Series - Maple Help

MathematicalFunctions

 Series
 compute a series expansion for a mathematical function

 Calling Sequence Series(math_function, eqn, n)

Parameters

 math_function - mathematical function call eqn - equation or name; specifies variable for the expansion (for example, z) and, if an equation (for example, z = z0), the expansion point n - (optional) non-negative integer; specifies the truncation order

Description

 • The Series function computes a truncated series expansion of the function call math_function, with respect to the variable z indicated in eqn, about the point z0 (default is z = 0), up to order n (default is n = 6). So, the call Series(math_func, z) is equivalent to Series(math_func, z = 0, 6).
 • The series returned is computed (outside singularities/branch cuts) using a unified approach, based on the knowledge of the series expansion around the origin (herein called the Sum form).
 • The Series function is not a replacement for the Maple series command, nor does it have the generality of series. As a function of the MathematicalFunctions package, the goal of Series is to return valuable mathematical information not returned by series (see the examples). As such, Series can be seen as a complement to the series command.
 • Unlike the Maple series command, the first argument in a call to Series must be a known mathematical function call or a Sum. When the expansion is around z = 0, Series computes the requested series expansion directly from the Sum form of the mathematical function. In this case, the output may consist of a sequence of two objects: first the series, then any restrictions on the function parameters known to the FunctionAdvisor such that the Sum representation and so the series expansion is valid. If the series cannot be computed directly from this sum representation, Series returns FAIL.
 • If the expansion point is specified as z = z0 with $\mathrm{z0}\ne 0$, Series first attempts to compute a Taylor series expansion. If that fails, it shifts the expansion variable $z↦z+\mathrm{z0}$, then attempts to compute the series around the origin from the Sum form.
 • In cases that the series is computed from the expansion around the origin, when a call of the form Series( f(g(z)), z ) is processed (where $f$ is a mathematical function and $g$ an arbitrary mathematical expression), the series computation proceeds directly from the Sum form whenever $g\left(0\right)=0$, and otherwise through a call to series with the Sum form.
 • For some mathematical functions, the series expansion can be computed only by first factoring out abstract powers of the expansion variable, $z$. In this case, the Series command attempts the factorization, and, if successful, returns a structure of the form ${\mathrm{p1}\left(z\right)}^{\mathrm{k1}}\mathrm{series_1}+{\mathrm{p2}\left(z\right)}^{\mathrm{k2}}\mathrm{series_2}+...$, where $\mathrm{series_1}$, $\mathrm{series_2}$, ... are of type/series and ${\mathrm{p1}\left(z\right)}^{\mathrm{k1}},{\mathrm{p2}\left(z\right)}^{\mathrm{k2}}$, ... are abstract powers of polynomials $\mathrm{p1}$, $\mathrm{p2}$, ... in $z$.

Examples

 > $\mathrm{with}\left(\mathrm{MathematicalFunctions},\mathrm{Series}\right)$
 $\left[{\mathrm{Series}}\right]$ (1)

The series expansion around the origin may have restrictions on the function parameters.

 > $\mathrm{Series}\left(\mathrm{sin}\left(z\right),z\right)$
 ${z}{-}\frac{{1}}{{6}}{}{{z}}^{{3}}{+}\frac{{1}}{{120}}{}{{z}}^{{5}}{+}{O}{}\left({{z}}^{{7}}\right)$ (2)
 > $\mathrm{Series}\left(\mathrm{arcsin}\left(z\right),z\right)$
 ${z}{+}\frac{{1}}{{6}}{}{{z}}^{{3}}{+}\frac{{3}}{{40}}{}{{z}}^{{5}}{+}{O}{}\left({{z}}^{{7}}\right){,}{\mathrm{And}}{}\left({{\mathrm{abs}}{}\left({z}\right)}^{{2}}{<}{1}\right)$ (3)

The information on the function parameters returned for the expansion of $\mathrm{arcsin}\left(z\right)$ is not returned by the more general series command.

 > $\mathrm{series}\left(\mathrm{arcsin}\left(z\right),z\right)$
 ${z}{+}\frac{{1}}{{6}}{}{{z}}^{{3}}{+}\frac{{3}}{{40}}{}{{z}}^{{5}}{+}{O}{}\left({{z}}^{{7}}\right)$ (4)

In general, in the case of entire functions, such as sin, the series does not have restrictions on the function parameters. The functions admitting a hypergeometric representation linear in 0F1 or 1F1 are typically entire functions, such as erf or BesselJ. For functions of these classes depending on more than one parameter, however, the expansion around $z=0$ may depend on conditions on the other function parameters (not the expansion variable).

Computing a series by first factoring out abstract powers of $z$:

 > $\mathrm{Series}\left(\mathrm{BesselJ}\left(n,z\right),z\right)$
 ${{z}}^{{n}}{}\left(\frac{{1}}{{{2}}^{{n}}{}{\mathrm{\Gamma }}{}\left({1}{+}{n}\right)}{-}\frac{{1}}{{4}}{}\frac{{1}}{{{2}}^{{n}}{}{\mathrm{\Gamma }}{}\left({n}{+}{2}\right)}{}{{z}}^{{2}}{+}\frac{{1}}{{32}}{}\frac{{1}}{{{2}}^{{n}}{}{\mathrm{\Gamma }}{}\left({3}{+}{n}\right)}{}{{z}}^{{4}}{+}{O}{}\left({{z}}^{{6}}\right)\right){,}{\mathrm{And}}{}\left({n}{::}\left({\mathrm{Not}}{}\left({\mathrm{negint}}\right)\right)\right)$ (5)
 > $\mathrm{Series}\left(\mathrm{BesselK}\left(n,z\right),z,3\right)$
 ${{z}}^{{n}}{}\left({-}\frac{{\mathrm{\pi }}}{{2}{}{\mathrm{\Gamma }}{}\left({1}{+}{n}\right){}{\mathrm{sin}}{}\left({n}{}{\mathrm{\pi }}\right){}{{2}}^{{n}}}{-}\frac{{1}}{{8}}{}\frac{{\mathrm{\pi }}}{{\mathrm{\Gamma }}{}\left({n}{+}{2}\right){}{\mathrm{sin}}{}\left({n}{}{\mathrm{\pi }}\right){}{{2}}^{{n}}}{}{{z}}^{{2}}{+}{O}{}\left({{z}}^{{4}}\right)\right){+}{{z}}^{{-}{n}}{}\left(\frac{{\mathrm{\pi }}{}{{2}}^{{n}}}{{2}{}{\mathrm{\Gamma }}{}\left({1}{-}{n}\right){}{\mathrm{sin}}{}\left({n}{}{\mathrm{\pi }}\right)}{+}\frac{{1}}{{8}}{}\frac{{\mathrm{\pi }}{}{{2}}^{{n}}}{{\mathrm{\Gamma }}{}\left({2}{-}{n}\right){}{\mathrm{sin}}{}\left({n}{}{\mathrm{\pi }}\right)}{}{{z}}^{{2}}{+}{O}{}\left({{z}}^{{4}}\right)\right){,}{\mathrm{And}}{}\left({n}{::}\left({\mathrm{Not}}{}\left({\mathrm{integer}}\right)\right)\right)$ (6)
 > $\mathrm{Series}\left(\mathrm{KummerU}\left(a,b,z\right),z,3\right)$
 $\left(\frac{{\mathrm{\pi }}}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{b}\right){}{\mathrm{\Gamma }}{}\left({a}{-}{b}{+}{1}\right){}{\mathrm{\Gamma }}{}\left({b}\right)}{+}\frac{{\mathrm{\pi }}{}{a}}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{b}\right){}{\mathrm{\Gamma }}{}\left({a}{-}{b}{+}{1}\right){}{\mathrm{\Gamma }}{}\left({b}{+}{1}\right)}{}{z}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{\pi }}{}{a}{}\left({a}{+}{1}\right)}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{b}\right){}{\mathrm{\Gamma }}{}\left({a}{-}{b}{+}{1}\right){}{\mathrm{\Gamma }}{}\left({b}{+}{2}\right)}{}{{z}}^{{2}}{+}{O}{}\left({{z}}^{{3}}\right)\right){+}{{z}}^{{-}{b}}{}\left({-}\frac{{\mathrm{\pi }}}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{b}\right){}{\mathrm{\Gamma }}{}\left({2}{-}{b}\right){}{\mathrm{\Gamma }}{}\left({a}\right)}{}{z}{-}\frac{{\mathrm{\pi }}{}\left({a}{-}{b}{+}{1}\right)}{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{b}\right){}{\mathrm{\Gamma }}{}\left({3}{-}{b}\right){}{\mathrm{\Gamma }}{}\left({a}\right)}{}{{z}}^{{2}}{+}{O}{}\left({{z}}^{{3}}\right)\right){,}{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left({b}{-}{1}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\left({1}{-}{b}\right){::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)$ (7)

Once the conditions on the parameters are known, in some cases, you can obtain expansions for the complementary conditions (see type/nonposint), for example:

 > $\mathrm{Series}\left(\mathrm{KummerU}\left(a,b,z\right),z,3\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a::'\mathrm{nonposint}'$
 ${\mathrm{pochhammer}}{}\left({a}{-}{b}{+}{1}{,}{-}{a}\right){+}\frac{{\mathrm{pochhammer}}{}\left({a}{-}{b}{+}{1}{,}{-}{a}\right){}{a}}{{b}}{}{z}{+}\frac{{1}}{{2}}{}\frac{{\mathrm{pochhammer}}{}\left({a}{-}{b}{+}{1}{,}{-}{a}\right){}{\mathrm{pochhammer}}{}\left({a}{,}{2}\right)}{{\mathrm{pochhammer}}{}\left({b}{,}{2}\right)}{}{{z}}^{{2}}{+}{O}{}\left({{z}}^{{3}}\right)$ (8)

Since the approach used to expand a function is to first express it as a Sum, Series can also directly expand the Sum. For example, for the Sum form of the elliptic integral of the first kind (trigonometric form), InverseJacobiAM:

 > $\mathrm{convert}\left(\mathrm{InverseJacobiAM}\left(z,k\right),\mathrm{Sum}\right)$
 ${\sum }_{{\mathrm{_m2}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_m1}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({{k}}^{{2}}\right)}^{{\mathrm{_m1}}}{}{{\mathrm{sin}}{}\left({z}\right)}^{{2}{}{\mathrm{_m2}}{+}{2}{}{\mathrm{_m1}}{+}{1}}{}{\mathrm{pochhammer}}{}\left(\frac{{1}}{{2}}{,}{\mathrm{_m2}}\right){}{\mathrm{pochhammer}}{}\left(\frac{{1}}{{2}}{,}{\mathrm{_m1}}\right)}{\left({2}{}{\mathrm{_m2}}{+}{2}{}{\mathrm{_m1}}{+}{1}\right){}{\mathrm{_m2}}{!}{}{\mathrm{_m1}}{!}}$ (9)
 > $\mathrm{Series}\left(,k\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}z::\mathrm{real}$
 ${\mathrm{arcsin}}{}\left(\left|{\mathrm{sin}}{}\left({z}\right)\right|\right){}{\mathrm{signum}}{}\left({\mathrm{sin}}{}\left({z}\right)\right){+}\left({-}\frac{{\mathrm{sin}}{}\left({z}\right){}\left|{\mathrm{cos}}{}\left({z}\right)\right|}{{4}}{+}\frac{{\mathrm{arcsin}}{}\left({\mathrm{sin}}{}\left({z}\right)\right)}{{4}}\right){}{{k}}^{{2}}{+}\left(\frac{{3}{}{\left|{\mathrm{cos}}{}\left({z}\right)\right|}^{{3}}{}{\mathrm{sin}}{}\left({z}\right)}{{32}}{-}\frac{{15}{}{\mathrm{sin}}{}\left({z}\right){}\left|{\mathrm{cos}}{}\left({z}\right)\right|}{{64}}{+}\frac{{9}{}{\mathrm{arcsin}}{}\left({\mathrm{sin}}{}\left({z}\right)\right)}{{64}}\right){}{{k}}^{{4}}{+}{O}{}\left({{k}}^{{6}}\right)$ (10)
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