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 wellknown 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) = (n1)!,
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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 Fall '09
 anetistis
 Derivative, Logarithm

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