We first study some properties of a regular function and then generalize Cauchy integral theorem and Cauchy integral formula in [A.sub.n](R).
In this paper, we studied some properties of a regular function in Clifford analysis and generalized several classical theorems in [A.sub.n](R),such as Liouville theorem, Plemelj formula, Cauchy integral theorem, and Cauchy integral formula. By means of these theorems and of the classical boundary value theory, we dealt with the solvability and the explicit solutions for some classes of BVPs and SIEs in Clifford analysis.
Then equality (3.1), the Cauchy integral formula and (3.4) show that
Then, using (3.6) and the Cauchy integral formula, we find that
Then, by the Cauchy integral formula
for a region [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we have:
Another representation for the fractional derivative is based on the Cauchy integral formula
. This representation, too, has been widely used in many interesting papers (see, e.g., the works of Osler [27-30]).
Emphasizing how complex analysis is a natural outgrowth of multivariable real calculus, this graduate textbook introduces the Cauchy integral formula
, the properties and behavior of holomorphic functions, harmonic functions, analytic continuation, topology, Mergelyan's theorem, Hilbert spaces, and the prime number theorem.
The values of the function f (z) for interior points z [member of] G can be computed by the Cauchy integral formula. Hence, the integral equation (2.3) can be used to solve the problems listed in Table 1.1.
Computing the interior values requires computing the Cauchy integral formula. For the convenience of the reader, we present a MATLAB function fcau for the fast computation of the Cauchy integral formula.
Computing the Cauchy integral formula. The solutions of the boundary integral equations (2.3) and (2.5) provide us with the values of the conformal mapping and the solution of the boundary value problem on the boundary [GAMMA].
Thus, the Cauchy integral formula (6.1) can then be written for z [member of] G as
Making use of formula (2.8), the Cauchy integral formula
for derivatives of an analytic function and reciprocal substitutions one can obtain the integral representation