Factorials medium

Factorials: Definition, History, Applications, and Properties

The factorial of a non-negative integer n is the product of all positive integers less than or equal to n.
Factorials have been discovered in several ancient cultures, including Indian mathematics and Jewish mysticism.
Factorials are used in combinatorics to count the possible distinct sequences of n distinct objects.
Factorials are also used in power series for the exponential function and other functions, and have applications in algebra, number theory, probability theory, and computer science.
The growth rate of factorials is faster than exponential growth but slower than a double exponential function.
Stirling's approximation provides an accurate approximation to the factorial of large numbers.
Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials.
The factorial function can be extended to a continuous function of complex numbers, except at the negative integers, the gamma function.
Other notable functions and number sequences are closely related to the factorials, including the binomial coefficients, double factorials, falling factorials, primorials, and subfactorials.
Implementations of the factorial function are commonly used as an example of different computer programming styles.
Factorials have applications in probability theory, quantum and statistical physics, and computer science.
The product formula for the factorial implies that n! is divisible by all prime numbers that are at most n, and by no larger prime numbers.The Factorial Function: Definition, Properties, and Computation
The factorial function is a mathematical function that takes a non-negative integer as input and returns the product of all positive integers up to and including that integer.
The factorial function is denoted by an exclamation mark (!) after the input integer, e.g., n! denotes the factorial of n.
The factorial function grows very quickly, as n! = 1 x 2 x 3 x ... x n, and its value for large n can be approximated by Stirling's formula.
The factorial function has many applications in combinatorics, probability theory, and number theory, such as counting permutations, combinations, and partitions.
The factorial function satisfies a recurrence relation, n! = n x (n-1)!, and has a closed-form expression in terms of gamma function, (n+1)! = Γ(n+2).
The factorial function has many properties, such as being an integer, being divisible by all integers up to n, and being related to the binomial coefficients.
The factorial function can be extended to non-integer values using the gamma function, which is a continuous interpolation of the factorials.
The gamma function satisfies a functional equation, Γ(z+1) = z Γ(z), and has a unique log-convexity property that distinguishes it from other continuous interpolations.
The factorial function can be computed in various ways, such as by iteration, recursion, memoization, dynamic programming, and prime factorization.
The computational complexity of the factorial function depends on the method used, and can range from O(n) to O(n log^2 n) in time and O(1) to O(n log n) in space.
The factorial function can be used to define and compute other related sequences and functions, such as the harmonic numbers, the digamma function, and the p-adic gamma function.
The factorial function is a common feature in scientific calculators and programming libraries, and its efficient computation is important in many scientific and engineering applications.

Factorials: Definition, History, Applications, and Properties

The factorial of a non-negative integer n is the product of all positive integers less than or equal to n.
Factorials have been discovered in several ancient cultures, including Indian mathematics and Jewish mysticism.
Factorials are used in combinatorics to count the possible distinct sequences of n distinct objects.
Factorials are also used in power series for the exponential function and other functions, and have applications in algebra, number theory, probability theory, and computer science.
The growth rate of factorials is faster than exponential growth but slower than a double exponential function.
Stirling's approximation provides an accurate approximation to the factorial of large numbers.
Legendre's formula describes the exponents of the prime numbers in a prime factorization of the factorials, and can be used to count the trailing zeros of the factorials.
The factorial function can be extended to a continuous function of complex numbers, except at the negative integers, the gamma function.
Other notable functions and number sequences are closely related to the factorials, including the binomial coefficients, double factorials, falling factorials, primorials, and subfactorials.
Implementations of the factorial function are commonly used as an example of different computer programming styles.
Factorials have applications in probability theory, quantum and statistical physics, and computer science.
The product formula for the factorial implies that n! is divisible by all prime numbers that are at most n, and by no larger prime numbers.The Factorial Function: Definition, Properties, and Computation
The factorial function is a mathematical function that takes a non-negative integer as input and returns the product of all positive integers up to and including that integer.
The factorial function is denoted by an exclamation mark (!) after the input integer, e.g., n! denotes the factorial of n.
The factorial function grows very quickly, as n! = 1 x 2 x 3 x ... x n, and its value for large n can be approximated by Stirling's formula.
The factorial function has many applications in combinatorics, probability theory, and number theory, such as counting permutations, combinations, and partitions.
The factorial function satisfies a recurrence relation, n! = n x (n-1)!, and has a closed-form expression in terms of gamma function, (n+1)! = Γ(n+2).
The factorial function has many properties, such as being an integer, being divisible by all integers up to n, and being related to the binomial coefficients.
The factorial function can be extended to non-integer values using the gamma function, which is a continuous interpolation of the factorials.
The gamma function satisfies a functional equation, Γ(z+1) = z Γ(z), and has a unique log-convexity property that distinguishes it from other continuous interpolations.
The factorial function can be computed in various ways, such as by iteration, recursion, memoization, dynamic programming, and prime factorization.
The computational complexity of the factorial function depends on the method used, and can range from O(n) to O(n log^2 n) in time and O(1) to O(n log n) in space.
The factorial function can be used to define and compute other related sequences and functions, such as the harmonic numbers, the digamma function, and the p-adic gamma function.
The factorial function is a common feature in scientific calculators and programming libraries, and its efficient computation is important in many scientific and engineering applications.