Notations. Primary definition. Specific values. General characteristics. Series representations. Traditional name. Traditional notation

Transcript

1 Pi Notations Traditional name Π Traditional notation Π Mathematica StandardForm notation Pi Primary definition.3... Π Specific values Π 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 r r 8 r Π r r 8 r 8 r ; r

2 Π log Π Π Π Π 3 Π log log Π Π Π Π Π Π

3 3 Π 5 Π F F F Π Π F Π tan F Ζ Π Π Π Π Π

4 Π Π Π Π Π Π Π Π Π j j Candido Otero Ramos (7) Expansions for Π Π

5 Π Π The above Chudnovsy`s formula is used for the numerical computation of Π in Mathematica. Expansions for Π Π Π Π Π Π Π Π Expansions for Π Π 3 3 3

6 6 Π G.Huvent (6) Expansions for Π Π Π 96 Π G.Huvent (6) Π Expansions for Π Π Π

7 7 Expansions for Π n Π n n n n B n ; n n Π n n n n B n n ; n Π n n n n B n Expansions for Π n Π n n n n E n ; n n ; n n Exponential Fourier series sin x Π x ; x x cos x Π ; x Π j j 3 Candido Otero Ramos (7) Π lim n n f n ; f f n Candido Otero Ramos (7) f n f n n Other series representations n Π n n ; n n n j l l Integral representations

8 8 On the real axis Of the direct function Π t t Π t t Π t t sint Π t t sin t Π t t Π 8 sin 3 t 3 t t sin t Π 3 t t Π 38 sin 5 t 5 t t Π sin 6 t t t Π n sin n n t n n t n t n n ; n Involving the direct function Π t t Gaussian probability density integral

9 Π sint t Fresnel integral Π cost t Fresnel integral Π log t t Π log t t Involving related functions cost Π t t Product representations Π Π Π sec Π Π 3sec Π Π Π

10 6 Π ; p p Π n lim Π Nest &,, n Limit representations Π lim n n n n n n Π lim n n n Π lim n n n n Π lim n Π lim n n n n n n n n n Π lim n n n n H ; n n n n n Π lim n n n 3 n Pete Koupriyanov n Π 6lim n n

11 Π lim n n log F 6 loglcmf, F,, F n ; n n Π lim n n cot Α logcos Α ; Α Π lim n n Α n sgncos Α,sgncos Α ; Α Π a Π lim ; n s a a b b a b s s c c a b a b s 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 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, (3) Π lim n An ; A B An Bn An Bn Bn Bn n Candido Otero Ramos (7) Continued fraction representations

12 Π Π Π 3, 6

13 Π Π 3, Π Π, Π Π, Π

14 Π, Π Π 6, Complex characteristics Real part ReΠ Π Imaginary part ImΠ Absolute value Π Π Argument argπ Conjugate value Π Π Signum value sgnπ Differentiation

15 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 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 c t Πz Π3 z Fourier sin transforms.3... s t Πz Π z Laplace transforms

16 t Πz Π z Inverse Laplace transforms t Πz Π z Representations through more general functions Through Meijer G Π Π G,, z Π G,, z, Through other functions Π tan 5 tan Π tan tan Π 8 tan 3 tan Π tan tan 5 tan Π 6 tan 8 tan Jeff Reid tan 39 Π tan 3 tan tan 5 tan Adam Bui (7) Π tan 3 tan tan 7 tan Adam Bui (7) 7 3

17 Π a a cot a tan a a a a a a a a a a a ; a Adam Bui &O.I. Marichev (7) Π 88 tan 8 8 tan 3 tan Π 8 tan 8 tan 7 tan Π 6 tan tan 39 6 tan 393 tan tan 6 tan tan tan tan tan 3 Π tan p q tan q p p q ; p q Π K Π E Π 6 Li Π Representations through equivalent functions Π Π log Π log Π.3.7..

18 8 identity due to L. Euler Π Π ; Π Π ; Inequalities Π 3 7 B.C. Archimedes Π q p q.65 ; p q Π Π 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.

19 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.

20 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 ; Egyptians (Rhind Papyrus, c. BC) gave Π as 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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