The conventional complex variable boundary integral equation (CVBIE) based
on the conventional Cauchy integral formula is powerful and suitable to solve
two-dimensional problems. In particular, the unknown function is a complex-valued
holomorphic function. In other words, the unknown function satisfies the
Cauchy-Riemann equations. However, the most part of practical engineering
problems are three-dimensional problems and do not necessarily satisfies
Cauchy-Riemann equations. Therefore, there are two targets in this dissertation. One
is to extend the conventional CVBIE to solve two-dimensional problems for which
the unknown function is not a complex-valued holomorphic function. The other is to
extend to three-dimensions and derive an extended BIE still preserving some
properties of complex variables in the three-dimensional state. For the extension of
the conventional CVBIE, we employ the Borel-Pompeiu formula to derive the
generalized CVBIE. In this way, the torsion problems can be solved in the state of
two shear stress fields directly. In addition, the torsional rigidity can also be
determined simultaneously. Since the theory of complex variables has a limitation
that is only suitable for 2-dimensional problems, we introduce Clifford algebra and
Clifford analysis to replace complex variables to deal with 3-dimensional problems.
Clifford algebra can be seen as an extension of complex or quaternionic algebras.
Clifford analysis is also known as hypercomplex analysis. We apply the Clifford
algebra valued Stokes' theorem to derive Clifford algebra valued BIEs with
Cauchy-type kernels. In this way, some three-dimensional problem with multiple
unknown fields may be solved straightforward. Finally, several electromagnetic
scattering problems are considered to check the validity of the derived Clifford
algebra valued BIEs.