TABLE OF CONTENTS A number of functions, somewhat more complex in nature than the standard elementary functions, arise from the mathematical analysis of many systems in the sciences. These include the gamma, beta, and zeta functions; Dirichlet integrals; the Dirac delta function; and other "named functions" represented by definite integrals such as the exponential, sine, cosine, Fresnel Sine and Cosine integrals. We define these functions, derive several of their important properties, and then show how they can be applied.
We also include a brief section on elliptic integrals and functions. The elliptic functions are a generalization of the trigonometric functions and satisfy many similar relations. The basic properties are derived for these functions along with the nonlinear, second-order differential equations to which they are solution. We also give their Fourier series representations.
The chapter ends with a topic that will be looked at somewhat differently in Chapter 9. We introduce integrals depending on parameters and show that many can be explicitly evaluated using various techniques involving differentiation with respect to these parameters. Application of the method to several examples demonstrates that it is a powerful method, when applicable, for evaluating integrals.
Define f( a) to be
where it is indicated that the integral exists for complex a, provided Re( a) is positive. In fact, f( a) can be easily computed and is found to be
If the following expression
is differentiated n-times with respect to a, then we...
