Abstract
In the article a new class of monogenic functions with (logarithmic) line singularities is studied, which naturally extend the classical system of outer solid spherical monogenics. These functions have special properties with respect to the hypercomplex derivative (generalized Appell property) and can be generated by a three-term recurrence relation. Furthermore, an explicit decomposition of the harmonic Newton kernel is given by means of these functions and a connection to the Cauchy kernel is shown.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, A., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55, Wiley, Newyork (1972)
Andrews, L.C.: Special Functions of Mathematics for Engineers. SPIE Oxford University Press, Oxford (1998)
Barber, J.R.: Elasticity, Solid Mechanics and its Applications, vol. 172, 3rd revised edition, Springer, Newyork (2010)
Bock, S., Gürlebeck, K.: On a generalized Appell system and monogenic power series. Mathematical Methods in the Applied Sciences 33, 394–411 (2010)
Bock, S.: On a three dimensional analogue to the holomorphic z-powers: power series and recurrence formulae. Complex Variables and Elliptic Equations 57(12), 1349–1370 (2012)
Bock, S.: On a three dimensional analogue to the holomorphic z-powers: Laurent series expansions. Complex Variables and Elliptic Equations 57(12), 1271–1287 (2012)
Bock, S.: On orthogonal series expansions in dimensions 2, 3 and 4. In: Proceedings of the 9th International Conference on Clifford Algebras and their Applications in Mathematical Physics (ICCA9), Weimar (2011)
Bock, S.: On monogenic series expansions with applications to linear elasticity. Advances in Applied Clifford Algebras 24(4), 931–943 (2014)
Bock, S.: On a hypercomplex version of the Kelvin solution in linear elasticity, to appear in the book Modern Problems in Applied Analysis. In: Proceedings of the 3rd Conference Boundary Value Problems, Functional Equations and Applications, Rzeszow (2016)
Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis, Pitman Research Notes Math. Ser. 76. Pitman, London (1982)
Chau, K.T.: Analytic Methods in Geomechanics. CRC Press, Boca Raton (2012)
Gürlebeck, K., Malonek, H.R.: A hypercomplex derivative of monogenic functions in \(\mathbb{R}^{n+1}\) and its applications. Complex Variables 39, 199–228 (1999)
Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic Functions in the Plane and n-dimensional Space. A Birkhäuser book (2008)
Luna-Elizarrarás, M.E., Shapiro, M.: A survey on the (hyper-) derivates in complex, quaternionic and Clifford analysis. Milan Journal of Mathematics 79, 521–542 (2011)
Malonek, H.: A new hypercomplex structure of the euclidean space \(\mathbb{R}^{m+1}\) and the concept of hypercomplex differentiability. Complex Variables, Theory and Application: An International Journal 14(1–4), 25–33 (1990)
Mindlin, R.D.: Force at a point in the interior of a semi-infinite solid. Physics 7(5), 195–202 (1936)
Szmytkowski, R.: On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). Journal of Mathematical Chemistry 49, 1436–1477 (2011)
Weisz-Patrault, D., Bock, S., Gürlebeck, K.: Three-dimensional elasticity based on quaternion-valued potentials. International Journal of Solids and Structures 51(19), 3422–3430 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Wolfgang Sprössig
Rights and permissions
About this article
Cite this article
Bock, S. On a Class of Monogenic Functions with (Logarithmic) Line Singularities. Adv. Appl. Clifford Algebras 28, 6 (2018). https://doi.org/10.1007/s00006-018-0823-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-018-0823-5