Pi Notations Traditional name Π Traditional notation Π Mathematica StandardForm notation Pi Primary definition.3... Π Specific values.3.3.. Π 3.5965358979338663383795889769399375589795937866868998683853 Above approximate numerical value of Π shows 9 decimal digits. General characteristics The pi Π is a constant. It is irrational and transcendental over positive real number. Series representations Generalized power series Expansions for Π.3.6.. r r 8 r Π 6 8 5 8 6 8 r 8 7 8 r 8 r 8 r 8 3 8 ; r
http://functions.wolfram.com.3.6.. Π log.3.6.3. Π.3.6.. Π.3.6.5. Π 6 5 39 Π 3 Π.3.6.6. 3 log 3 3.3.6.7. log Π.3.6.8..3.6.9. Π 3 3 3 6 3 85 667 53.3.6.. Π 8 6 3 6 5 6.3.6.. 8 Π 8 8 7.3.6.. Π 6 8 8 5 8 6 8.3.6.3. Π 35 65 38 3 5
http://functions.wolfram.com 3 Π 5 Π.3.6.. 5 F 6.3.6.5. 3 F 5 5 3.3.6.6. F Π 3.3.6.7. Π 5 3 6 3 5 F.3.6.8. Π tan F.3.6.9. 3 Ζ Π.3.6.. Π 3 3 9 3 Π 8 3.3.6.5. 5 Π 35 3 37 7.3.6.6. 8 7 3 3 3 Π 3 3 9.3.6.7. 8 38 3
http://functions.wolfram.com Π 3 3 67.3.6.8. 8 938.3.6.9. 3 8 3 Π 6565 3 5 3 3 3 3 6.3.6.5. 795 3 Π 797 8 3 59 3 7.3.6.5. 355 56 3 Π 69 3 79 98 6 3 8.3.6.5. 8 3 75 3 Π 3 38 583.3.6.53. 365 36 7 3 Π 7 965 87 6 637 66 3 9 87 3 93 3.3.6.5. 99 76 5 3 Π 7 5 87 656 5 5 5 3 3.3.6.55. 5 73 37 8 3 Π 67 63 35 957.3.6.56. Π j j 3 9 683 35 6 7 9 3 Candido Otero Ramos (7) Expansions for Π.3.6.. Π 98 6 39 3 396
http://functions.wolfram.com 5.3.6.. Π 5 3.3.6.. 6 55 3 3 59 9 Π 3 3 6 3 3 The above Chudnovsy`s formula is used for the numerical computation of Π in Mathematica. Expansions for Π.3.6.3. Π 6.3.6.. Π 8.3.6.5. Π.3.6.6. Π 8.3.6.7. Π 36.3.6.8. Π 5 38 3 5 3.3.6.9. 3 Π 8 3 6 6 3 8 6 6 6 5 6 6 Expansions for Π 3.3.6.3. Π 3 3 3
http://functions.wolfram.com 6 Π 3 6.3.6.57. 8 3 3 3 3 3 5 9 88 8 8 6 3 3 3 5 3 6 3 G.Huvent (6) Expansions for Π.3.6.3. Π 9.3.6.3. Π 96 Π 7 6 3 7 3 9 3 3 3 3.3.6.58. G.Huvent (6) Π 3 7 38 9 3 8 9 3 6 8 358 5 5 3 55 6 3 68 3 8 78 8 368 5 6 7 8 9 8 6 38 8 3.3.6.59. Expansions for Π 6.3.6.33. Π 6 95 6.3.6.3. Π 6 96 6 56 7 7 68 8 9
http://functions.wolfram.com 7 Expansions for Π n.3.6.35. Π n n n n B n ; n n.3.6.6. Π n n n n B n n ; n.3.6.36. Π n n n n B n Expansions for Π n.3.6.6. Π n n n n E n ; n n ; n n Exponential Fourier series.3.6.. sin x Π x ; x x.3.6.3. cos x Π ; x.3.6.6. Π j j 3 Candido Otero Ramos (7).3.6.63. Π lim n n f n ; f f n Candido Otero Ramos (7) f n f n n Other series representations.3.6.. n Π n n ; n n n j l l Integral representations
http://functions.wolfram.com 8 On the real axis Of the direct function.3.7.. Π t t.3.7.. Π t t.3.7.3. Π t t.3.7.. sint Π t t.3.7.5. sin t Π t t.3.7.6. Π 8 sin 3 t 3 t t 3.3.7.7. sin t Π 3 t t.3.7.8. Π 38 sin 5 t 5 t t 5.3.7.9. Π sin 6 t t t 6.3.7.. Π n sin n n t n n t n t n n ; n Involving the direct function.3.7.. Π t t Gaussian probability density integral
http://functions.wolfram.com 9.3.7.. Π sint t Fresnel integral.3.7.3. Π cost t Fresnel integral.3.7.. Π log t t.3.7.5. Π log t t Involving related functions.3.7.6. cost Π t t Product representations.3.8.. Π.3.8.. Π.3.8.3. Π sec Π.3.8.. Π 3sec Π.3.8.8. Π Π.3.8.9. 6
http://functions.wolfram.com.3.8.5. 6 Π ; p p.3.8.6. Π.3.8.7. n lim Π Nest &,, n Limit representations.3.9.. Π lim n n n n n.3.9.. n Π lim n n n.3.9.3. Π lim n n n n Π lim n Π lim n.3.9.. n n n n.3.9.6. n n n.3.9.. n Π lim n n n n H ; n.3.9.3. n n n n Π lim n n n 3 n Pete Koupriyanov.3.9.7. n Π 6lim n n
http://functions.wolfram.com.3.9.. Π lim n n log F 6 loglcmf, F,, F n ; n.3.9.8. n Π lim n n cot Α logcos Α ; Α Π lim n.3.9.9. n Α n sgncos Α,sgncos Α ; Α Π.3.9.. a Π lim ; n s a a b b a b s s c c a b a b s.3.9.. Π lim ; n Α n Α n Β n Β n Α n n3 Β n Β n Β n Β n Β n Α 6 Β.3.9.5. Π lim n n b b n b n b n b sin Π b ; b n b n b n b b L. D. Servi: Nested Square Roots of American Mathematical Monthly, 36-39 (3).3.9.6. Π lim n An ; A B An Bn An Bn Bn Bn n Candido Otero Ramos (7) Continued fraction representations
http://functions.wolfram.com.3... Π 3 7 5 9 3.3... Π 3 9 6 5 6 9 6 8 6 6 6.3..3. Π 3, 6
http://functions.wolfram.com 3.3... Π 6 3 3 3 3 3.3..5. Π 3,.3..6. Π 3 9 5 6 7 5 9 36 3.3..7. Π,.3..8. Π.3..9. Π, 9 5 9 8.3... Π 6 3 8 5 56 7 65 9 96 3
http://functions.wolfram.com.3... Π,.3... 6 Π 6 6 9.3..3. 6 Π 6, Complex characteristics Real part.3.9.. ReΠ Π Imaginary part.3.9.. ImΠ Absolute value Π Π.3.9.3. Argument.3.9.. argπ Conjugate value Π Π.3.9.5. Signum value.3.9.6. sgnπ Differentiation
http://functions.wolfram.com 5 Low-order differentiation.3... Π z Fractional integro-differentiation Α Π z Α.3... zα Π Α Integration Indefinite integration Π z Π z.3....3... z Α Π z zα Π Α Integral transforms Fourier exp transforms.3... t Πz Π 3 z Inverse Fourier exp transforms.3... t Πz Π 3 z Fourier cos transforms.3..3. c t Πz Π3 z Fourier sin transforms.3... s t Πz Π z Laplace transforms
http://functions.wolfram.com 6.3..5. t Πz Π z Inverse Laplace transforms.3..6. t Πz Π z Representations through more general functions Through Meijer G.3.6.. Π Π G,, z Π G,, z, Through other functions.3.6.. Π tan 5 tan.3.6.8. 39 Π tan tan 3.3.6.9. Π 8 tan 3 tan 7.3.6.. Π tan tan 5 tan 8.3.6.3. Π 6 tan 8 tan Jeff Reid.3.6.5. 57 tan 39 Π tan 3 tan tan 5 tan Adam Bui (7).3.6.6. Π tan 3 tan tan 7 tan Adam Bui (7) 7 3
http://functions.wolfram.com 7.3.6.7. Π a a cot a tan a a a a a a a a a a a ; a Adam Bui &O.I. Marichev (7).3.6.. Π 88 tan 8 8 tan 3 tan.3.6.. Π 8 tan 8 tan 7 tan.3.6.. Π 6 tan tan 39 6 tan 393 tan 8 99 3 tan 6 tan 5 5 6 tan 6 575 8 tan 37 55 3 tan 3 5 5 3 tan 3 Π tan p q.3.6.3. tan q p p q ; p q.3.6.. Π K.3.6.5. Π E.3.6.6. Π 6 Li.3.6.7. Π Representations through equivalent functions.3.7.. Π 8.3.7.. Π log.3.7.3. Π log Π.3.7..
http://functions.wolfram.com 8 identity due to L. Euler.3.7.5. Π.3.7.6. Π ;.3.7.7. Π.3.7.8. Π ; Inequalities.3.9.. 3 7 Π 3 7 B.C. Archimedes.3.9.. Π q p q.65 ; p q Π Π.3.9.3. Theorems Volume of an n-dimensional sphere Volume V n of an n-dimensional sphere of radius r: V Π r ; V Π r. For instance, the area of a circle with radius r is Π r and the volume of a sphere with radius r is Π 3 r3. Above general formulas can be joined into one V n Surface area of an n-dimensional sphere Surface area S n of n-dimensional sphere of radius r: Πn n r n. S Π r ; S Π r. For instance, the circumference of circle with radius r is Π r, and the surface area of a sphere with radius r is Π r.
http://functions.wolfram.com 9 Above general formulas can be joined into one S n Πn n rn. Volume of an n-dimensional cylinder?? Volume V n of an n-dimensional cylinder of radius r and height h : V Π r h; V Π r h. For instance, the volume of a cylinder with radius r and height h is 3 Π r3 h. Above general formulas can be joined into one V n n Π n n Surface area of an n-dimensional cylinder?? r n h. Surface area S n of n-dimensional cone of radius r and height h : S Π r h r; S Π r h r. For instance, the volume of a cylinder with radius r and height h is 3 Π r h. For instance, the surface area of a cylinder with radius r and height h is Π r r h r. n Above general formulas can be joined into one S n Π Volume of an n-dimensional cone n r h r r n. Volume V n of an n-dimensional cone of radius r and height h : V Π r h; Π V r h. For instance, the volume of a cone with radius r and height h is 3 Π r h.
http://functions.wolfram.com Above general formulas can be joined into one V n Surface area of an n-dimensional cone n Π n n r n h. Surface area S n of n-dimensional cone of radius r and height h : S Π Π r h r r h r r ; S r. For instance, the volume of a cone with radius r and height h is 3 Π r h. For instance, the surface area of a cone with radius r and height h is Π r r h r. n Above general formulas can be joined into one S n Π n r h r r n. Probability of two random integers being relatively prime The probability that two integers piced at random are relatively prime is 6 Π. History The design of Egyptian pyramids (c. 3 BC) incorporated Π as 3 7 3.857 ; Egyptians (Rhind Papyrus, c. BC) gave Π as 6 9 3.65 China (c. BC) gave Π as 3 The Biblical verse I Kings 7:3 (c. 95 BC) gave Π as 3 3. Archimedes (Greece, c. BC) new that 3 7 Π 3 7 and gave Π as 3.8 W. Jones (76) introduced the symbol Π C. Goldbach (7) also used the symbol Π J. H. Lambert (76) established that Π is an irrational number F. Lindemann (88) proved that Π is transcendental The constant Π is the most frequently encountered classical constant in mathematics and the natural sciences.
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