Topics on Hyperbolic Function Theory in Geometric Algebra with a Positive Signature

2010 ◽  
Vol 10 (1) ◽  
pp. 249-263 ◽  
Author(s):  
Sirkka-Liisa Eriksson ◽  
Heikki Orelma
Keyword(s):  
Geometric Algebra ◽  
Function Theory ◽  
Hyperbolic Function ◽  
Positive Signature

Hyperbolic Function Theory

2007 ◽  
Vol 17 (3) ◽  
pp. 437-450 ◽  
Author(s):  
Sirkka-Liisa Eriksson ◽  
Heinz Leutwiler
Keyword(s):  
Function Theory ◽  
Hyperbolic Function

On Hyperbolic Function Theory

2008 ◽  
Vol 18 (3-4) ◽  
pp. 587-598 ◽  
Author(s):  
Sirkka-Liisa Eriksson ◽  
Heinz Leutwiler
Keyword(s):  
Function Theory ◽  
Hyperbolic Function

scholarly journals Hyperbolic Function Theory in the Skew-Field of Quaternions

2019 ◽  
Vol 29 (5) ◽  
Author(s):  
Sirkka-Liisa Eriksson ◽  
Heikki Orelma
Keyword(s):  
Half Space ◽  
Conjugate Gradient ◽  
Harmonic Functions ◽  
Function Theory ◽  
Integral Formula ◽  
Hyperbolic Function ◽  
Hyperbolic Distance ◽  
Cauchy Type ◽  
Skew Field ◽  
Hyperbolic Harmonic Functions

Abstract We are studying hyperbolic function theory in the total skew-field of quaternions. Earlier the theory has been studied for quaternion valued functions depending only on three reduced variables. Our functions are depending on all four coordinates of quaternions. We consider functions, called $$\alpha $$α-hyperbolic harmonic, that are harmonic with respect to the Riemannian metric $$\begin{aligned} ds_{\alpha }^{2}=\frac{dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}{x_{3}^{\alpha }} \end{aligned}$$dsα2=dx02+dx12+dx22+dx32x3αin the upper half space $${\mathbb {R}}_{+}^{4}=\{\left( x_{0},x_{1},x_{2} ,x_{3}\right) \in {\mathbb {R}}^{4}:x_{3}>0\}$$R+4={x0,x1,x2,x3∈R4:x3>0}. If $$\alpha =2$$α=2, the metric is the hyperbolic metric of the Poincaré upper half-space. Hempfling and Leutwiler started to study this case and noticed that the quaternionic power function $$x^{m}\,(m\in {\mathbb {Z}})$$xm(m∈Z), is a conjugate gradient of a 2-hyperbolic harmonic function. They researched polynomial solutions. Using fundamental $$\alpha $$α-hyperbolic harmonic functions, depending only on the hyperbolic distance and $$x_{3}$$x3, we verify a Cauchy type integral formula for conjugate gradient of $$\alpha $$α-hyperbolic harmonic functions. We also compare these results with the properties of paravector valued $$\alpha $$α-hypermonogenic in the Clifford algebra $${{\,\mathrm{{\mathcal {C}}\ell }\,}}_{0,3}$$Cℓ0,3.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

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