چكيده به لاتين
Abstract
In this thesis, we examine the concepts of quaternion calculus including multiplication, addition, inverse and conjugate, rotation of quaternions, exponential function and logarithm of quaternions and differential and integral equations of quaternions.
In the following, we compare complex numbers and quaternion numbers and prove the Cauchy integral formula in the complex plane and its extension to the Cauchy-Fouter formula in the space of quaternions. For example, we show that multiplication by a unit complex number implements a plane rotation, while the conjugate by a unit quaternion over a vector in space (a quaternion with zero real part) implements a rotation about an axis in space. Furthermore, we show that because quaternions are noncommutative (or noncommutative), there are two definitions for the derivative (left and right) of the function F∶ U ⊆ H → H, while there is only one definition of the derivative of the function F∶ U ⊆C→C. This means that there are two concepts of the regular function F∶ U ⊆ H → H (left-regular and right-regular) and are respectively two different generalizations of the Cauchy integral formula.
A key element in the proof of the Cauchy integral formula is Green's theorem for 1-form integrals on 1-chains bounding open sets in R^2 , and similarly for the Cauchy-Footer formula, we derive from the generalized Stokes theorem for the integral We use 3-forms on 3-chains that bound open sets in R^4 .
