13.3.4 Special Functions

13.3.4  Special Functions

The Math::Cephes module provides many functions that are based on advanced calculus. Thus, the following discussion assumes that the reader has a good background in calculus. The material discussed is frequently needed by many engineers and scientists. Undergraduate students in engineering disciplines such as mechanical, civil and electrical, and students in physics are familiar with the use of such functions. In earlier days, the values of these functions were computed painstakingly and large tables of values made available in the form of printed books for scientists and engineers to use. Computing the values with desired accuracy requires careful work. These days, the values are computed sufficiently accurately using computers.

Math::Cephes contains definitions of many of the so-called special functions found in standard mathematical or engineering handbooks. The term special functions is a common name used for functions that usually arise from solving differential equations of order two or higher. Although they arise mostly in solving differential equations, it is not necessary to talk about differential equations to understand some of the simple special functions. In technical literature, there are often several conflicting definitions of many special functions. Therefore, when one uses a special function from Math::Cephes, one should look at the definition given in the documentation to confirm that it is exactly what one wants. The special functions defined in Math::Cephes
take only real arguments.

We start with the discussion of a few useful special functions that are obtained by integrating certain functions within pre-specified limits. The functions we look at are the gamma function, the beta function, the exponential function, the sine and cosine integrals, the hyperbolic sine and cosine integrals, the Fresnel integral and the error function. These are all commonly used in scientific and engineering computation. In each case, we simply illustrate how the values of the functions can be computed. We follow with a more detailed discussion of the Bessel’s functions. Interested readers are advised to consult one of the following books or a similar book for detailed discussions on these functions and the corresponding integral: Advanced Engineering Mathematics [Kre93], Advanced Engineering Mathematics [BMW77], Advanced Engineering Mathematics with MATLAB [HDR00].