In earlier chapters, complex-valued functions appeared in connection with Fourier series expansions. In this context, while the function assumes complex values, the argument of the function is real-valued. There is a highly developed theory of (complex-valued) functions of a complex-valued argument. This theory contains some remarkably powerful results which are applicable to a variety of problems.

Functions of a Complex Variable

Multifold ground coverage by seismic techniques such as the common reflection point method provides a multiplicity of wave travel path information which allows direct determination of root‐mean‐square velocities associated with such paths. Hyperbolic searches for semblance among appropriately gathered arrays of traces form the basis upon which velocities are estimated. Measured semblances are presented as a velocity spectral display. Interpretation of this information can give velocities with meaningful accuracy for primary as well as multiple events. In addition, the velocity data can help correctly label events. This paper outlines the fundamental principles for calculating velocity spectra displays. Examples are included which demonstrate the depth and detail of geological information which may be obtained from the interpretation of such displays.

Velocity spectra-digital computer derivation and applications of velocity functions

Krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper. We derive error bounds via a functional calculus of Arnoldi and Lanczos methods that reduces the study of Krylov subspace approximations of functions of matrices to that of linear systems of equations. As a side result, we obtain error bounds for Galerkin-type Krylov methods for linear equations, namely, the biconjugate gradient method and the full orthogonalization method. For Krylov approximations to matrix exponentials, we show superlinear error decay from relatively small iteration numbers onwards, depending on the geometry of the numerical range, the spectrum, or the pseudospectrum. The convergence to exp$(\tau A)v$ is faster than that of corresponding Krylov methods for the solution of linear equations $(I-\tau A)x=v$, which usually arise in the numerical solution of stiff ordinary differential equations (ODEs). We therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use Krylov approximations to the exponential function of the Jacobian instead of solving linear or nonlinear systems of equations in every time step.

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On Krylov Subspace Approximations to the Matrix Exponential Operator

IN contagious processes (e.g. measles, hijacking, etc.) the occurrence of events increases the probability of further events occurring in the near future. Also several series of events may interact with each other, for example one might consider notifications of some disease in a number of adjacent regions which would interact through infectives or carriers moving between the regions. In this paper we postulate a model for such processes and derive a general expression for the point spectral matrices. These theoretical spectra are useful for comparison with spectra estimated from data and thus provide a means of evaluating the fit of such a model in the manner of Bartlett (1963). The model studied was put forward in an earlier paper (Hawkes, 1971) but the solution was obtained only in special cases. In this paper an elegant solution is obtained for the. general case. Consider a stationary k-variate point process N(t), where Ni(t) represents the cumulative number of events in the ith process up to time t, with intensity vector X = of{dN(t)}/dt and covariance density matrix

Point Spectra of Some Mutually Exciting Point Processes

In this paper we investigate the Painleve property for partial differential equations. By application to several well‐known partial differential equations (Burgers, KdV, MKdV, Bousinesq, higher‐order KdV and KP equations) it is shown that consideration of the ‘‘singular manifold’’ leads to a formulation of these equations in terms of the ‘‘Schwarzian derivative.’’ This formulation is invariant under the Moebius group (acting on dependent variables) and is shown to obtain the appropriate Lax pair (linearization) for the underlying nonlinear pde.

The painleve property for partial differential equations. ii. backlund transformation, lax pairs, and the schwarzian derivative

Volume I, Part 1: Basic Concepts: I.1 Introduction I.2 Complex numbers I.3 Sets and functions. Limits and continuity I.4 Connectedness. Curves and domains I.5. Infinity and stereographic projection I.6 Homeomorphisms Part 2: Differentiation. Elementary Functions: I.7 Differentiation and the Cauchy-Riemann equations I.8 Geometric interpretation of the derivative. Conformal mapping I.9 Elementary entire functions I.10 Elementary meromorphic functions I.11 Elementary multiple-valued functions Part 3: Integration. Power Series: I.12 Rectifiable curves. Complex integrals I.13 Cauchy's integral theorem I.14 Cauchy's integral and related topics I.15 Uniform convergence. Infinite products I.16 Power series: rudiments I.17 Power series: ramifications I.18 Methods for expanding functions in Taylor series Volume II, Part 1: Laurent Series. Calculus of Residues: II.1 Laurent's series. Isolated singular points II.2 The calculus of residues and its applications II.3 Inverse and implicit functions II.4 Univalent functions Part 2: Harmonic and Subharmonic Functions: II.5 Basic properties of harmonic functions II.6 Applications to fluid dynamics II.7 Subharmonic functions II.8 The Poisson-Jensen formula and related topics Part 3: Entire and Meromorphic Functions: II.9 Basic properties of entire functions II.10 Infinite product and partial fraction expansions Volume III, Part 1: Conformal Mapping. Approximation Theory: III.1 Conformal mapping: rudiments III.2 Conformal mapping: ramifications III.3 Approximation by rational functions and polynomials Part 2: Periodic and Elliptic Functions: III.4 Periodic meromorphic functions III.5 Elliptic functions: Weierstrass' theory III.6 Elliptic functions: Jacobi's theory Part 3: Riemann Surfaces. Analytic Continuation: III.7 Riemann surfaces III.8 Analytic continuation III.9 The symmetry principle and its applications Bibliography Index.

Richard A. Silverman

Papers

Theory of Functions of a Complex Variable