type/puiseuxseries¶

formal Puiseux series type¶

Calling sequence:¶

$\operatorname{type}(f, {\mathit puiseuxseries})$ $\operatorname{type}(f, {\mathit puiseuxseries}(K))$ $\operatorname{type}(f, {\mathit puiseuxseries}(K, v))$

Parameters:¶

  • $f$, a Maple expression
  • $K$, type name for coefficient domain
  • $v$, a variable, given as a name, a list or set of one name

Description:¶

  • $\operatorname{type}(f, {\mathit puiseuxseries}(K, v))$ answers $\mathit true$ iff $f$ is a Puiseux series in the variable $v$ with coefficients in $K$ which is NOT a Laurent series.
  • A Puiseux series is an object of the form $\operatorname{PuiseuxSeries}(q, s)$, where $q$ is a positive integer $\geq 2$ and $s$ is a Laurent series.

Example:¶

In [1]:
libname := libname, FileTools:-JoinPath(["maple","lib","dcfun.mla"],base=homedir):
In [2]:
 with(dcfun):
Out[2]:

A series whose exponents are half-integers is not a Puiseux series, in the sense of dcfun.

In [3]:
 f := series(sin(x)^(1/2)/x^5, x);
Out[3]:

$$\frac{1}{x^{\frac{9}{2}}}-\frac{1}{12 x^{\frac{5}{2}}}+\frac{1}{1440 \sqrt{x}}+O\! \left(x^{\frac{3}{2}}\right)$$

In [4]:
 type(f, puiseuxseries);
Out[4]:

$$\mathit{false}$$

Let us remove the ramification by changing x into x^2.

In [5]:
 f := simplify(eval(f, x = x^2), power, symbolic);
Out[5]:

$$\frac{1}{x^{9}}-\frac{1}{12 x^{5}}+\frac{1}{1440 x}+O\! \left(x^{3}\right)$$

In [6]:
 f := series(f, x);
Out[6]:

$$x^{-9}-\frac{1}{12} x^{-5}+\frac{1}{1440} x^{-1}+\mathrm{O}\! \left(x^{3}\right)$$

We can define a Puiseux series in the sense of dcfun using the PuiseuxSeries procedure.

In [7]:
 f := PuiseuxSeries(2, f);
Out[7]:

$$\frac{1}{x^{\frac{9}{2}}}-\frac{1}{12 x^{\frac{5}{2}}}+\frac{1}{1440 \sqrt{x}}+O\! \left(x^{\frac{3}{2}}\right)$$

Now we have a Puiseux series in the sense of dcfun.

In [8]:
 type(f, puiseuxseries(anything,x));
Out[8]:

$$\mathit{true}$$

In [9]:
 type(f, puiseuxseries(rational, x));
Out[9]:

$$\mathit{true}$$

In [10]:
 type(f, puiseuxseries(polynom(anything, y), x));
Out[10]:

$$\mathit{true}$$