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The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series ...
It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n + 1) : n ∈ , are irrational. [1] There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ (5), ζ (7), ζ (9), or ζ ...
[a] The constant satisfies the quadratic equation = + and is an irrational number with a value of [1] φ = 1 + 5 2 = {\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=} 1.618 033 988 749 .... The golden ratio was called the extreme and mean ratio by Euclid , [ 2 ] and the divine proportion by Luca Pacioli , [ 3 ] and also goes by several other ...
A Gram point is a point on the critical line 1/2 + it where the zeta function is real and non-zero. Using the expression for the zeta function on the critical line, ζ(1/2 + it) = Z(t)e − iθ(t), where Hardy's function, Z, is real for real t, and θ is the Riemann–Siegel theta function, we see that zeta is real when sin(θ(t)) = 0.
One half is a rational number that lies midway between nil and unity (which are the elementary additive and multiplicative identities) as the quotient of the first two non-zero integers, . It has two different decimal representations in base ten, the familiar and the recurring , with a similar pair of expansions in any even base; while in odd ...
The circumference of a circle with diameter 1 is π. A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter ), or by mathematicians' names to facilitate using it across multiple mathematical problems. [ 1 ]
Gematria. Gematria ( / ɡəˈmeɪtriə /; Hebrew: גמטריא or gimatria גימטריה, plural גמטראות or גימטריות, gimatriot) [ 1] is the practice of assigning a numerical value to a name, word or phrase by reading it as a number, or sometimes by using an alphanumerical cipher. The letters of the alphabets involved have ...
The absorption value for normal-incident light on graphene in vacuum would then be given by πα / (1 + πα/2) 2 or 2.24%, and the transmission by 1 / (1 + πα/2) 2 or 97.75% (experimentally observed to be between 97.6% and 97.8%). The reflection would then be given by π 2 α 2 / 4 (1 + πα/2) 2 .