Pi

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

General characteristics

The pi π is a constant. It is irrational and transcendental over ℚ positive real number.

Series representations

Generalized power series

Expansions for π

Candido Otero Ramos (2007)

Expansions for 1/π

Expansions for (π^2)

Expansions for (π^3)

G.Huvent (2006)

Expansions for (π^4)

G.Huvent (2006)

Expansions for (π^6)

Expansions for (π^(2n))

Expansions for (π^(2n-1))

Exponential Fourier series

Candido Otero Ramos (2007)

Candido Otero Ramos (2007)

Other series representations

Integral representations

On the real axis

Of the direct function

Involving the direct function

Involving related functions

Product representations

Limit representations

Pete Koupriyanov

L. D. Servi: Nested Square Roots of 2 American Mathematical Monthly 110, 326-329 (2003)

Candido Otero Ramos (2007)

Continued fraction representations

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Inverse Laplace transforms

Representations through more general functions

Through Meijer G

Through other functions

Jeff Reid

Adam Bui (2007)

Adam Bui (2007)

Adam Bui &O.I. Marichev (2007)

Representations through equivalent functions

Inequalities

Theorems

Volume of an n-dimensional sphere

Volume (Subscript[V, n]) of an n-dimensional sphere of radius r:

(Subscript[V, 2k])==((π^k)/(k!))(r^(2k));                                       (Subscript[V, 2k+1])==(((π^k)(2^(2k+1))k!)/((2k+1)!))(r^(2k+1)).

For instance, the area of a circle with radius r is π (r^2) and the volume of a sphere with radius r is ((4 π)/3)(r^3).

Above general formulas can be joined into one (Subscript[V, n])==((π^(n/2))/(Γ(((n+2)/2))))(r^n).

Surface area of an n-dimensional sphere

Surface area (Subscript[S, n]) of n-dimensional sphere of radius r:

(Subscript[S, 2k])==((2(π^k))/((k-1)!))(r^(2k-1));                            (Subscript[S, 2k+1])==(((π^k)(2^(2k+1))k!)/((2k)!))(r^(2k)).

For instance, the circumference of circle with radius r is 2π r, and the surface area of a sphere with radius r is 4 π (r^2).

Above general formulas can be joined into one (Subscript[S, n])==((2 (π^(n/2)))/(Γ((n/2))))(r^(n-1)).

Volume of an n-dimensional cylinder ??

Volume (Subscript[V, n]) of an n-dimensional cylinder of radius r and height h:

(Subscript[V, 2k])==((π^k)/(k!))(r^(2k))h;                                       (Subscript[V, 2k+1])==(((π^k)(2^(2k+1))k!)/((2k+1)!))(r^(2k+1))h.

For instance, the volume of a cylinder with radius r and height h is (4/3)π (r^3)h.

Above general formulas can be joined into one (Subscript[V, n])==((π^((n-1)/2))/(n Γ(((n+1)/2))))(r^(n-1))h.

Surface area of an n-dimensional cylinder ??

Surface area (Subscript[S, n]) of n-dimensional cone of radius r and height h:

(Subscript[S, 2k])==((2 (π^k))/(k!)) (r^(2 k-1))(h k+r);                            (Subscript[S, 2k+1])==(((2^(2 k+1)) (π^k) k!)/((2 k+1)!))(r^(2 k)) (h(2k+1)+2 r) .

For instance, the volume of a cylinder with radius r and height h is (1/3)π (r^2)h.

For instance, the surface area of a cylinder with radius r and height h is π r (r+(Sqrt[(h^2)+(r^2)])).

Above general formulas can be joined into one (Subscript[S, n])==(((π^((n-1)/2)) )/(Γ(((n+1)/2))))(r+(Sqrt[(h^2)+(r^2)]))(r^(n-2)).

Volume of an n-dimensional cone

Volume (Subscript[V, n]) of an n-dimensional cone of radius r and height h:

(Subscript[V, 2k])==(((2^(2 k-1))(π^(k-1))(k-1)!)/((2 k)!))(r^(2k-1))h;                                       (Subscript[V, 2k+1])==((π^k)/((2 k+1) k!))(r^(2k))h.

For instance, the volume of a cone with radius r and height h is (1/3)π (r^2)h.

Above general formulas can be joined into one (Subscript[V, n])==((π^((n-1)/2))/(n Γ(((n+1)/2))))(r^(n-1))h.

Surface area of an n-dimensional cone

Surface area (Subscript[S, n]) of n-dimensional cone of radius r and height h:

(Subscript[S, 2k])==(((4^k) (π^(k-1)) k! )/((2 k)!))(r+(Sqrt[(h^2)+(r^2)])) (r^(2 k-2));                            (Subscript[S, 2k+1])==(((π^k) (r+(Sqrt[(h^2)+(r^2)])))/(k!)) (r^(2 k-1)).

For instance, the volume of a cone with radius r and height h is (1/3)π (r^2)h.

For instance, the surface area of a cone with radius r and height h is π r (r+(Sqrt[(h^2)+(r^2)])).

Above general formulas can be joined into one (Subscript[S, n])==(((π^((n-1)/2)) )/(Γ(((n+1)/2))))(r+(Sqrt[(h^2)+(r^2)]))(r^(n-2)).

Probability of two random integers being relatively prime

The probability that two integers picked at random are relatively prime is (6/(π^2)) .

History

– The design of Egyptian pyramids (c. 3000 BC) incorporated π as 3+1/7 (∼3.142857…);
– Egyptians (Rhind Papyrus, c. 2000 BC) gave π as (((16/9))^2)∼3.16045…
– China (c. 1200 BC) gave π as 3
– The Biblical verse I Kings 7:23 (c. 950 BC) gave π as 30/10==3.0
– Archimedes (Greece, c. 240 BC) knew that 3+10/71<π<3+1/7 and gave π as 3.1418…
– W. Jones (1706) introduced the symbol π
– C. Goldbach (1742) also used the symbol π
– J. H. Lambert (1761) established that π is an irrational number
– F. Lindemann (1882) 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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