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Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function and the Fourier coefficients of the J-invariant (OEIS: A000521): ∑ n = − 1 ∞ j n q n = 256 ( 1 − z + z 2 ) 3 z 2 ( 1 − z ) 2 , {\displaystyle \sum _{n=-1}^{\infty }\mathrm {j} _{n}q^{n}=256{\dfrac {(1-z+z^{2})^{3}}{z ...
For the first term in the Taylor series, all digits must be processed. In the last term of the Taylor series, however, there's only one digit remaining to be processed. In all of the intervening terms, the number of digits to be processed can be approximated by linear interpolation. Thus the total is given by:
Another proof that uses triangles considers the area enclosed by a circle to be made up of an infinite number of triangles (i.e. the triangles each have an angle of d𝜃 at the center of the circle), each with an area of 1 / 2 · r 2 · d𝜃 (derived from the expression for the area of a triangle: 1 / 2 · a · b · sin𝜃 ...
The number π (/ p aɪ / ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
(Pi function) – the gamma function when offset to coincide with the factorial Rectangular function π ( n ) {\displaystyle \pi (n)\,\!} – the Pisano period
The constant π (pi) has a natural definition in Euclidean geometry as the ratio between the circumference and diameter of a circle. It may be found in many other places in mathematics: for example, the Gaussian integral, the complex roots of unity, and Cauchy distributions in probability. However, its ubiquity is not limited to pure mathematics.
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Descartes's theorem on the "total defect" of a polyhedron states that if the polyhedron is homeomorphic to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720° or 4 π ...
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