My own unipodal papers:

An Instructive Unipodal Integral A unipodal integration of the integral

∫ (a cosh x - sinh x)cosh bx dx.

See my own unipodal papers and web content at

Unipodal Algebra/Numbers Content.


Unipodal and Bicomplex Internet References

This page presents a number of Internet references to papers on the topics of unipodal algebra (or unipodal numbers), bicomplex numbers, hyperbolic numbers, split-complex numbers, perplex numbers, hypercomplex numbers. These number systems (mostly) have in common that they are (usually commutative) rings with one unipotent number u, that is, a number that is not a real number, but that squares to unity (u2= 1). Most of these systesms have two or more distinct unit imaginary numbers. The unipodal algebra is formed from the linear combinations of {1, u} over the complex numbers. The bicomplex numbers are formed by the linear combinations of {1, i, j, ij} over the real numbers, where ij=u. Then, appropriate multiplications are defined on these elements to ensure closure.

In no particular order are the Internet references I've found so far.

B0-VALUED MONOGENIC FUNCTIONS TO THE THEORY OF PLANE ANISOTROPY , by S. V. Gryshchuk.

On algebraic properties of bicomplex and hyperbolic numbers, by D. Rochon and M. Shapiro.

The Poincare mass operator in terms of a hyperbolic algebra, by S. Ulrych.

Perplex numbers and quaternions, by A. K. T. ASSIS.

Bicomplex Numbers and their Elementary Functions, by M.E. Luna-Elizarraras, M. Shapiro, D.C. Struppa1, and A. Vajiac.

Understanding bicomplex numbers, by various contributors.

Structure of Hypercomplex Units and Exotic Numbers as Sections of Bi-Quaternions, by Alexander P. Yefremov.

Solutions of Higher-Degree Reduced Polynomial Equations by Using Unipodal Numbers , by Mohammed R. Karim.

Functions of multivectors in 3D Euclidean geometric algebra via spectral decomposition (for physicists and engineers) , by Miroslav Josipović.

Fundamental Theorems of Algebra for the Perplexes , by Rob Poodiack and Kevin J. LeClair.

Bicomplex Version of Laplace Transform , by Mr. Anand Kumar, Mr. Pravindra Kumar.

ON HYPERBOLIC COMPLEX LTI DIGITAL SYSTEMS , by Daniel Alfsmann and Heinz G. Gockler.

A Bicomplex Riemann Zeta Function , by Dominic Rochon.

Gravitoelectromagnetism in a complex Clifford algebra , by S. Ulrych.

Relativistic quantum physics with hyperbolic numbers , by S. Ulrych.

Symmetries in the hyperbolic Hilbert space , by S. Ulrych.

Bicomplex Functional Analysis , by Dominic Rochon.

Pythagorean Triples, Complex Numbers, and Perplex Numbers , by Howard Sporn.

Some Qualities of the Unipodal Numbers , by Yang Fan.

Perplex number , by nLab.

The Geometric Concept of Number , by G. Sobczyk.

Considerations on the hyperbolic complex Klein-Gordon equation , by S. Ulrych.

Infinite Product of Bicompcomplex Numbers , by Chinmay Ghosh.

Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis , by D. Alpay, Ben Gurion, M. E. Luna-Elizarrarás, M. Shapiro, Daniele C. Struppa.

GEOMETRIC and CLIFFORD ALGEBRA , links provided.

On the set of zeros of bihyperbolic polynomials , by A.A. Pogorui, R.M. Rodríguez-Dagnino & R.D. Rodríguez-Said.

The Bicomplex Heisenberg Uncertainty Principle , by Raphael Gervais Lavoie and Dominic Rochon.

SOLUTION OF MAXWELL'S WAVE EQUATIONS IN BICOMPLEX SPACE , by RITU AGARWAL, MAHESH PURI GOSWAMI, RAVI P. AGARWAL, KAMMA KESA, VENKATARATNAM, DUMITRU BALEANU.

Bicomplex Holomorphic Function , by M. Elena, Luna-Elizarraras, Michael Shapiro, Daniele C. Struppa, Adrian Vajiac.

Bicomplex algebra and function theory , by Stefan Ronn.

Certain results on entire functions defined by bicomplex Dirichlet series , by Jogendra Kumar.

Understanding bicomplex numbers , by various contributors.

Monogenic Functions in a Finite-Dimensional Semi-Simple Commutative Algebra , by S. A. Plaksa and R. P. Pukhtaievych.

Certain Results on Bicomplex Matrices , by Anjali & Amita, Dr. B. R. Ambedkar University .

Self-Inversive Bicomplex Polynomials , by Sara Ahmadi and Mahmood Bidkham.

Certain Study on Eigen Values of Bicomplex Matrices , Dhruva Dixit

Bicomplex Fibonacci quaternions , by F. Torunbalci Aydin.

Bicomplex number explained , by author.

A note on Bicomplex Linear operators on bicomplex Hilbert spaces , by Khalid Manzoor.

Maximal Ideals in a Bicomplex Algebra and Bicomplex Gelfand-Mazur Theorem , by Singh, Kulbir; Kumar, Romesh.

Generalization of the Gauss-Lucas theorem for bicomplex polynomials , by Mahmood BIDKHAM, Sara AHMADI.

A NOTE ON BICOMPLEX FIBONACCI AND LUCAS NUMBERS , by Semra Kaya Nurkan and Ilkay Arslan Guven.

Bicomplex Hermitian Clifford analysis , by Lin CHEN, Guangbin REN,Haiyan WANG.

BICOMPLEX NUMBERS -- Found in: Crux , by HERBERT E. SALZER.

Homothetic Motions at (E8) with Bicomplex Numbers (C3) , by Faik BABADAG, Yusuf YAYLI, Nejat EKMEKCI.

Certain Study of Bicomplex Matrices and a New Composition of Bicomplex Matrices , by Prabhat Kumar & Akhil Prakash.

Infinite-dimensional bicomplex Hilbert spaces. , by Gervais Lavoie, Raphael; Marchildon, Louis; Rochon, Dominic.

ON DUAL BICOMPLEX NUMBERS AND THEIR SOME ALGEBRAIC PROPERTIES. , by SERPIL HALICI, SULE CURUK.

Singularities of functions of one and several bicomplex variables , by Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, Adrian Vajiac and Mihaela B. Vajiac.

REMARKS ON BICOMPLEX VARIABLES AND OTHER SIMILAR VARIABLES , by S. DIMIEV, R. LAZOV and S. SLAVOVA.

On bicomplex numbers with coefficients from the complex Fibonacci sequence , by Serpil Halici and Sule Curuk.

Relation between Dirac spinors, quaternions, and bicomplex numbers , by various contributors.

Topological Bicomplex Modules , by Romesh Kumar and Heera Saini.

Bicomplex Version of Cayley Hamilton Theorem , by Dhruva Dixit.

Bicomplex Waves in Electromagnetic Scattering and Diffraction Problems , by Masahiro Hashimoto.

Bicomplex holomorphic functional calculus , by Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa.

Note on Bicomplex Matrices , by Canan Olcek and Semra Kaya Nurkan.


Garret Sobczyk publications:

Various articles .

Matrix Gateway to Geometric Algebra, Spacetime and Spinors .