A Collocation Method Based on the Bernoulli Operational Matrix for Solving High-Order Linear Complex Differential Equations in a Rectangular Domain

and Applied Analysis 3 Remark 5 (complex partial differential operators). Thepartial differential operators ∂/∂x and ∂/∂y are applied to a complexvalued function f = u + iV in the natural way: ∂f ∂x = ∂u ∂x + i ∂V ∂x , ∂f ∂y = ∂u ∂y + i ∂V ∂y . (7) We define the complex partial differential operators ∂/∂z and ∂/∂z by ∂ ∂z = 1 2 ( ∂ ∂x − i ∂ ∂y ) , ∂ ∂z = 1 2 ( ∂ ∂x + i ∂ ∂y ) . (8) Thus, ∂/∂x = ∂/∂z + ∂/∂z and ∂/∂y = i(∂/∂z − ∂/∂z). Intuitively one can think of a holomorphic function as a complex-valued function in an open subset ofC that depends only on z, that is, independent of z. We can make this notion precisely as follows. Suppose the functionf = u+iV is defined and differentiable in an open set. One then has ∂f ∂z = 1 2 ( ∂u ∂x + ∂V ∂y ) + i 2 ( ∂V ∂x − ∂u ∂y ) , ∂f ∂z = 1 2 ( ∂u ∂x − ∂V ∂y ) + i 2 ( ∂V ∂x + ∂u ∂y ) . (9) The Cauchy-Riemann equations thus can be written ∂f/∂z = 0. As this is the condition forf to be holomorphic, it provides a precise meaning for the statement: a holomorphic function is one that is independent of z. If f is holomorphic, then (not surprisingly) f = ∂f/∂z, as the following calculation shows: f = ∂f ∂x = ∂f ∂z + ∂f ∂z = ∂f ∂z . (10) 2.2. The Bernoulli Polynomials and Their Operational Matrix. TheBernoulli polynomials play an important role in different areas of mathematics, including number theory and the theory of finite differences. The classical Bernoulli polynomials Bn(x) are usually defined by means of the exponential generating functions (see [21])