Special functions

You know that familiar functions such as sin(x), cos(x) and exp(x) are defined as infinite series

There are other, lesser known, functions that can be defined as definite integrals. While this might seem like a strange thing to do, these so-called special functions appear often enough in engineering analysis that understanding their use in Maple is important. One such function is the gamma function defined as

Note that the independent variable x is a parameter in the integral. The Gamma function has remarkable properties such as

for all positive real numbers x. Thus it acts like a factorial, only it works for all positive real numbers and not just integers.

Another common special function found in statistical analysis is the so-called Error function, erf(x), defined as

In this case the independent variable x, is the upper limit of the integral.

These special functions can be differentiated and integrated and manipulated just as you would more familiar functions such as sin(x), cos(x) or exp(x). So if Maple produces one of these functions as part of a solution, that is just fine.