Notation:

• Where possible, we have tried to use LaTex-generated images for mathematical notation on the web documents.  This is sometimes tedious, so we often write out names, such as using alpha for the symbol .
• There are a few constants which appear repeatedly throughout sinc theory which we'd like to cover:
• Sinc theory is applied on the interval to functions which exponentially decay on both ends, which is often not the interval we are interested in.  We use a conformal map, phi, to map from the domain of interest to .  We refer to the problem domain as gamma.  The inverse of phi is labelled psi.  The left side of gamma is psi(-infty,0) and the right side of gamma is psi(0,infty).
  
• The constant alpha measures the convergence rate on the left side of the domain
• The constant beta measures the convergence rate on the right side of the domain.
• The constant d is the width of the domain D_s.  This is a rather technical choice (usually, the default d=pi/2 is fine), but integral to the convergence theory of Sinc methods.
• The constant M is the number of Sinc nodes for the left side of gamma.  The constant N is the number of Sinc nodes for the right side of gamma.  Usually, the two are related with the following formula:
  
  
  
  This is required for convergence theorems.
• The constant h is the the mesh-size; the distance between sinc nodes.  It is usually decided by the following formula:
  
  
  
  This is required for convergence theorems.
• The above notation usually results in convergence theorems which read like this:
  
  
  
• Second-order differential equations usually require three special matrices, I0, I1, and I2.  Their elements are defined as:
  
  
  
  

Conformal Mappings

The conformal mappings we use in SincLib are outlined in this section.  In order for proper convergence of sinc methods, functions should behave in a certain way as they approach the endpoints of the domain.  They need to either converge polynomially to the zero or decay exponentially.  Where noted, SincLib uses augmented methods so all the problems don't need to be held to zero at the endpoints.

• none:  The conformal map is the identity.  Maps to , and functions are expected to converge exponentially.
  
• eyelet:  The conformal map phi is .  Maps to the interval (0,1).  Functions are expected to converge polynomially.
  
• wedge: The conformal mapping is .  Maps to the interval (0,infty).  Functions are expected to converge polynomially at 0 and exponentially at infty.
  
• hyperbola: The conformal map is .  Maps to the interval .  Functions are expected to converge polynomially at the endpoints.
  
• bullet: A conformal map of .  Maps to the interval (0,infty).  Functions are expected to converge polynomially at both endpoints.
  

Table of Functions

This section outlines the various functions and domains associated with the conformal mappings we use. a is the left endpoint, b is the right endpoint.  We define the functions f₁, f₂, and f₃, for convenience.  We additionally will need the function psi, which is the inverse of phi.

So, after the constructor is called, you should be able to call self.phi(x), self.phip(x), etc, if your object's name is self (and the correct function out of the table will be selected).