This thesis is concerned with the development, design, and analysis of simple and efficient numerical quadrature methods for integrals on finite intervals with endpoint singularities, for integrals on the real line of steepest descent type, for integrals on finite intervals with branch point singularities near the interval of integration, and for integrals on the real line of Laplace type with branch point singularities near the path of integration.

In Chapter 1 we develop and analyse a numerical quadrature method, known as the

variable transformation method, for integrals on finite intervals