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L2041-04 - Worksheet 4 Simple Functions and Calculus in...

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Worksheet 4: Simple Functions and Calculus in Maple Standard Simple Functions The Maple algebra engine accesses a library of preprogrammed simple and some special functions. Under the simple functions there are the circle functions, and the exponential and logarithmic functions. The special functions include the more well-known functions such as the gamma function. Example: > exp(1); exp(1.); The result is e^1, or the value of the constant e up to 10 Digits or 9 decimal places. One may find more accurate values for e by increasing the number of digits. This is done as follows, and is a helpful tool in numerical computations, and checking the accuracy of results. > Digits := 40; exp(1.); The logarithm has base e; in other words, log = ln. This is the way it is supposed to be, since the change of base formula for logarithms shows that there is one logarithm. All other logarithms are just scaled versions of the logarithm by a constant (an analogy is that there is just one linear function y = x ; all other linear functions is a constant multiple of this: eg. y = Cx). In particular, the change of base formula for logarithms read log_A (x) = log_A(e) . log(x) = C_A log(x) where C_A is a constant multiplier. Any log in base e will be translated to a log in base A by multiplying with C_A. The ln function or log base 10 functions can also be accessed.
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> log(2.); ln(2.); log10(2.); To compute logarithms to base 7, the syntax is > log[7](2.); ----------------------------------------------------------- Circle Funcions The circle functions sin, cos and tan are similarly defined, as well as arcsin, arctan and arccos. > sin(1.); cos(1.); tan(1.); > arcsin(0.8414709848); arccos(.5403023059); arctan(1.557407725); arcsec(1.557407725); Example of a special function: the Gamma function The Gamma function can similarly be computed: Notice that Gamma(n) = (n-1)!, but that Gamma is really a function of complex numbers, defined everywhere in the complex plane, except at negative integers... Type ?GAMMA to find the definitions of GAMMA(x) and GAMMA(x,y).
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