Maple/Using Maple in Analysis

There are many various kinds of analysis, and in very few of them Maple may help. One of the "maplable" analyses is the Asymptotical analysis. It is typical, that at calculus of cumbersome expressions, small values of some parameters are especially interesting. One can evaluate derivatives and then approximate the function with truncated Taylor series. As these operations often appear together, there is special service called series.

Examples of series
series(sin(x), x, 4)

returns

x-1/6*x^3+O(x^4) because the function sin has known derivatives. If the function is not known, the derivatives remain in general form:

s(0)+(D(s))(0)*x+1/2*((`@@`(D, 2))(s))(0)*x^2+1/6*((`@@`(D, 3))(s))(0)*x^3+O(x^4)

One can guess that the sequence

`@@`(D, 2))(s))(0)

means second derivative of function whose name is s.

In general, arguments of series can be: expression, variable which is supposed to be small, and the minimal power of this argument dropped out.

If no obvious dependency of the expression on the variable is found, the result is just expression:

series(a, b)

returns just

a

Not all functions can be expressed as Tailor expansion. Nevertheless Maple series tries to do its best, for example,

series((1+1/x^2)/(1+2/x+x*log(x)), x, 2)

returns

1/2/x-1/4+(5/8-1/4*ln(x))*x+O(x^2)

which may have sense in asymptotic analysis.

It is not difficult to write a function when the series cannot help, for example,

series(exp(1/x^2), x, 3)

returns

Error, (in series/exp) unable to compute series

although it is easy to see, that, at least for the real values of the argument, all the derivatives in zero are just zero. By default, Maple considers all variables as complex; in the complex plane, behavior of the function in the last example is pretty singular and no simple expansion is possible.