Factorials of real negative and imaginary numbers - A new perspective.
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Similarly, the factorials of imaginary numbers are complex numbers.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.
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PubMed Central - PubMed
Affiliation: Department of Botanical & Environmental Sciences, Guru Nanak Dev University, Amritsar, 143005 India.
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. No MeSH data available. |
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Mentions: Analogous to factorials of real positive numbers, the factorials of real negative numbers, Π(-1,z) may be given by the notation(-z)!. Figure 2 gives the curves for the integral functions of factorials of real negative integers, (-1), (-2), (-3), on the real negative axis. The area between a curve and the X-axis gives the factorial of that number.Figure 2 |
View Article: PubMed Central - PubMed
Affiliation: Department of Botanical & Environmental Sciences, Guru Nanak Dev University, Amritsar, 143005 India.
No MeSH data available.