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Let z_0 be a point in a simply connected region R!=C, where C is the complex plane. Then there is a unique analytic function w=f(z) mapping R one-to-one onto the disk |w|<1 ...

The method for solving the Goursat problem and Cauchy problem for linear hyperbolic partial differential equations using a Riemann function.

Let f:D(z_0,r)\{z_0}->C be analytic and bounded on a punctured open disk D(z_0,r), then lim_(z->z_0)f(z) exists, and the function defined by f^~:D(z_0,r)->C f^~(z)={f(z) for ...

The word "place" has a special meaning in complex variables, where it roughly corresponds to a point in the complex plane (except that it reflects the Riemann sheet structure ...

A manifold possessing a metric tensor. For a complete Riemannian manifold, the metric d(x,y) is defined as the length of the shortest curve (geodesic) between x and y. Every ...

Find an analytic parameterization of the compact Riemann surfaces in a fixed homeomorphism class. The Ahlfors-Bers theorem proved that Riemann's moduli space gives the ...

Riemann's moduli space R_p is the space of analytic equivalence classes of Riemann surfaces of fixed genus p.

Every continuous linear functional U[f] for f in C[a,b] can be expressed as a Stieltjes integral U[f]=int_a^bf(x)dw(x), where w(x) is determined by U and is of bounded ...

Consider a countable subgroup H with elements h_i and an element x not in H, then h_ix for i=1, 2, ... constitute the right coset of the subgroup H with respect to x.

A right eigenvector is defined as a column vector X_R satisfying AX_R=lambda_RX_R. In many common applications, only right eigenvectors (and not left eigenvectors) need be ...