This treatise investigates holomorphic functions defined on the space of bicomplex numbers introduced by Segre. The theory of these functions is associated with Fueter’s theory of regular, quaternionic functions. The algebras of quaternions and bicomplex numbers are developed by making use of so-called complex pairs. Special attention is paid to singular bicomplex numbers that lack an inverse. ...

We introduce the set of bicomplex numbers T which is a commutative ring with zero divisors defined by T = {w0 + w1i1 + w2i2 + w3j| w0, w1, w2, w3 ∈ R} where i21 = −1, i22 = −1, j = 1, i1i2 = j = i2i1. We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schrödinger equation and found the continuity equations. The discrete symmetries of the ...

Using the bicomplex numbers T which is a commutative ring with zero divisors defined by T = {w0+w1i1+w2i2+ w3j | w0, w1, w2, w3 ∈ R} where i21 = −1, i22 = −1, j = 1, i1i2 = j = i2i1, we construct hyperbolic and bicomplex Hilbert spaces. Linear functionals and dual spaces are considered on these spaces and properties of linear operators are obtain; in particular it is established that the eigenv...

The algebra B of bicomplex numbers is viewed as a complexification the Archimedean f-algebra hyperbolic D. This lattice-theoretic approach allows us to establish new properties so-called D-norms. In particular, we show that D-norms generate same topology in B. We develop D-trigonometric form number which leads geometric interpretation nth roots terms polyhedral tori. use concepts developed, par...