 | Square root of e | [math]\displaystyle{ \sqrt e }[/math] | [math]\displaystyle{ \sum_{n = 0}^\infty \frac{1}{2^n n!} = \sum_{n = 0}^\infty \frac{1}{(2n)!!} = \frac{1}{1}+\frac{1}{2}+\frac{1}{8}+\frac{1}{48}+\cdots }[/math] | sum[n=0 to ∞]{1/(2^n n!)} | T | A019774 | [1;1,1,1,5,1,1,9,1,1,13,1,1,17,1,1,21,1,1,...] = [1;1,(1,1,4p+1)], p∈ℕ |
| i | i, imaginary unit | [math]\displaystyle{ {i} }[/math] | [math]\displaystyle{ \sqrt{-1} = \frac{\ln(-1)}{\pi} \qquad\qquad \mathrm{e}^{i\,\pi} = -1 }[/math] | sqrt(-1) | C | ||
| 262537412640768743.999999999999250073 | Hermite-Ramanujan constant | [math]\displaystyle{ {R} }[/math] | [math]\displaystyle{ e^{\pi\sqrt{163}} }[/math] | e^(π sqrt(163)) | T | A060295 | [262537412640768743;1,1333462407511,1,8,1,1,5,...] |
| 4.81047738096535165547303566670383313 | John constant | [math]\displaystyle{ \gamma }[/math] | [math]\displaystyle{ \sqrt[i]{i} = i^{-i} = i^{\frac{1}{i}} = (i^i)^{-1} = e^{\frac{\pi}{2}} }[/math] | e^(π/2) | T | A042972 | [4;1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,3,...] |
| 4.53236014182719380962768294571666681 | Constante de Van der Pauw | [math]\displaystyle{ \alpha }[/math] | [math]\displaystyle{ \frac{\pi}{ln(2)} = \frac{\sum_{n = 0}^\infty \frac{4(-1)^n}{2n+1}} {\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}} = \frac{\frac{4}{1} {-} \frac{4}{3} {+} \frac{4}{5} {-} \frac{4}{7} {+} \frac{4}{9} - \dots} {\frac{1}{1}{-}\frac{1}{2}{+}\frac{1}{3}{-}\frac{1}{4}{+}\frac{1}{5}- \dots} }[/math] | π/ln(2) | T | A163973 | [4;1,1,7,4,2,3,3,1,4,1,1,4,7,2,3,3,12,2,1,...] |
| 0.76159415595576488811945828260479359 | Hyperbolic tangent (1) | [math]\displaystyle{ th \, 1 }[/math] | [math]\displaystyle{ \frac{e-\frac{1}{e}}{e+\frac{1}{e}} = \frac{e^2-1}{e^2+1} }[/math] | (e-1/e)/(e+1/e) | T | A073744 | [0;1,3,5,7,9,11,13,15,17,19,21,23,25,27,...] = [0;(2p+1)], p∈ℕ |
| 0.69777465796400798200679059255175260 | Continued Fraction constant | [math]\displaystyle{ {C}_{CF} }[/math] | [math]\displaystyle{ \underset{J_{k}() {Bessel}}\underset{{Function}}\frac{J_1(2)}{J_0(2)} = \frac{ \sum\limits_{n = 0}^{\infty} \frac{n}{n!n!}} {{ \sum\limits_{n = 0}^{\infty} \frac{1}{n!n!}}} = \frac{\frac{0}{1}+\frac{1}{1}+\frac{2}{4}+\frac{3}{36}+\frac{4}{576}+ \dots} {\frac{1}{1}+\frac{1}{1}+\frac{1}{4}+\frac{1}{36}+\frac{1}{576}+ \dots} }[/math] | (sum {n=0 to inf} n/(n!n!)) /(sum {n=0 to inf} 1/(n!n!)) | A052119 | [0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...] = [0;(p+1)], p∈ℕ | |
| 0.36787944117144232159552377016146086 | Inverse Napier constant | [math]\displaystyle{ \frac{1}{e} }[/math] | [math]\displaystyle{ \sum_{n = 0}^\infty \frac{(-1)^n}{n!} = \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \dots }[/math] | sum[n=2 to ∞]{(-1)^n/n!} | T | A068985 | [0;2,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...] = [0;2,1,(1,2p,1)], p∈ℕ |
| 2.71828182845904523536028747135266250 | Napier constant | [math]\displaystyle{ e }[/math] | [math]\displaystyle{ \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \cdots }[/math] | Sum[n=0 to ∞]{1/n!} | T | A001113 | [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...] = [2;(1,2p,1)], p∈ℕ |
| 0.49801566811835604271369111746219809 - 0.15494982830181068512495513048388 i |
Factorial of i | [math]\displaystyle{ i\,! }[/math] | [math]\displaystyle{ \Gamma (1+i) = i \, \Gamma (i) }[/math] | Gamma(1+i) | C | A212877 A212878 |
[0;6,2,4,1,8,1,46,2,2,3,5,1,10,7,5,1,7,2,...] - [0;2,125,2,18,1,2,1,1,19,1,1,1,2,3,34,...] i |
| 0.43828293672703211162697516355126482 + 0.36059247187138548595294052690600 i |
Infinite Tetration of i |
[math]\displaystyle{ {}^\infty i }[/math] | [math]\displaystyle{ \lim_{n \to \infty} {}^n i = \lim_{n \to \infty} \underbrace{i^{i^{\cdot^{\cdot^{i}}}}}_n }[/math] | i^i^i^... | C | A077589 A077590 |
[0;2,3,1,1,4,2,2,1,10,2,1,3,1,8,2,1,2,1, ...] + [0;2,1,3,2,2,3,1,5,5,1,2,1,10,10,6,1,1...] i |
| 0.56755516330695782538461314419245334 | Module of Infinite Tetration of i |
[math]\displaystyle{ |{}^\infty i | }[/math] | [math]\displaystyle{ \lim_{n \to \infty} \left | {}^n i \right | =\left | \lim_{n \to \infty} \underbrace{i^{i^{\cdot^{\cdot^{i}}}}}_n \right | }[/math] | Mod(i^i^i^...) | A212479 | [0;1,1,3,4,1,58,12,1,51,1,4,12,1,1,2,2,3,...] | |
| 0.26149721284764278375542683860869585 | Meissel-Mertens constant | [math]\displaystyle{ M }[/math] | [math]\displaystyle{ \lim_{n \rightarrow \infty } \left( \sum_{p \leq n} \frac{1}{p} - \ln(\ln(n)) \right) }[/math] ..... p: primes | A077761 | [0;3,1,4,1,2,5,2,1,1,1,1,13,4,2,4,2,1,33,...] | ||
| 1.9287800... | Wright constant | [math]\displaystyle{ \omega }[/math] | [math]\displaystyle{ \left \lfloor 2^{2^{2^{\cdot^{\cdot^{2^{\omega}}}}}} \right \rfloor }[/math] = primos: [math]\displaystyle{ \quad }[/math] [math]\displaystyle{ \left\lfloor 2^\omega\right\rfloor }[/math] =3, [math]\displaystyle{ \left\lfloor 2^{2^\omega} \right\rfloor }[/math] =13, [math]\displaystyle{ \left\lfloor 2^{2^{2^\omega}} \right\rfloor }[/math] =16381, [math]\displaystyle{ \dots }[/math] | A086238 | [1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3] | ||
| 0.37395581361920228805472805434641641 | Artin constant | [math]\displaystyle{ C_{Artin} }[/math] | [math]\displaystyle{ \prod_{n=1}^{\infty} \left(1-\frac{1}{p_n(p_n-1)}\right) }[/math] ...... pn: primo | T | A005596 | [0;2,1,2,14,1,1,2,3,5,1,3,1,5,1,1,2,3,5,46,...] | |
| 4.66920160910299067185320382046620161 | Feigenbaum constant δ | [math]\displaystyle{ {\delta} }[/math] | [math]\displaystyle{ \lim_{n \to \infty}\frac {x_{n+1}-x_n}{x_{n+2}-x_{n+1}} \qquad \scriptstyle x \in (3,8284;\, 3,8495) }[/math]
[math]\displaystyle{ \scriptstyle x_{n+1}=\,ax_n(1-x_n)\quad {o} \quad x_{n+1}=\,a\sin(x_n) }[/math] |
T | A006890 | [4;1,2,43,2,163,2,3,1,1,2,5,1,2,3,80,2,5,...] | |
| 2.50290787509589282228390287321821578 | Feigenbaum constant α | [math]\displaystyle{ \alpha }[/math] | [math]\displaystyle{ \lim_{n \to \infty}\frac {d_n}{d_{n+1}} }[/math] | T | A006891 | [2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...] | |
| 5.97798681217834912266905331933922774 | Hexagonal Madelung Constant 2 | [math]\displaystyle{ H_{2}(2) }[/math] | [math]\displaystyle{ \pi \ln(3) \sqrt 3 }[/math] | Pi Log[3]Sqrt[3] | T | A086055 | [5;1,44,2,2,1,15,1,1,12,1,65,11,1,3,1,1,...] |
| 0.96894614625936938048363484584691860 | Beta(3) | [math]\displaystyle{ \beta (3) }[/math] | [math]\displaystyle{ \frac{\pi^3}{32} = \sum_{n=1}^\infty\frac{-1^{n+1}}{(-1+2n)^3} = \frac{1}{1^3} {-} \frac{1}{3^3} {+} \frac{1}{5^3} {-} \frac{1}{7^3} {+} \dots }[/math] | Sum[n=1 to ∞]{(-1)^(n+1)/(-1+2n)^3} | T | A153071 | [0;1,31,4,1,18,21,1,1,2,1,2,1,3,6,3,28,1,...] |
| 1.902160583104 | Brun constant 2 = Σ inverse twin primes | [math]\displaystyle{ B_{\,2} }[/math] | [math]\displaystyle{ \textstyle \sum \underset{p,\, p+2: \, {primos}}{(\frac1{p}+\frac1{p+2})} = (\frac1{3} {+} \frac1{5}) + (\tfrac1{5} {+} \tfrac1{7}) + (\tfrac1{11} {+} \tfrac1{13}) + \dots }[/math] | A065421 | [1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2] | ||
| 0.870588379975 | Brun constant 4 = Σ inverse of twin prime | [math]\displaystyle{ B_{\,4} }[/math] | [math]\displaystyle{ \underset{p,\, p+2,\, p+4,\, p+6: \, {primes}} {\left(\tfrac1{5} + \tfrac1{7} + \tfrac1{11} + \tfrac1{13}\right)}+ \left(\tfrac1{11} + \tfrac1{13} + \tfrac1{17} + \tfrac1{19}\right)+ \dots }[/math] | A213007 | [0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1] | ||
| 22.4591577183610454734271522045437350 | pi^e | [math]\displaystyle{ \pi^{e} }[/math] | [math]\displaystyle{ \pi^{e} }[/math] | pi^e | A059850 | [22;2,5,1,1,1,1,1,3,2,1,1,3,9,15,25,1,1,5,...] | |
| 3.14159265358979323846264338327950288 | Pi, Archimedes constant | [math]\displaystyle{ \pi }[/math] | [math]\displaystyle{ \lim_{n\to \infty }\, 2^{n} \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+ \dots +\sqrt{2}}}}}_n }[/math] | Sum[n=0 to ∞]{(-1)^n 4/(2n+1)} | T | A000796 | [3;7,15,1,292,1,1,1,2,1,3,1,14,...] |
| 0.06598803584531253707679018759684642 | [math]\displaystyle{ e^{-e} }[/math] | [math]\displaystyle{ e^{-e} }[/math] ... Lower limit of Tetration | T | A073230 | [0;15,6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,...] | ||
| 0.20787957635076190854695561983497877 | i^i | [math]\displaystyle{ i^i }[/math] | [math]\displaystyle{ e^ \frac{-\pi}{2} }[/math] | e^(-pi/2) | T | A049006 | [0;4,1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,...] |
| 0.28016949902386913303643649123067200 | Bernstein constant | [math]\displaystyle{ \beta }[/math] | [math]\displaystyle{ \frac {1}{2\sqrt {\pi}} }[/math] | T | A073001 | [0;3,1,1,3,9,6,3,1,3,13,1,16,3,3,4,…] | |
| 0.28878809508660242127889972192923078 | Flajolet and Richmond | [math]\displaystyle{ Q }[/math] | [math]\displaystyle{ \prod_{n=1}^{\infty} \left(1 - \frac{1}{2^n}\right) = \left(1{-}\frac{1}{2^1}\right) \left(1{-}\frac{1}{2^2} \right)\left(1{-}\frac{1}{2^3} \right) \dots }[/math] | prod[n=1 to ∞]{1-1/2^n} | A048651 | ||
| 0.31830988618379067153776752674502872 | Inverse of Pi, Ramanujan | [math]\displaystyle{ \frac{1}{\pi} }[/math] | [math]\displaystyle{ \frac{2\sqrt{2}}{9801} \sum^\infty_{n=0} \frac{(4n)!(1103+26390n)}{(n!)^4 396^{4n}} }[/math] | T | A049541 | [0;3,7,15,292,1,1,1,2,1,3,1,14,2,1,1,...] | |
| 0.47494937998792065033250463632798297 | Weierstraß constant | [math]\displaystyle{ W_{_{WE}} }[/math] | [math]\displaystyle{ \frac{e^{\frac{\pi}{8}}\sqrt{\pi}}{4*2^{3/4} {(\frac {1}{4}!)^2}} }[/math] | (E^(Pi/8) Sqrt[Pi])/(4 2^(3/4) (1/4)!^2) | T | A094692 | [0;2,9,2,11,1,6,1,4,6,3,19,9,217,1,2,...] |
| 0.56714329040978387299996866221035555 | Omega constant | [math]\displaystyle{ \Omega }[/math] | [math]\displaystyle{ W(1)=\sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!} = 1 {-} 1 {+} \frac{3}{2} {-} \frac{8}{3} {+} \frac{125}{24} - \dots }[/math] | sum[n=1 to ∞]{(-n)^(n-1)/n!} | T | A030178 | [0;1,1,3,4,2,10,4,1,1,1,1,2,7,306,1,5,1,...] |
| 0.57721566490153286060651209008240243 | Euler's number | [math]\displaystyle{ \gamma }[/math] | [math]\displaystyle{ -\psi(1) = \sum_{n=1}^\infty \sum_{k=0}^\infty \frac{(-1)^k}{2^n+k} }[/math] | sum[n=1 to ∞]|sum[k=0 to ∞]{((-1)^k)/(2^n+k)} | ? | A001620 | [0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,...] |
| 0.60459978807807261686469275254738524 | Dirichlet serie | [math]\displaystyle{ \frac{\pi}{3 \sqrt 3} }[/math] | [math]\displaystyle{ \sum_{n = 1}^\infty \frac{1}{n{2n \choose n}} = 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{5} + \frac{1}{7} - \frac{1}{8} + \cdots }[/math] | Sum[1/(n Binomial[2 n, n]), {n, 1, ∞}] | T | A073010 | [0;1,1,1,1,8,10,2,2,3,3,1,9,2,5,4,1,27,27,...] |
| 0.63661977236758134307553505349005745 | 2/Pi, François Viète | [math]\displaystyle{ \frac{2}{\pi} }[/math] | [math]\displaystyle{ \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots }[/math] | T | A060294 | [0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...] | |
| 0.66016181584686957392781211001455577 | Twin prime constant | [math]\displaystyle{ C_{2} }[/math] | [math]\displaystyle{ \prod_{p=3}^\infty \frac{p(p-2)}{(p-1)^2} }[/math] | prod[p=3 to ∞]{p(p-2)/(p-1)^2 | A005597 | [0;1,1,1,16,2,2,2,2,1,18,2,2,11,1,1,2,4,1,...] | |
| 0.66274341934918158097474209710925290 | Laplace Limit constant | [math]\displaystyle{ \lambda }[/math] | A033259 | [0;1,1,1,27,1,1,1,8,2,154,2,4,1,5,...] | |||
| 0.69314718055994530941723212145817657 | Logarithm de 2 | [math]\displaystyle{ Ln(2) }[/math] | [math]\displaystyle{ \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = \frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots }[/math] | Sum[n=1 to ∞]{(-1)^(n+1)/n} | T | A002162 | [0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10,...] |
| 0.78343051071213440705926438652697546 | Sophomore's Dream 1 J.Bernoulli | [math]\displaystyle{ I_{1} }[/math] | [math]\displaystyle{ \sum_{n = 1}^\infty \frac{(-1)^{n+1}}{n^n} = 1 - \frac{1}{2^2} + \frac{1}{3^3} - \frac{1}{4^4} + \frac{1}{5^5} + \dots }[/math] | Sum[ -(-1)^n /n^n] | T | A083648 | [0;1,3,1,1,1,1,1,1,2,4,7,2,1,2,1,1,1,...] |
| 0.78539816339744830961566084581987572 | Dirichlet beta(1) | [math]\displaystyle{ \beta(1) }[/math] | [math]\displaystyle{ \frac{\pi}{4} = \sum_{n = 0}^\infty \frac{(-1)^n}{2n+1} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots }[/math] | Sum[n=0 to ∞]{(-1)^n/(2n+1)} | T | A003881 | [0; 1,3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,...] |
| 0.82246703342411321823620758332301259 | Traveling Salesman Nielsen-Ramanujan | [math]\displaystyle{ \frac{\zeta(2)}{2} }[/math] | [math]\displaystyle{ \frac{\pi^2}{12} = \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^2} = \frac{1}{1^2} {-} \frac{1}{2^2} {+} \frac{1}{3^2} {-} \frac{1}{4^2} {+} \frac{1}{5^2} - \dots }[/math] | Sum[n=1 to ∞]{((-1)^(k+1))/n^2} | T | A072691 | [0;1,4,1,1,1,2,1,1,1,1,3,2,2,4,1,1,1,...] |
| 0.91596559417721901505460351493238411 | Catalan constant | [math]\displaystyle{ C }[/math] | [math]\displaystyle{ \sum_{n = 0}^\infty \frac{(-1)^n}{(2n+1)^2} = \frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots }[/math] | Sum[n=0 to ∞]{(-1)^n/(2n+1)^2} | I | A006752 | [0;1,10,1,8,1,88,4,1,1,7,22,1,2,...] |
| 1.05946309435929526456182529494634170 | Ratio of the distance between semi-tones | [math]\displaystyle{ \sqrt[12]{2} }[/math] | [math]\displaystyle{ \sqrt[12]{2} }[/math] | 2^(1/12) | I | A010774 | [1;16,1,4,2,7,1,1,2,2,7,4,1,2,1,60,1,3,1,2,...] |
| 1,.08232323371113819151600369654116790 | Zeta(04) | [math]\displaystyle{ \zeta{4} }[/math] | [math]\displaystyle{ \frac{\pi^4}{90} = \sum_{n=1}^\infty\frac{{1}}{n^4} = \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \frac{1}{5^4} + \dots }[/math] | Sum[n=1 to ∞]{1/n^4} | T | A013662 | [1;12,6,1,3,1,4,183,1,1,2,1,3,1,1,5,4,2,7,...] |
| 1.1319882487943 ... | Viswanaths Archived 2013-04-13 at the Wayback Machine constant | [math]\displaystyle{ C_{Vi} }[/math] | [math]\displaystyle{ \lim_{n \to \infty}|a_n|^\frac{1}{n} }[/math] | A078416 | [1;7,1,1,2,1,3,2,1,2,1,8,1,5,1,1,1,9,1,...] | ||
| 1.20205690315959428539973816151144999 | Apéry constant | [math]\displaystyle{ \zeta(3) }[/math] | [math]\displaystyle{ \sum_{n=1}^\infty\frac{1}{n^3} = \frac{1}{1^3}+\frac{1}{2^3} + \frac{1}{3^3} + \frac{1}{4^3} + \frac{1}{5^3} + \cdots\,\! }[/math] | Sum[n=1 to ∞]{1/n^3} | I | A010774 | [1;4,1,18,1,1,1,4,1,9,9,2,1,1,1,2,...] |
| 1.22541670246517764512909830336289053 | Gamma(3/4) | [math]\displaystyle{ \Gamma(\tfrac34) }[/math] | [math]\displaystyle{ \left(-1+\frac{3}{4}\right)! }[/math] | (-1+3/4)! | T | A068465 | [1;4,2,3,2,2,1,1,1,2,1,4,7,1,171,3,2,3,1,1,...] |
| 1.23370055013616982735431137498451889 | Favard constant | [math]\displaystyle{ \tfrac34\zeta(2) }[/math] | [math]\displaystyle{ \frac{\pi^2}{8} = \sum_{n = 0}^\infty \frac{1}{(2n-1)^2} = \frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+ \dots }[/math] | sum[n=1 to ∞]{1/((2n-1)^2)} | T | A111003 | [1;4,3,1,1,2,2,5,1,1,1,1,2,1,2,1,10,4,3,1,1,...] |
| 1.25992104989487316476721060727822835 | Cube root of 2, constante Delian | [math]\displaystyle{ \sqrt[3]{2} }[/math] | [math]\displaystyle{ \sqrt[3]{2} }[/math] | 2^(1/3) | I | A002580 | [1;3,1,5,1,1,4,1,1,8,1,14,1,10,...] |
| 1.29128599706266354040728259059560054 | Sophomore's Dream 2 J.Bernoulli | [math]\displaystyle{ I_{2} }[/math] | [math]\displaystyle{ \sum_{n = 1}^\infty \frac{1}{n^n} = 1 + \frac{1}{2^2} + \frac{1}{3^3} + \frac{1}{4^4} + \frac{1}{5^5} + \frac{1}{6^6} + \dots }[/math] | Sum[1/(n^n]), {n, 1, ∞}] | A073009 | [1;3,2,3,4,3,1,2,1,1,6,7,2,5,3,1,2,1,8,1,...] | |
| 1.32471795724474602596090885447809734 | Plastic number | [math]\displaystyle{ \rho }[/math] | [math]\displaystyle{ \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \cdots}}}} }[/math] | I | A060006 | [1;3,12,1,1,3,2,3,2,4,2,141,80,2,5,1,2,8,...] | |
| 1.41421356237309504880168872420969808 | Square root of 2, Pythagoras constant | [math]\displaystyle{ \sqrt{2} }[/math] | [math]\displaystyle{ \prod_{n=1}^\infty 1+\frac{(-1)^{n+1}}{2n-1} = \left(1{+}\frac{1}{1}\right) \left(1{-}\frac{1}{3} \right)\left(1{+}\frac{1}{5} \right) ... }[/math] | prod[n=1 to ∞]{1+(-1)^(n+1)/(2n-1)} | I | A002193 | [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...] = [1;(2),...] |
| 1.44466786100976613365833910859643022 | Steiner number | [math]\displaystyle{ e^{\frac{1}{e}} }[/math] | [math]\displaystyle{ e^{1/e} }[/math] ... Upper Limit of Tetration | A073229 | [1;2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...] | ||
| 1.53960071783900203869106341467188655 | Lieb's Square Ice constant | [math]\displaystyle{ W_{2D} }[/math] | [math]\displaystyle{ \lim_{n \to \infty}(f(n))^{n^{-2}}=\left(\frac{4}{3}\right)^\frac{3}{2} }[/math] | (4/3)^(3/2) | I | A118273 | [1;1,1,5,1,4,2,1,6,1,6,1,2,4,1,5,1,1,2,...] |
| 1.57079632679489661923132169163975144 | Wallis product | [math]\displaystyle{ \pi/2 }[/math] | [math]\displaystyle{ \prod_{n=1}^{\infty} \left(\frac{4n^2}{4n^2 - 1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots }[/math] | T | A019669 | [1;1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1...] | |
| 1.60669515241529176378330152319092458 | Erdős–Borwein constant | [math]\displaystyle{ E_{\,B} }[/math] | [math]\displaystyle{ \sum_{n=1}^{\infty}\frac{1}{2^n-1} = \frac{1}{1} + \frac{1}{3} + \frac{1}{7} + \frac{1}{15} + \cdots\,\! }[/math] | sum[n=1 to ∞]{1/(2^n-1)} | I | A065442 | [1;1,1,1,1,5,2,1,2,29,4,1,2,2,2,2,6,1,7,1,...] |
| 1.61803398874989484820458633436563812 | Phi, Golden ratio | [math]\displaystyle{ \varphi }[/math] | [math]\displaystyle{ \frac{1 + \sqrt{5}}{2} = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}} }[/math] | (1+5^(1/2))/2 | I | A001622 | [0;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...] = [0;(1),...] |
| 1.64493406684822643647241516664602519 | Zeta(2) | [math]\displaystyle{ \zeta(\,2) }[/math] | [math]\displaystyle{ \frac{\pi^2}{6} = \sum_{n=1}^\infty\frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots }[/math] | Sum[n=1 to ∞]{1/n^2} | T | A013661 | [1;1,1,1,4,2,4,7,1,4,2,3,4,10 1,2,1,1,1,15,...] |
| 1.66168794963359412129581892274995074 | Somos' quadratic recurrence constant | [math]\displaystyle{ \sigma }[/math] | [math]\displaystyle{ \sqrt {1 \sqrt {2 \sqrt{3 \cdots}}} = 1^{1/2} ; 2^{1/4} ; 3^{1/8} \cdots }[/math] | T | A065481 | [1;1,1,1,21,1,1,1,6,4,2,1,1,2,1,3,1,13,13,...] | |
| 1.73205080756887729352744634150587237 | Theodorus constant | [math]\displaystyle{ \sqrt{3} }[/math] | [math]\displaystyle{ \sqrt{3} }[/math] | 3^(1/2) | I | A002194 | [1;1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,...] = [1;(1,2),...] |
| 1.75793275661800453270881963821813852 | Kasner number | [math]\displaystyle{ R }[/math] | [math]\displaystyle{ \sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \cdots}}}} }[/math] | A072449 | [1;1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...] | ||
| 1.77245385090551602729816748334114518 | Carlson-Levin constant | [math]\displaystyle{ \Gamma(\tfrac12) }[/math] | [math]\displaystyle{ \sqrt{\pi} = \left(-\frac{1}{2}\right)! }[/math] | sqrt (pi) | T | A002161 | [1;1,3,2,1,1,6,1,28,13,1,1,2,18,1,1,1,83,1,...] |
| 2.29558714939263807403429804918949038 | P, Universal parabolic constant | [math]\displaystyle{ P }[/math] | [math]\displaystyle{ \ln(1 + \sqrt2) + \sqrt2 }[/math] | ln(1+sqrt 2)+sqrt 2 | T | A103710 | [2;3,2,1,1,1,1,3,3,1,1,4,2,3,2,7,1,6,1,8,...] |
| 2.30277563773199464655961063373524797 | Bronze Number | [math]\displaystyle{ \sigma_{\,Rr} }[/math] | [math]\displaystyle{ \frac {3+\sqrt{13}}{2} = 1+ \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}} }[/math] | (3+sqrt 13)/2 | I | A098316 | [3;3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...] = [3;(3),...] |
| 2.37313822083125090564344595189447424 | Lévy constant2 | [math]\displaystyle{ 2\,\ln\,\gamma }[/math] | [math]\displaystyle{ \frac{\pi^2}{6\ln(2)} }[/math] | Pi^(2)/(6*ln(2)) | T | A174606 | [2;2,1,2,8,57,9,32,1,1,2,1,2,1,2,1,2,1,3,2,...] |
| 2.50662827463100050241576528481104525 | square root of 2 pi | [math]\displaystyle{ \sqrt{2 \pi} }[/math] | [math]\displaystyle{ \sqrt{2 \pi} = \lim_{n \to \infty} \frac {n! \; e^n}{n^n \sqrt{n}} }[/math] | sqrt (2*pi) | T | A019727 | [2;1,1,37,4,1,1,1,1,9,1,1,2,8,6,1,2,2,1,3,...] |
| 2.66514414269022518865029724987313985 | Gelfond-Schneider constant | [math]\displaystyle{ G_{_{\,GS}} }[/math] | [math]\displaystyle{ 2^{\sqrt{2}} }[/math] | 2^sqrt{2} | T | A007507 | [2;1,1,1,72,3,4,1,3,2,1,1,1,14,1,2,1,1,3,1,...] |
| 2.68545200106530644530971483548179569 | Khintchin constant | [math]\displaystyle{ K_{\,0} }[/math] | [math]\displaystyle{ \prod_{n=1}^\infty \left[{1+{1\over n(n+2)}}\right]^{\ln n/\ln 2} }[/math] | prod[n=1 to ∞]{(1+1/(n(n+2)))^((ln(n)/ln(2))} | ? | A002210 | [2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...] |
| 3.27582291872181115978768188245384386 | Khinchin-Lévy constant | [math]\displaystyle{ \gamma }[/math] | [math]\displaystyle{ e^{\pi^2/(12\ln2)} }[/math] | e^(\pi^2/(12 ln(2)) | A086702 | [3;3,1,1,1,2,29,1,130,1,12,3,8,2,4,1,3,55,...] | |
| 3.35988566624317755317201130291892717 | Reciprocal Fibonacci constant | [math]\displaystyle{ \Psi }[/math] | [math]\displaystyle{ \sum_{n=1}^{\infty} \frac{1}{F_n} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \cdots }[/math] | A079586 | [3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...] | ||
| 4.13273135412249293846939188429985264 | Root of 2 e pi | [math]\displaystyle{ \sqrt{2e \pi} }[/math] | [math]\displaystyle{ \sqrt{2e \pi} }[/math] | sqrt(2e pi) | T | A019633 | [4;7,1,1,6,1,5,1,1,1,8,3,1,2,2,15,2,1,1,2,4,...] |
| 6.58088599101792097085154240388648649 | Froda constant | [math]\displaystyle{ 2^{\,e} }[/math] | [math]\displaystyle{ 2^e }[/math] | 2^e | [6;1,1,2,1,1,2,3,1,14,11,4,3,1,1,7,5,5,2,7,...] | ||
| 9.86960440108935861883449099987615114 | Pi Squared | [math]\displaystyle{ \pi ^2 }[/math] | [math]\displaystyle{ 6 \sum_{n=1}^\infty \frac{1}{n^2} = \frac{6}{1^2} + \frac{6}{2^2} + \frac{6}{3^2} + \frac{6}{4^2}+ \cdots }[/math] | 6 Sum[n=1 to ∞]{1/n^2} | T | A002388 | [9;1,6,1,2,47,1,8,1,1,2,2,1,1,8,3,1,10,5,...] |
| 23.1406926327792690057290863679485474 | Gelfond constant | [math]\displaystyle{ e^{\pi} }[/math] | [math]\displaystyle{ \sum_{n=0}^\infty \frac{\pi^{n}}{n!} = \frac{\pi^{1}}{1} + \frac{\pi^{2}}{2!} + \frac{\pi^{3}}{3!} + \frac{\pi^{4}}{4!}+ \cdots }[/math] | Sum[n=0 to ∞]{(pi^n)/n!} | T | A039661 | [23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...] |
Mathematical Constant Media
The circumference of a circle with diameter 1 is π.
The square root of 2 is equal to the length of the hypotenuse of a right-angled triangle with legs of length 1.
The area of the circle equals π times the shaded area. The area of the unit circle is π.
Exponential growth (green) describes many physical phenomena.
The imaginary unit i in the complex plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis
Golden rectangles in a regular icosahedron
The area between the two curves (red) tends to a limit, namely the Euler-Mascheroni constant.
Bifurcation diagram of the logistic map.
This Babylonian clay tablet gives an approximation of the square root of 2 in four sexagesimal figures: 1; 24, 51, 10, which is accurate to about six decimal figures.
Related pages
References
- ↑ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-26.
- ↑ "Constant | mathematics and logic". Encyclopedia Britannica. Retrieved 2020-08-26.
- ↑ Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-26.
Books
- Finch, Steven (2003). Mathematical constants. Cambridge University Press. ISBN 0-521-81805-2.
- Daniel Zwillinger (2012). Standard Mathematical Tables and Formulae. Imperial College Press. ISBN 978-1-4398-3548-7.
- Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics. Chapman & Hall/CRC. ISBN 1-58488-347-2.
- Lloyd Kilford (2008). Modular Forms, a Classical and Computational Introduction. Imperial College Press. ISBN 978-1-84816-213-6.