Presently, factorials of real negative numbers and imaginary numbers, except for zero and negative integers are interpolated using the Euler’s gamma function.Graphmatica Help - Introduction to Graphmatica In the present paper, the concept of factorials has been generalised as applicable to real and imaginary numbers, and multifactorials. New functions based on Euler’s factorial function have been proposed for the factorials of real negative and imaginary numbers. As per the present concept, the factorials of real negative numbers, are complex numbers.
The factorials of real negative integers have their imaginary part equal to zero, thus are real numbers. Similarly, the factorials of imaginary numbers are complex numbers. The moduli of the complex factorials of real negative numbers, and imaginary numbers are equal to their respective real positive number factorials. Fractional factorials and multifactorials have been defined in a new perspective. The proposed concept has also been extended to Euler’s gamma function for real negative numbers and imaginary numbers, and beta function. The other notable contributors to the field of factorials are J. Mollerup, and others (Wolfram Research 2014b). Dutka ( 1991) gave an account of the early history of the factorial function. Bhargava ( 2000) gave an expository account of the factorials, gave several new results and posed certain problems on factorials. Ibrahim ( 2013) defined the factorial of negative integer n as the product of first n negative integers. There are some other factorial like products and functions, such as, primordial, double factorial, multifactorials, superfactorial, hyperfactorials etc. It is seen that till now the definition of the factorials of real negative numbers is sought from the extrapolation of gamma and other functions. 1) have been defined for the factorials of real negative numbers and imaginary numbers.įractional factorials and multifactorials of real positive numbersįractional factorials and multifactorials of real negative numbersįractional factorials and multifactorials of imaginary positive numbersįractional factorials and multifactorials of imaginary negative numbers In the present paper, the Eularian concept of factorials has been revisited, and new functions based on Euler’s factorial function (Eqn. Graph for B( x ,0.5) and B(- x ,-0.5) as per the present concept. It is seen from the historical account that the Euler’s contributions to logarithms and gamma function have revolutionized developments in science and technology (Lefort 2002 Lexa 2013). Factorial function was first defined for the positive real axis.
Later its argument was shifted down by 1, and the factorial function was extended to negative real axis and imaginary numbers. Recently, the author (Thukral and Parkash 2014 Thukral 2014) gave a new concept on the logarithms of real negative and imaginary numbers. Earlier the logarithms of real negative numbers were defined on the basis of hyperbola defined for the first quadrant and extended to the negative real axis, but the author defined the logarithms for the real negative axis on the basis of hyperbola located in the third quadrant. Similarly, the author in this paper has defined the factorial function for the real negative axis. The factorials of real and imaginary numbers thus defined show uniformity in magnitude and satisfy the basic factorial equation ( c) n n ! = c( c2)( c3) … ( cn).
Another lacuna in the existing Eularian concept of factorials is that although the factorials of negative integers are not defined, the double factorial of any negative odd integer may be defined, e.g., (-1)!! =1, (-3)!! = -1, (-5)!! =1/3 etc. Another strange behaviour of double factorials is that as an empty product, 0!! =1 but for non-negative even integer values. The present concept will remove anomalies in factorials and double factorials. The present concept generalizes factorials as applicable to real and imaginary numbers, and fractional and mutifactorials. The present paper examines the Eularian concept of factorials from basic principles and gives a new concept, based on the Eularian concept for factorials of real negative and imaginary numbers. The factorials of positive and negative integers, and positive and negative imaginary number integers ( z), may be defined as Π( c, z) = c z z ! = c( c2)( c3) … ( cn), where c is a constant (+1, -1, + i or – i), and z >0. The factorials can be interpolated using the Euler’s modified integral equation, for real and imaginary numbers.
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