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Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function ...
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Double Factorial
In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the double factorial is :n!! = \prod_^\frac (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,, and for odd it is :n!! = \prod_^\frac (2k-1) = n(n-2)(n-4)\cdots 3\cdot 1 \,. For example, . The zero double factorial as an empty product. The sequence of double factorials for even = starts as : 1, 2, 8, 48, 384, 3840, 46080, 645120,... The sequence of double factorials for odd = starts as : 1, 3, 15, 105, 945, 10395, 135135,... The term odd factorial is sometimes used for the double factorial of an odd number. History and usage In a 1902 paper, the physicist Arthur Schuster wrote: states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of t ...
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Falling Factorial
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial :\begin (x)_n = x^\underline &= \overbrace^ \\ &= \prod_^n(x-k+1) = \prod_^(x-k) \,. \end The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, (A reprint of the 1950 edition by Chelsea Publishing Co.) rising sequential product, or upper factorial) is defined as :\begin x^ = x^\overline &= \overbrace^ \\ &= \prod_^n(x+k-1) = \prod_^(x+k) \,. \end The value of each is taken to be 1 (an empty product) when . These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , where is a non-negative integer. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely t ...
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Gamma Function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer , \Gamma(n) = (n-1)!\,. Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: \Gamma(z) = \int_0^\infty t^ e^\,dt, \ \qquad \Re(z) > 0\,. The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles. The gamma function has no zeroes, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function: \Gamma(z) = \mathcal M ...
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Legendre's Formula
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime ''p'' that divides the factorial ''n''!. It is named after Adrien-Marie Legendre. It is also sometimes known as de Polignac's formula, after Alphonse de Polignac. Statement For any prime number ''p'' and any positive integer ''n'', let \nu_p(n) be the exponent of the largest power of ''p'' that divides ''n'' (that is, the ''p''-adic valuation of ''n''). Then :\nu_p(n!) = \sum_^ \left\lfloor \frac \right\rfloor, where \lfloor x \rfloor is the floor function. While the sum on the right side is an infinite sum, for any particular values of ''n'' and ''p'' it has only finitely many nonzero terms: for every ''i'' large enough that p^i > n, one has \textstyle \left\lfloor \frac \right\rfloor = 0. This reduces the infinite sum above to :\nu_p(n!) = \sum_^ \left\lfloor \frac \right\rfloor \, , where L = \lfloor \log_ n \rfloor. Example For ''n'' = 6, one has 6! = 720 = 2 ...
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Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310. The first five primorials are: : 2, 6, 30, 210, 2310 . The sequence also includes as empty product. Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. Definition for natural numbers In general, for a positive integer , its p ...
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Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ...
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Sefer Yetzirah
''Sefer Yetzirah'' ( ''Sēp̄er Yəṣīrā'', ''Book of Formation'', or ''Book of Creation'') is the title of a book on Jewish mysticism, although some early commentators treated it as a treatise on mathematical and linguistic theory as opposed to Kabbalah. ''Yetzirah'' is more literally translated as "Formation"; the word ''Briah'' is used for "Creation". The book is traditionally ascribed to the patriarch Abraham, although others attribute its writing to Rabbi Akiva. Modern scholars have not reached consensus on the question of its origins. According to Rabbi Saadia Gaon, the objective of the book's author was to convey in writing how the things of our universe came into existence. Conversely, Judah Halevi asserts that the main objective of the book, with its various examples, is to give to man the means by which he is able to understand the unity and omnipotence of God, which are multiform on one side and, yet, uniform on the other. The famous opening words of the book are as ...
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Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of s ...
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Googol
A googol is the large number 10100. In decimal notation, it is written as the digit 1 followed by one hundred zeroes: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. Etymology The term was coined in 1920 by 9-year-old Milton Sirotta (1911–1981), nephew of U.S. mathematician Edward Kasner. He may have been inspired by the contemporary comic strip character Barney Google. Kasner popularized the concept in his 1940 book '' Mathematics and the Imagination''. Other names for this quantity include ''ten duotrigintillion'' on the short scale, ''ten thousand sexdecillion'' on the long scale, or ''ten sexdecilliard'' on the Peletier long scale. Size A googol has no special significance in mathematics. However, it is useful when comparing with other very large quantities such as the number of subatomic particles in the visible universe or the number of hypothetical possibilities in a chess ...
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Stirling's Approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the logarithm of the factorial: \ln(n!) = n\ln n - n +O(\ln n), where the big O notation means that, for all sufficiently large values of n, the difference between \ln(n!) and n\ln n-n will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to use instead the binary logarithm, giving the equivalent form \log_2 (n!) = n\log_2 n - n\log_2 e +O(\log_2 n). The error term in either base can be expressed more precisely as \tfrac12\log(2\pi n)+O(\tfrac1n), corresponding to an approximate formula for the factorial itself, n! \sim \sqrt\left(\frac\righ ...
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Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with rea ...
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Interpolate
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance ...
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