ABSTRACT
Since the domain of a complex function is a plane (Section 11.1) its derivative at any given point may be obtained by approaching the point from any direction or along any curve (Section 11.2). The derivative only exists if it is independent of the direction of approach, and for this reason a differentiable complex function is called holomorphic; the greek word “holos” stands for symmetry, for example, the most symmetric class in each chrystalographic group is called holoedric. The derivative of a complex function of a complex variable involves all four real partial derivatives and it is independent of direction only if these satisfy the Cauchy-Riemann conditions (Section 11.3). The Cauchy-Riemann conditions can be interpreted as stating that the real and imaginary parts of a complex holomorphic function: (i) specify two families of orthogonal curves (Section 11.8); (ii) satisfy Laplace’s equation (Section 11.6). The former (i) can be used to introduce orthogonal curvilinear coordinates in the plane (Section 11.9); these can be used together with (ii) to specify the invariant differential operators (gradient, divergence, curl, and Laplacian) in plane orthogonal curvilinear coordinates, including as a particular case polar coordinates (Section 11.7). The Cauchy-Riemann conditions (Section 11.4) and hence the derivative of a holomorphic function (Section 11.5) can also be expressed not only in Cartesian (Section 11.3), but also in polar (Section 11.4) and other orthogonal curvilinear coordinates (Section 11.9). A nonholomorphic function may have specific properties in particular directions (Example 20.1) that are excluded by the “isotropy” of holomorphic functions; some of the consequences of isotropy are explored next.
