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A related uniqueness theorem of Helmut Wielandt states that the complex gamma function and its scalar multiples are the only holomorphic functions on the positive complex half-plane that obey the functional equation and remain bounded for complex numbers with real part between 1 and 2.
The factorial number system is sometimes defined with the 0! place omitted because it is always zero (sequence A007623 in the OEIS). In this article, a factorial number representation will be flagged by a subscript "!". In addition, some examples will have digits delimited by a colon. For example, 3:4:1:0:1:0! stands for
Multiplicative partitions of factorials are expressions of values of the factorial function as products of powers of prime numbers. They have been studied by Paul Erdős and others. [1] [2] [3] The factorial of a positive integer is a product of decreasing integer factors, which can in turn be factored into prime numbers.
The entry 4+2i = −i(1+i) 2 (2+i), for example, could also be written as 4+2i= (1+i) 2 (1−2i). The entries in the table resolve this ambiguity by the following convention: the factors are primes in the right complex half plane with absolute value of the real part larger than or equal to the absolute value of the imaginary part.
The interaction of two factors with s 1 and s 2 levels, respectively, has (s 1 −1)(s 2 −1) degrees of freedom. The formula for more than two factors follows this pattern. In the 2 × 3 example above, the degrees of freedom for the two main effects and the interaction — the number of columns for each — are 1, 2 and 2, respectively.
Before performing a Yates analysis, the data should be arranged in "Yates' order". That is, given k factors, the k th column consists of 2 (k - 1) minus signs (i.e., the low level of the factor) followed by 2 (k - 1) plus signs (i.e., the high level of the factor). For example, for a full factorial design with three factors, the design matrix is
A factorial x! is the product of all numbers from 1 to x. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included. A k-smooth number (for a natural number k) has its prime factors ≤ k (so it is also j-smooth for any j > k).
The factorial function, generalized to all complex numbers except negative integers. For example, 0! = 1! = 1, (−0.5)! = √π, (0.5)! = √π/2. Besides nonnegative integers, the factorial function can also be defined for non-integer values, but this requires more advanced tools from mathematical analysis.