Functions on Power Series

Once you have created a power series, you can apply transcendental functions (for example, exp, log, sin, tan, cosh, etc.) to it. To demonstrate this, we first create the power series expansion of the rational function x^2/(1-6*x+x^2) about x=0.

If you want to compute the series expansion of sin(x^2/1-6*x+x^2) you simply compute the sine of rat.

Warning: the type of the coefficients of a power series may affect the kind of computations that you can do with that series. This can only happen when you have made a declaration to specify a series domain with a certain type of coefficient.

If you evaluate then you have declared that y is a one variable Taylor series (UTS is the abbreviation for UnivariateTaylorSeries) in the variable y with FRAC INT (that is, fractions of integers) coefficients, centered about 0.

You can now compute certain power series in y, provided that these series have rational coefficients.

You can get examples of such series by applying transcendental functions to series in y that have no constant terms.

Similarly, you can compute the logarithm of a power series with rational coefficients if the constant coefficient is 1.

If you wanted to apply, say, the operation exp to a power series with a nonzero constant coefficient a0, then the constant coefficient of the result would be exp(a0), which is not a rationa number. Therefore, evaluating exp(2+tan(y)) would generate an error message. If you want to compute the Taylor expansion of exp(2+tan(y)), you must ensure that the coefficient domain has an operation exp defined for it. An example of such a domain is Expression Integer, the type of formal functional expressions over the integers. When working with coefficients of this type

this presents no problems.

Another way to create Taylor series whose coefficients are expressions over the integers is to use taylor which works similarly to series. This is equivalent to the previous computation, except that now we are using the variable w instead of z.