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A simple pole of an analytic function f is a pole of order one. That is, (z-z_0)f(z) is an analytic function at the pole z=z_0. Alternatively, its principal part is c/(z-z_0) ...

Let phi(z)=cz+c_0+c_1z^(-1)+c_2z^(-2)+... be an analytic function, regular and univalent for |z|>1, that maps |z|>1 conformally onto the region T preserving the point at ...

Let U subset= C be a domain, and let f be an analytic function on U. Then if there is a point z_0 in U such that |f(z_0)|>=|f(z)| for all z in U, then f is constant. The ...

Consider a power series in a complex variable z g(z)=sum_(n=0)^inftya_nz^n (1) that is convergent within the open disk D:|z|<R. Convergence is limited to within D by the ...

In its simplest form, the principle of permanence states that, given any analytic function f(z) defined on an open (and connected) set U of the complex numbers C, and a ...

Let K subset= C be compact, let f be analytic on a neighborhood of K, and let P subset= C^*\K contain at least one point from each connected component of C^*\K. Then for any ...

Linear Algebra

Topology

Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions ...

The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are ...