The Pochhammer symbol(1)(2)(Abramowitz and Stegun 1972, p. 256; Spanier 1987; Koepf 1998, p. 5) for is an unfortunate notation used in the theory of special functions for the rising factorial, also known as the rising factorial power (Graham et al. 1994, p. 48) or ascending Factorial (Boros and Moll 2004, p. 16). The Pochhammer symbol is implemented in the Wolfram Language as Pochhammer[x, n].In combinatorics, the notation (Roman 1984, p. 5), (Comtet 1974, p. 6), or (Graham et al. 1994, p. 48) is used for the rising factorial, while or denotes the falling factorial (Graham et al. 1994, p. 48). Extreme caution is therefore needed in interpreting the notations and .The first few values of for nonnegative integers are(3)(4)(5)(6)(7)(OEIS A054654).In closed form, can be written(8)where is a Stirling number of the first kind.The Pochhammer symbol satisfies(9)the dimidiation formulas(10)(11)and..
The (complete) gamma function is defined to be an extension of the factorial to complex and real number arguments. It is related to the factorial by(1)a slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simpler (Gauss 1812; Edwards 2001, p. 8).It is analytic everywhere except at , , , ..., and the residue at is(2)There are no points at which .The gamma function is implemented in the WolframLanguage as Gamma[z].There are a number of notational conventions in common use for indication of a power of a gamma functions. While authors such as Watson (1939) use (i.e., using a trigonometric function-like convention), it is also common to write .The gamma function can be defined as a definite integral for (Euler's integral form)(3)(4)or(5)The complete gamma function can be generalized to the upper incomplete gamma function and lower incomplete gamma function . Min Max Re Im Plots of the real and imaginary..
The factorial is defined for a positive integer as(1)So, for example, . An older notation for the factorial was written (Mellin 1909; Lewin 1958, p. 19; Dudeney 1970; Gardner 1978; Conway and Guy 1996).The special case is defined to have value , consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set ).The factorial is implemented in the Wolfram Language as Factorial[n] or n!.The triangular number can be regarded as the additive analog of the factorial . Another relationship between factorials and triangular numbers is given by the identity(2)(K. MacMillan, pers. comm., Jan. 21, 2008).The factorial gives the number of ways in which objects can be permuted. For example, , since the six possible permutations of are , , , , , . The first few factorials for , 1, 2, ... are 1, 1, 2, 6, 24, 120, ... (OEIS A000142).The..
Let any finite or infinite set of points having no finite limit point be prescribed, and associate with each of its points a definite positive integer as its order. Then there exists an entire function which has zeros to the prescribed orders at precisely the prescribed points, and is otherwise different from zero. Moreover, this function can be represented as a product from which one can read off again the positions and orders of the zeros. Furthermore, if is one such function, thenis the most general function satisfying the conditions of the problem, where denotes an arbitrary entire function.This theorem is also sometimes simply known as Weierstrass's theorem. A spectacularexample is given by the Hadamard product.
The Hadamard product is a representation for the Riemann zeta function as a product over its nontrivial zeros ,(1)where is the Euler-Mascheroni constant and is the Gamma function (Titchmarsh 1987, Voros 1987). The constant in the exponent is given by(2)(3)(OEIS A077142). Hadamard used the Weierstrass product theorem to derive this result. The plot above shows the convergence of the formula along the real axis using the first 100 (red), 500 (yellow), 1000 (green), and 2000 (blue) Riemann zeta function zeros.The product can also be stated in the alternate form(4)where is the xi-function and(5)(Havil 2003, p. 204).
There are (at least) two mathematical objects known as Weierstrass forms. The first is a general form into which an elliptic curve over any field can be transformed, given bywhere , , , , and are elements of .The second is the definition of the gamma functionaswhere is the Euler-Mascheroni constant (Krantz 1999, p. 157).