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Factorials of real negative and imaginary numbers - A new perspective
- Ashwani K Thukral^{1}Email author
- Received: 8 October 2014
- Accepted: 23 October 2014
- Published: 6 November 2014
Abstract
Presently, factorials of real negative numbers and imaginary numbers, except for zero and negative integers are interpolated using the Euler’s gamma function. 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.
Keywords
- Factorials of negative numbers
- Factorials of imaginary numbers
- Pi function
- Fractional factorials
- Multifactorials
- Gamma function
- Beta function
Background
Roman factorials
n | Roman factorial ⌊ n⌉! |
---|---|
0 | 1 |
-1 | 1 |
-2 | -1 |
-3 | 1/2 |
-4 | -1/6 |
-5 | 1/24 |
-6 | -1/120 |
The other notable contributors to the field of factorials are J. Stirling, F.W. Newman, B. Riemann, H. Hankel, O. Holder, H. Bohr and 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. (Wikipedia 2014a,[b]; Weistein 2014a).
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. In the present paper, the Eularian concept of factorials has been revisited, and new functions based on Euler’s factorial function (Eqn. 1) have been defined for the factorials of real negative numbers and imaginary numbers.
1. Factorials of real negative numbers
Let a_{n} be a sequence of positive integers, a_{n=1,2,3,…,n.} Therefore,
n! = 1.2.3…n.
Factorials of some integers as per present concept
n | n! | -n | (-n)! = [(-1)^{ n }n!] |
---|---|---|---|
1 | 1 | -1 | -1 |
2 | 2 | -2 | 2 |
3 | 6 | -3 | -6 |
4 | 24 | -4 | 24 |
5 | 120 | -5 | -120 |
where x is the real part and y is imaginary. The factorial of 0 is 1. At real negative integers the imaginary part is zero and the real part has alternating – and + signs, with the factorial of (-1) being (-1). The most important property that justifies the present concept is that the moduli of the complex factorials of real negative numbers are equal to the factorials of real positive numbers.
Complex factorials of some real negative numbers
Real | Imaginary | Modulus | Im/Re | |
---|---|---|---|---|
z | Complex factorial of (-z) | |||
0 | 1 | 0 | 1 | 0 |
0.25 | 0.640 | 0.640i | 0.906 | 1 |
0.5 | 0 | 0.886i | 0.886 | Comp Inf |
0.75 | -0.649 | 0.649i | 0.919 | -1 |
1 | -1 | 0 | 1 | 0 |
1.25 | -0.801 | -0.801i | 1.133 | 1 |
1.5 | 0 | -1.329i | 1.329 | Comp Inf |
1.75 | 1.137 | -1.137i | 1.608 | -1 |
2 | 2 | 0 | 2 | 0 |
2.25 | 1.802 | 1.802i | 2.549 | 1 |
2.5 | 0 | 3.323i | 3.323 | Comp Inf |
2.75 | -3.127 | 3.127i | 4.422 | -1 |
3 | -6 | 0 | 6 | 0 |
1.1 Factorials of half fractions of real negative numbers
1.2 Exponential function
2. Factorials of imaginary numbers
Complex factorials of some imaginary numbers
z | Complex factorial of ( iz ) | |||
---|---|---|---|---|
Real | Imaginary | Modulus | Im/Re | |
0 | 1 | 0 | 1 | 0 |
0.25 | 0.837 | 0.346i | 0.906 | 2.414 |
0.5 | 0.626 | 0.626i | 0.886 | 1 |
0.75 | 0.351 | 0.849i | 0.919 | 2.414 |
1 | 0 | i | 1 | Comp Inf |
1.25 | -0.433 | 1.046i | 1.133 | -0.414 |
1.5 | -0.939 | 0.939i | 1.329 | -1 |
1.75 | -1.485 | 0.615i | 1.608 | -2.414 |
2 | -2 | 0 | 2 | 0 |
2.25 | -2.355 | -0.975i | 2.549 | 2.414 |
2.5 | -2.349 | -2.349i | 3.323 | 1 |
2.75 | -1.692 | -4.086i | 4.422 | 0.414 |
3 | 0 | -6i | 6 | Comp Inf |
z | Complex factorial of (- iz ) | |||
0 | 1 | 0 | 1 | 0 |
0.25 | 0.837 | -0.346i | 0.906 | -0.414 |
0.5 | 0.626 | -0.626i | 0.886 | -1 |
0.75 | 0.351 | -0.849i | 0.919 | -2.414 |
1 | 0 | -i | 1 | Comp Inf |
1.25 | -0.433 | -1.046i | 1.133 | 2.414 |
1.5 | -0.939 | -0.939i | 1.329 | 1 |
1.75 | -1.485 | -0.615i | 1.608 | 0.414 |
2 | -2 | 0 | 2 | 0 |
2.25 | -2.355 | 0.975i | 2.549 | -2.414 |
2.5 | -2.349 | 2.349i | 3.323 | -1 |
2.75 | -1.692 | 4.086i | 4.422 | -0.414 |
3 | 0 | 6i | 6 | Comp Inf |
Periodicity of factorials of imaginary numbers
Z | (iz) ! | Coeff. | (-iz) ! | Coeff. |
---|---|---|---|---|
1 | Π(i, 1) = iΠ(1) = i | i | Π(-i, 1) = - iΠ(1) = - i | -i |
2 | Π(i, 2) = - Π(2) = - 2 | -1 | Π(-i, 2) = - Π(2) = - 2 | -1 |
3 | Π(i, 3) = - iΠ(3) = - 6i | -i | Π(-i, 3) = iΠ(3) = 6i | i |
4 | Π(i, 4) = Π(4) = 24 | 1 | Π(-i, 4) = Π(4) = 24 | 1 |
2.1 Factorials of half fractions of imaginary numbers
3. Multifactorials and fractional factorials
Fractional factorials and multifactorials
Fractional factorials and multifactorials of real positive numbers | |||||
---|---|---|---|---|---|
Fractional factorials | Multifactorials | ||||
z | z! | (0.5z)! | (1.5z)! | (2z)!! | (3z)!!! |
0 | 1 | 1 | 1 | 1 | 1 |
0.5 | 0.886 | 0.626 | 1.085 | 1.253 | 1.534 |
1 | 1 | 0.5 | 1.5 | 2 | 3 |
1.5 | 1.329 | 0.469 | 2.442 | 3.759 | 6.907 |
2 | 2 | 0.5 | 4.5 | 8 | 18 |
2.5 | 3.323 | 0.587 | 9.158 | 18.799 | 51.805 |
3 | 6 | 0.75 | 20.25 | 48 | 162 |
Fractional factorials and multifactorials of real negative numbers | |||||
Fractional factorials | Multifactorials | ||||
z | (-z)! | (-0.5)z! | (-1.5z)! | (-2z)!! | (-3z)!!! |
0 | 1 | 1 | 1 | 1 | 1 |
0.5 | 0.886i | 0.626i | 1.085 | 1.253i | 1.534i |
1 | -1 | -0.5 | -1.5 | -2 | -3 |
1.5 | -0.139i | -0.469i | -2.442 | -3.759i | -6.907i |
2 | 2 | 0.5 | 4.5 | 8 | 18 |
2.5 | 3.323i | 0.587 | 9.157 | 18.799i | 51.805i |
3 | -6 | -0.75 | -20.25 | -48 | -162 |
Fractional factorials and multifactorials of imaginary positive numbers | |||||
Fractional factorials | Multifactorials | ||||
z | (iz!) | (0.5iz)! | (1.5iz)! | (2iz)!! | (3iz)!!! |
0 | 1 | 1 | 1 | 1 | 1 |
0.5 | 0.626 + 0.626i | 0.443 + 0.443i | 0.767 + 0.767i | 0.886 + 0.886i | 1.085 + 1.085i |
1 | i | 0.5i | 1.5i | 2i | 3i |
1.5 | -0.939 + 0.939i | -0.332 + 0.332i | -1.726 + 1.726i | -2.658 + 2.658i | -4.862 + 4.862i |
2 | -2 | -0.5 | -4.5 | -8 | -18 |
2.5 | -2.349-2.349i | -0.415-0.415i | -6.475-6.475i | -13.29-13.29i | 36.6-36.6i |
3 | 6i | -0.75i | -20.25i | -48i | -162i |
Fractional factorials and multifactorials of imaginary negative numbers | |||||
Fractional factorials | Multifactorials | ||||
z | (-iz)! | (-0.5iz)! | (-1.5iz)! | (-2iz)!! | (-3iz)!!! |
0 | 1 | 1 | 1 | 1 | 1 |
0.5 | 0.626-0.626i | 0.443-0.443i | 0.767-0.767i | 0.886-0.886i | 1.085-1.085i |
1 | -i | -0.5i | -1.5i | -2i | -3i |
1.5 | -0.939-0.939i | -0.332-0.332i | -1.726-1.726i | -2.658-2.658i | -4.862-4.862i |
2 | -2 | -0.5 | -4.5 | -8 | -18 |
2.5 | -2.349 + 2.349i | -0.415 + 0.415i | -6.475 + 6.475i | -13.29 + 13.29i | -36.6 + 36.6i |
3 | 6i | 0.75i | 20.25i | 48i | 162i |
4. Gamma function
Complex gamma of real negative and imaginary numbers
Complex gamma of (-z) | ||||
---|---|---|---|---|
z | Real | Imaginary | Modulus | Im/Re |
0.05 | -19.230 | -3.045i | 19.470 | 0.158 |
0.25 | -2.563 | -2.563i | 3.625 | 1 |
0.5 | 0 | -1.772i | 1.772 | Comp Inf |
0.75 | 0.866 | -0.866i | 1.225 | -1 |
1 | 1 | 0 | 1 | 0 |
Complex gamma of ( iz ) | ||||
z | Real | Imaginary | Modulus | Im/Re |
0.05 | 1.527 | -19.410i | 19.470 | -12.706 |
0.25 | 1.387 | -3.349i | 3.625 | -2.414 |
0.5 | 1.253 | -1.253i | 1.772 | -1 |
0.75 | 1.132 | -0.468i | 1.225 | -0.414 |
1 | 1 | 0 | 1 | 0 |
Complex gamma of (- iz ) | ||||
z | Real | Imaginary | Modulus | Im/Re |
0.05 | 19.410 | 1.527i | 19.470 | 12.706 |
0.25 | 3.349 | 1.387i | 3.625 | 2.414 |
0.5 | 1.253 | 1.253i | 1.772 | 1 |
0.75 | 0.468 | 1.132i | 1.225 | 0.414 |
1 | 0 | i | 1 | Comp Inf |
5. Beta function
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(c 2)(c 3) … (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. (Wikipedia 2014b). Another strange behaviour of double factorials is that as an empty product, 0!! =1 but for non-negative even integer values, $0!!=\sqrt{\frac{2}{\pi}}\phantom{\rule{0.2em}{0ex}}\approx \phantom{\rule{0.2em}{0ex}}0.7978$. 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.
Conclusions
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(c 2)(c 3) … (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, $\Pi \left(c,z\right)=\left({c}^{z}\right)z!={c}^{z}{\displaystyle \underset{0}{\overset{\infty}{\int}}{t}^{z}{e}^{-t}\mathrm{dt}},z>0$ for real and imaginary numbers. The factorials for real negative numbers may be defined by the integral equation, $\Pi \left(-1,z\right)={\left(-1\right)}^{z}z!={\displaystyle \underset{-\infty}{\overset{0}{\int}}{t}^{z}{e}^{t}\mathrm{dt}},z>0$. The factorials of negative real numbers are complex numbers. At negative integers the imaginary part of complex factorials is zero, and the factorials for -1, -2, -3, -4 are -1, 2, -6, 24 respectively. Similarly, the factorials of imaginary numbers are complex numbers. The moduli of negative real number factorials and imaginary number factorials are equal to the factorials of respective real positive numbers. The present paper also provides a general definition of fractional factorials and multifactorials. The factorials follow recurrence relations. Similarly, the Euler’s gamma function has been redefined for negative real and imaginary numbers in a new perspective. Beta function on the real negative axis has also been redefined in the context of new concept. The present concept on factorials will be an improvement in the Euler’s factorial and gamma functions.
Software used
- 1.
Wolfram Alpha Examples: Complex Numbers (http://www.wolframalpha.com/examples/ComplexNumbers.html)
- 2.
Draw Function Graphs – Recheronline. (http://rechneronline.de/function-graphs/)
- 3.
Definite integral calculator from Wolfram Alpha Widgets by Evan added in 2010 (http://www.wolframalpha.com/input/?i=definite%20integral%20calculator)
- 4.
Integral calculator from Wolfram Mathematica Online integrator (http://integrals.wolfram.com/index.jsp)
- 5.
Gamma Function Evaluator – The Wolfram Functions site http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=Gamma
- 6.
Function calculator by XIAO gang 2012 (http://wims.unice.fr/wims/en_tool~analysis~function.en.html)
- 7.
Microsoft Excel.
Declarations
Acknowledgements
Thanks are due to the Head, Department of Botanical & Environmental Sciences, Guru Nanak Dev University, Amritsar, for research facilities.
Authors’ Affiliations
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