We present the results of theoretical research of the generalized hypersherical function (HS) by generalizing two known functions related to the sphere hypersurface and hypervolume and the recurrent relation between them. By introducing two-dimensional degrees of freedom and (and the third, radius ), we develop the derivative functions for all three arguments and the possibilities of their use. The symbolical evolution, numerical experiment, and graphical presentation of functions are realized using the Mathcad Professional and Mathematica softwares. 1. Introduction The hyperspherical function (HS) is a hypothetical function related to multidimensional space and generalization of the sphere geometry. This function is primarily formed on the basis of the interpolating power of the gamma function. It belongs to the group of special functions, so its testing, besides gamma, is performed on the basis of the related functions, such as -gamma, -psi, -beta, erf, and so forth. Its most significant value is in its generalizing from discrete to continuous [1]. In addition, we move from the field of natural integers of the geometry sphere dimensions—degrees of freedom, to the set of real and nonintegral values, thus, we obtain the prerequisites for a more concise analysis of this function. In this paper the analysis is focused on the infinitesimal calculus application of the hyperspherical function that is given in its generalized form. For the development of hyperspherical and other functions of the multidimensional objects, see: Bishop and Whitlock [2], Collins [3], Conway [4], Dodd and Coll [5], Hinton [6], Hocking and Young [1], Manning [7], Maunder [8], Neville [9], Rohrmann and Santos [10], Rucker [11], Maeda et al. [12], Sloane [13], Sommerville [14], Weels [15], and others; see [16–22]. Nowadays, the research of the hypersperical functions is represented both in Euclid’s and Riemann’s geometry and topology (Riemann’s and Poincare’s sphere, multidimensional potentials, theory of fluids, nuclear (atomic) physics, hyperspherical black holes, etc.) 2. The Hyperspherical Functions of a Derivative 2.1. The Hyperspherical Funcional Matrix The former results, as it is known, [5, 23], give the functions of the hyperspherical surface ( ) and volume ( ) therefore, we have In general, we have Thus, we give the definition of the hyperspherical function [24]. Definition 2.1. The hyperspherical function with two degrees of freedom and is defined as This is a function of three variables and two degrees of freedom and and the radius of the hypersphere . In real spherical
References
[1]
J. G. Hocking and G. S. Young, Topology, Dover, New York, NY, USA, 2nd edition, 1988.
[2]
M. Bishop and P. A. Whitlock, “The equation of state of hard hyperspheres in four and five dimensions,” Journal of Chemical Physics, vol. 123, no. 1, p. 14507, 2005.
[3]
G. P. Collins, “The shapes of space,” Scientific American, vol. 291, no. 1, pp. 94–103, 2004.
[4]
J. H. Conway, Sphere Packings, Lattices and Groups, vol. 290, Springer, New York, NY, USA, 2nd edition, 1993.
[5]
J. Dodd and V. Coll, “Generalizing the equal area zones property of the sphere,” Journal of Geometry, vol. 90, no. 1-2, pp. 47–55, 2008.
[6]
H. C. Hinton, The Fourth Dimension, Health Research, Pomeroy, Wash, USA, 2nd edition, 1993.
[7]
H. P. Manning, Geometry of Four Dimensions, Phillips Press, 2010.
[8]
F. R. C. Maunder, Algebraic Topology, Dover, New York, NY, USA, 1997.
[9]
H. E. Neville, The Fourth Dimension, Cambridge, England, Cambridge University Press, 1921.
[10]
R. D. Rohrmann and A. Santos, “Structure of hard-hypersphere fluids in odd dimensions,” Physical Review E, vol. 76, no. 5, p. 051202, 17, 2007.
[11]
R. Rucker, The Fourth Dimension: A Guided Tour of the Higher Universes, Houghton Mifflin, Boston, Mass, USA, 1984.
[12]
S. Maeda, Y. Watanabe, and K. Ohno, “A scaled hypersphere interpolation technique for efficient construction of multidimensional potential energy surfaces,” Chemical Physics Letters, vol. 414, no. 4–6, pp. 265–270, 2005.
[13]
A. J. N. Sloane, “Sequences, A072478, A072479, A072345, A072346, A087299, A087300 and A074457,” in The On-Line Encyclopedia of Integer Sequences.
[14]
D. M. Y. Sommerville, An Introduction to the Geometry of Dimensions, Dover, New York, NY, USA, 1958.
[15]
D. Wells, The Penguin Dictionary Of Curious and Interesting Numbers, Penguin Books, Middlesex, England, 1986.
[16]
C. M. J. Joshi and S. Sadan, “Random walk over a hypersphere,” International Journal of Mathematics and Mathematical Sciences, vol. 8, no. 4, pp. 683–688, 1985.
[17]
G. A. Kabatiansky and V. I. Levenshtein, “Bounds for packings on a sphere and in space,” Problemy Peredachi Informatsii, vol. 14, no. 1, 1978.
[18]
D. Leti？ and N. Caki？, Srinivasa Ramanujan, The Prince of Numbers, Computer Library, Belgrade, Serbia, 2010.
[19]
D. Leti？, Generalisation of the Hyperspherical Function, monograph, Technical Faculty M. Pupin, Zrenjanin, Serbia, 2010.
[20]
D. Leti？, B. Davidovi？,, I. Berkovi？, and T. Petrov, Mathcad 13 in Mathematics and Visualization, Computer Library, Belgrade, Serbia, 2007.
[21]
P. Loskot and N. C. Beaulieu, “On monotonicity of the hypersphere volume and area,” Journal of Geometry, vol. 87, no. 1-2, pp. 96–98, 2007.
[22]
T. Sasaki, “Hyperbolic affine hyperspheres,” Nagoya Mathematical Journal, vol. 77, pp. 107–123, 1980.
[23]
http://mathworld.wolfram.com/Hypersphere.html.
[24]
D. Letic, Ed., Three Archimedes' Bodies, Belgrade, Serbia, Technical Faculty M. Pupin, Zrenjanin, Electrotechnical Faculty, 2010.
