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<h1 class="chapter"> 30. Series </h1>
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<h2 class="section"> 30.1 Introduction to Series </h2>
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<h2 class="section"> 30.2 Functions and Variables for Series </h2>
<p><a name="Item_003a-cauchysum"></a>
</p><dl>
<dt><u>Option variable:</u> <b>cauchysum</b>
<a name="IDX1106"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When multiplying together sums with <code>inf</code> as their upper limit,
if <code>sumexpand</code> is <code>true</code> and <code>cauchysum</code> is <code>true</code>
then the Cauchy product will be used rather than the usual
product.
In the Cauchy product the index of the inner summation is a
function of the index of the outer one rather than varying
independently.
</p>
<p>Example:
</p>
<pre class="example">(%i1) sumexpand: false$
(%i2) cauchysum: false$
(%i3) s: sum (f(i), i, 0, inf) * sum (g(j), j, 0, inf);
inf inf
==== ====
\ \
(%o3) ( > f(i)) > g(j)
/ /
==== ====
i = 0 j = 0
(%i4) sumexpand: true$
(%i5) cauchysum: true$
(%i6) ''s;
inf i1
==== ====
\ \
(%o6) > > g(i1 - i2) f(i2)
/ /
==== ====
i1 = 0 i2 = 0
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Sums-and-products">Sums and products</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-deftaylor"></a>
</p><dl>
<dt><u>Function:</u> <b>deftaylor</b><i> (<var>f_1</var>(<var>x_1</var>), <var>expr_1</var>, ..., <var>f_n</var>(<var>x_n</var>), <var>expr_n</var>)</i>
<a name="IDX1107"></a>
</dt>
<dd><p>For each function <var>f_i</var> of one variable <var>x_i</var>,
<code>deftaylor</code> defines <var>expr_i</var> as the Taylor series about zero.
<var>expr_i</var> is typically a polynomial in <var>x_i</var> or a summation;
more general expressions are accepted by <code>deftaylor</code> without complaint.
</p>
<p><code>powerseries (<var>f_i</var>(<var>x_i</var>), <var>x_i</var>, 0)</code>
returns the series defined by <code>deftaylor</code>.
</p>
<p><code>deftaylor</code> returns a list of the functions
<var>f_1</var>, ..., <var>f_n</var>.
<code>deftaylor</code> evaluates its arguments.
</p>
<p>Example:
</p>
<pre class="example">(%i1) deftaylor (f(x), x^2 + sum(x^i/(2^i*i!^2), i, 4, inf));
(%o1) [f]
(%i2) powerseries (f(x), x, 0);
inf
==== i1
\ x 2
(%o2) > -------- + x
/ i1 2
==== 2 i1!
i1 = 4
(%i3) taylor (exp (sqrt (f(x))), x, 0, 4);
2 3 4
x 3073 x 12817 x
(%o3)/T/ 1 + x + -- + ------- + -------- + . . .
2 18432 307200
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-maxtayorder"></a>
</p><dl>
<dt><u>Option variable:</u> <b>maxtayorder</b>
<a name="IDX1108"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>maxtayorder</code> is <code>true</code>, then during algebraic
manipulation of (truncated) Taylor series, <code>taylor</code> tries to retain
as many terms as are known to be correct.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-niceindices"></a>
</p><dl>
<dt><u>Function:</u> <b>niceindices</b><i> (<var>expr</var>)</i>
<a name="IDX1109"></a>
</dt>
<dd><p>Renames the indices of sums and products in <var>expr</var>.
<code>niceindices</code> attempts to rename each index to the value of <code>niceindicespref[1]</code>,
unless that name appears in the summand or multiplicand,
in which case <code>niceindices</code> tries
the succeeding elements of <code>niceindicespref</code> in turn, until an unused variable is found.
If the entire list is exhausted,
additional indices are constructed by appending integers to the value of
<code>niceindicespref[1]</code>, e.g., <code>i0</code>, <code>i1</code>, <code>i2</code>, ....
</p>
<p><code>niceindices</code> returns an expression.
<code>niceindices</code> evaluates its argument.
</p>
<p>Example:
</p>
<pre class="example">(%i1) niceindicespref;
(%o1) [i, j, k, l, m, n]
(%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
inf inf
/===\ ====
! ! \
(%o2) ! ! > f(bar i j + foo)
! ! /
bar = 1 ====
foo = 1
(%i3) niceindices (%);
inf inf
/===\ ====
! ! \
(%o3) ! ! > f(i j l + k)
! ! /
l = 1 ====
k = 1
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Sums-and-products">Sums and products</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-niceindicespref"></a>
</p><dl>
<dt><u>Option variable:</u> <b>niceindicespref</b>
<a name="IDX1110"></a>
</dt>
<dd><p>Default value: <code>[i, j, k, l, m, n]</code>
</p>
<p><code>niceindicespref</code> is the list from which <code>niceindices</code>
takes the names of indices for sums and products.
</p>
<p>The elements of <code>niceindicespref</code> are typically names of variables,
although that is not enforced by <code>niceindices</code>.
</p>
<p>Example:
</p>
<pre class="example">(%i1) niceindicespref: [p, q, r, s, t, u]$
(%i2) product (sum (f (foo + i*j*bar), foo, 1, inf), bar, 1, inf);
inf inf
/===\ ====
! ! \
(%o2) ! ! > f(bar i j + foo)
! ! /
bar = 1 ====
foo = 1
(%i3) niceindices (%);
inf inf
/===\ ====
! ! \
(%o3) ! ! > f(i j q + p)
! ! /
q = 1 ====
p = 1
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Sums-and-products">Sums and products</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-nusum"></a>
</p><dl>
<dt><u>Function:</u> <b>nusum</b><i> (<var>expr</var>, <var>x</var>, <var>i_0</var>, <var>i_1</var>)</i>
<a name="IDX1111"></a>
</dt>
<dd><p>Carries out indefinite hypergeometric summation of <var>expr</var> with
respect to <var>x</var> using a decision procedure due to R.W. Gosper.
<var>expr</var> and the result must be expressible as products of integer powers,
factorials, binomials, and rational functions.
</p>
<p>The terms "definite"
and "indefinite summation" are used analogously to "definite" and
"indefinite integration".
To sum indefinitely means to give a symbolic result
for the sum over intervals of variable length, not just e.g. 0 to
inf. Thus, since there is no formula for the general partial sum of
the binomial series, <code>nusum</code> can't do it.
</p>
<p><code>nusum</code> and <code>unsum</code> know a little about sums and differences of finite products.
See also <code>unsum</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) nusum (n*n!, n, 0, n);
Dependent equations eliminated: (1)
(%o1) (n + 1)! - 1
(%i2) nusum (n^4*4^n/binomial(2*n,n), n, 0, n);
4 3 2 n
2 (n + 1) (63 n + 112 n + 18 n - 22 n + 3) 4 2
(%o2) ------------------------------------------------ - ------
693 binomial(2 n, n) 3 11 7
(%i3) unsum (%, n);
4 n
n 4
(%o3) ----------------
binomial(2 n, n)
(%i4) unsum (prod (i^2, i, 1, n), n);
n - 1
/===\
! ! 2
(%o4) ( ! ! i ) (n - 1) (n + 1)
! !
i = 1
(%i5) nusum (%, n, 1, n);
Dependent equations eliminated: (2 3)
n
/===\
! ! 2
(%o5) ! ! i - 1
! !
i = 1
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Sums-and-products">Sums and products</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-pade"></a>
</p><dl>
<dt><u>Function:</u> <b>pade</b><i> (<var>taylor_series</var>, <var>numer_deg_bound</var>, <var>denom_deg_bound</var>)</i>
<a name="IDX1112"></a>
</dt>
<dd><p>Returns a list of
all rational functions which have the given Taylor series expansion
where the sum of the degrees of the numerator and the denominator is
less than or equal to the truncation level of the power series, i.e.
are "best" approximants, and which additionally satisfy the specified
degree bounds.
</p>
<p><var>taylor_series</var> is a univariate Taylor series.
<var>numer_deg_bound</var> and <var>denom_deg_bound</var>
are positive integers specifying degree bounds on
the numerator and denominator.
</p>
<p><var>taylor_series</var> can also be a Laurent series, and the degree
bounds can be <code>inf</code> which causes all rational functions whose total
degree is less than or equal to the length of the power series to be
returned. Total degree is defined as <code><var>numer_deg_bound</var> + <var>denom_deg_bound</var></code>.
Length of a power series is defined as
<code>"truncation level" + 1 - min(0, "order of series")</code>.
</p>
<pre class="example">(%i1) taylor (1 + x + x^2 + x^3, x, 0, 3);
2 3
(%o1)/T/ 1 + x + x + x + . . .
(%i2) pade (%, 1, 1);
1
(%o2) [- -----]
x - 1
(%i3) t: taylor(-(83787*x^10 - 45552*x^9 - 187296*x^8
+ 387072*x^7 + 86016*x^6 - 1507328*x^5
+ 1966080*x^4 + 4194304*x^3 - 25165824*x^2
+ 67108864*x - 134217728)
/134217728, x, 0, 10);
2 3 4 5 6 7
x 3 x x 15 x 23 x 21 x 189 x
(%o3)/T/ 1 - - + ---- - -- - ----- + ----- - ----- - ------
2 16 32 1024 2048 32768 65536
8 9 10
5853 x 2847 x 83787 x
+ ------- + ------- - --------- + . . .
4194304 8388608 134217728
(%i4) pade (t, 4, 4);
(%o4) []
</pre>
<p>There is no rational function of degree 4 numerator/denominator, with this
power series expansion. You must in general have degree of the numerator and
degree of the denominator adding up to at least the degree of the power series,
in order to have enough unknown coefficients to solve.
</p>
<pre class="example">(%i5) pade (t, 5, 5);
5 4 3
(%o5) [- (520256329 x - 96719020632 x - 489651410240 x
2
- 1619100813312 x - 2176885157888 x - 2386516803584)
5 4 3
/(47041365435 x + 381702613848 x + 1360678489152 x
2
+ 2856700692480 x + 3370143559680 x + 2386516803584)]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-powerdisp"></a>
</p><dl>
<dt><u>Option variable:</u> <b>powerdisp</b>
<a name="IDX1113"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>powerdisp</code> is <code>true</code>,
a sum is displayed with its terms in order of increasing power.
Thus a polynomial is displayed as a truncated power series,
with the constant term first and the highest power last.
</p>
<p>By default, terms of a sum are displayed in order of decreasing power.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Display-flags-and-variables">Display flags and variables</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-powerseries"></a>
</p><dl>
<dt><u>Function:</u> <b>powerseries</b><i> (<var>expr</var>, <var>x</var>, <var>a</var>)</i>
<a name="IDX1114"></a>
</dt>
<dd><p>Returns the general form of the power series expansion for <var>expr</var> in the
variable <var>x</var> about the point <var>a</var> (which may be <code>inf</code> for infinity):
</p><pre class="example"> inf
====
\ n
> b (x - a)
/ n
====
n = 0
</pre>
<p>If <code>powerseries</code> is unable to expand <var>expr</var>,
<code>taylor</code> may give the first several terms of the series.
</p>
<p>When <code>verbose</code> is <code>true</code>,
<code>powerseries</code> prints progress messages.
</p>
<pre class="example">(%i1) verbose: true$
(%i2) powerseries (log(sin(x)/x), x, 0);
can't expand
log(sin(x))
so we'll try again after applying the rule:
d
/ -- (sin(x))
[ dx
log(sin(x)) = i ----------- dx
] sin(x)
/
in the first simplification we have returned:
/
[
i cot(x) dx - log(x)
]
/
inf
==== i1 2 i1 2 i1
\ (- 1) 2 bern(2 i1) x
> ------------------------------
/ i1 (2 i1)!
====
i1 = 1
(%o2) -------------------------------------
2
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-psexpand"></a>
</p><dl>
<dt><u>Option variable:</u> <b>psexpand</b>
<a name="IDX1115"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>psexpand</code> is <code>true</code>,
an extended rational function expression is displayed fully expanded.
The switch <code>ratexpand</code> has the same effect.
</p>
<p>When <code>psexpand</code> is <code>false</code>,
a multivariate expression is displayed just as in the rational function package.
</p>
<p>When <code>psexpand</code> is <code>multi</code>,
then terms with the same total degree in the variables are grouped together.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Display-flags-and-variables">Display flags and variables</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-revert"></a>
</p><dl>
<dt><u>Function:</u> <b>revert</b><i> (<var>expr</var>, <var>x</var>)</i>
<a name="IDX1116"></a>
</dt>
<dt><u>Function:</u> <b>revert2</b><i> (<var>expr</var>, <var>x</var>, <var>n</var>)</i>
<a name="IDX1117"></a>
</dt>
<dd><p>These functions return the reversion of <var>expr</var>, a Taylor series about zero in the variable <var>x</var>.
<code>revert</code> returns a polynomial of degree equal to the highest power in <var>expr</var>.
<code>revert2</code> returns a polynomial of degree <var>n</var>,
which may be greater than, equal to, or less than the degree of <var>expr</var>.
</p>
<p><code>load ("revert")</code> loads these functions.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) load ("revert")$
(%i2) t: taylor (exp(x) - 1, x, 0, 6);
2 3 4 5 6
x x x x x
(%o2)/T/ x + -- + -- + -- + --- + --- + . . .
2 6 24 120 720
(%i3) revert (t, x);
6 5 4 3 2
10 x - 12 x + 15 x - 20 x + 30 x - 60 x
(%o3)/R/ - --------------------------------------------
60
(%i4) ratexpand (%);
6 5 4 3 2
x x x x x
(%o4) - -- + -- - -- + -- - -- + x
6 5 4 3 2
(%i5) taylor (log(x+1), x, 0, 6);
2 3 4 5 6
x x x x x
(%o5)/T/ x - -- + -- - -- + -- - -- + . . .
2 3 4 5 6
(%i6) ratsimp (revert (t, x) - taylor (log(x+1), x, 0, 6));
(%o6) 0
(%i7) revert2 (t, x, 4);
4 3 2
x x x
(%o7) - -- + -- - -- + x
4 3 2
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-taylor"></a>
</p><dl>
<dt><u>Function:</u> <b>taylor</b><i> (<var>expr</var>, <var>x</var>, <var>a</var>, <var>n</var>)</i>
<a name="IDX1118"></a>
</dt>
<dt><u>Function:</u> <b>taylor</b><i> (<var>expr</var>, [<var>x_1</var>, <var>x_2</var>, ...], <var>a</var>, <var>n</var>)</i>
<a name="IDX1119"></a>
</dt>
<dt><u>Function:</u> <b>taylor</b><i> (<var>expr</var>, [<var>x</var>, <var>a</var>, <var>n</var>, 'asymp])</i>
<a name="IDX1120"></a>
</dt>
<dt><u>Function:</u> <b>taylor</b><i> (<var>expr</var>, [<var>x_1</var>, <var>x_2</var>, ...], [<var>a_1</var>, <var>a_2</var>, ...], [<var>n_1</var>, <var>n_2</var>, ...])</i>
<a name="IDX1121"></a>
</dt>
<dt><u>Function:</u> <b>taylor</b><i> (<var>expr</var>, [<var>x_1</var>, <var>a_1</var>, <var>n_1</var>], [<var>x_2</var>, <var>a_2</var>, <var>n_2</var>], ...)</i>
<a name="IDX1122"></a>
</dt>
<dd><p><code>taylor (<var>expr</var>, <var>x</var>, <var>a</var>, <var>n</var>)</code> expands the expression <var>expr</var>
in a truncated Taylor or Laurent series in the variable <var>x</var>
around the point <var>a</var>,
containing terms through <code>(<var>x</var> - <var>a</var>)^<var>n</var></code>.
</p>
<p>If <var>expr</var> is of the form <code><var>f</var>(<var>x</var>)/<var>g</var>(<var>x</var>)</code>
and <code><var>g</var>(<var>x</var>)</code> has no terms up to degree <var>n</var>
then <code>taylor</code> attempts to expand <code><var>g</var>(<var>x</var>)</code> up to degree <code>2 <var>n</var></code>.
If there are still no nonzero terms, <code>taylor</code> doubles the
degree of the expansion of <code><var>g</var>(<var>x</var>)</code>
so long as the degree of the expansion is less than or equal to <code><var>n</var> 2^taylordepth</code>.
</p>
<p><code>taylor (<var>expr</var>, [<var>x_1</var>, <var>x_2</var>, ...], <var>a</var>, <var>n</var>)</code>
returns a truncated power series
of degree <var>n</var> in all variables <var>x_1</var>, <var>x_2</var>, ...
about the point <code>(<var>a</var>, <var>a</var>, ...)</code>.
</p>
<p><code>taylor (<var>expr</var>, [<var>x_1</var>, <var>a_1</var>, <var>n_1</var>], [<var>x_2</var>, <var>a_2</var>, <var>n_2</var>], ...)</code>
returns a truncated power series in the variables <var>x_1</var>, <var>x_2</var>, ...
about the point <code>(<var>a_1</var>, <var>a_2</var>, ...)</code>,
truncated at <var>n_1</var>, <var>n_2</var>, ....
</p>
<p><code>taylor (<var>expr</var>, [<var>x_1</var>, <var>x_2</var>, ...], [<var>a_1</var>, <var>a_2</var>, ...], [<var>n_1</var>, <var>n_2</var>, ...])</code>
returns a truncated power series in the variables <var>x_1</var>, <var>x_2</var>, ...
about the point <code>(<var>a_1</var>, <var>a_2</var>, ...)</code>,
truncated at <var>n_1</var>, <var>n_2</var>, ....
</p>
<p><code>taylor (<var>expr</var>, [<var>x</var>, <var>a</var>, <var>n</var>, 'asymp])</code>
returns an expansion of <var>expr</var> in negative powers of <code><var>x</var> - <var>a</var></code>.
The highest order term is <code>(<var>x</var> - <var>a</var>)^<var>-n</var></code>.
</p>
<p>When <code>maxtayorder</code> is <code>true</code>, then during algebraic
manipulation of (truncated) Taylor series, <code>taylor</code> tries to retain
as many terms as are known to be correct.
</p>
<p>When <code>psexpand</code> is <code>true</code>,
an extended rational function expression is displayed fully expanded.
The switch <code>ratexpand</code> has the same effect.
When <code>psexpand</code> is <code>false</code>,
a multivariate expression is displayed just as in the rational function package.
When <code>psexpand</code> is <code>multi</code>,
then terms with the same total degree in the variables are grouped together.
</p>
<p>See also the <code>taylor_logexpand</code> switch for controlling expansion.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) taylor (sqrt (sin(x) + a*x + 1), x, 0, 3);
2 2
(a + 1) x (a + 2 a + 1) x
(%o1)/T/ 1 + --------- - -----------------
2 8
3 2 3
(3 a + 9 a + 9 a - 1) x
+ -------------------------- + . . .
48
(%i2) %^2;
3
x
(%o2)/T/ 1 + (a + 1) x - -- + . . .
6
(%i3) taylor (sqrt (x + 1), x, 0, 5);
2 3 4 5
x x x 5 x 7 x
(%o3)/T/ 1 + - - -- + -- - ---- + ---- + . . .
2 8 16 128 256
(%i4) %^2;
(%o4)/T/ 1 + x + . . .
(%i5) product ((1 + x^i)^2.5, i, 1, inf)/(1 + x^2);
inf
/===\
! ! i 2.5
! ! (x + 1)
! !
i = 1
(%o5) -----------------
2
x + 1
(%i6) ev (taylor(%, x, 0, 3), keepfloat);
2 3
(%o6)/T/ 1 + 2.5 x + 3.375 x + 6.5625 x + . . .
(%i7) taylor (1/log (x + 1), x, 0, 3);
2 3
1 1 x x 19 x
(%o7)/T/ - + - - -- + -- - ----- + . . .
x 2 12 24 720
(%i8) taylor (cos(x) - sec(x), x, 0, 5);
4
2 x
(%o8)/T/ - x - -- + . . .
6
(%i9) taylor ((cos(x) - sec(x))^3, x, 0, 5);
(%o9)/T/ 0 + . . .
(%i10) taylor (1/(cos(x) - sec(x))^3, x, 0, 5);
2 4
1 1 11 347 6767 x 15377 x
(%o10)/T/ - -- + ---- + ------ - ----- - ------- - --------
6 4 2 15120 604800 7983360
x 2 x 120 x
+ . . .
(%i11) taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6);
2 2 4 2 4
k x (3 k - 4 k ) x
(%o11)/T/ 1 - ----- - ----------------
2 24
6 4 2 6
(45 k - 60 k + 16 k ) x
- -------------------------- + . . .
720
(%i12) taylor ((x + 1)^n, x, 0, 4);
2 2 3 2 3
(n - n) x (n - 3 n + 2 n) x
(%o12)/T/ 1 + n x + ----------- + --------------------
2 6
4 3 2 4
(n - 6 n + 11 n - 6 n) x
+ ---------------------------- + . . .
24
(%i13) taylor (sin (y + x), x, 0, 3, y, 0, 3);
3 2
y y
(%o13)/T/ y - -- + . . . + (1 - -- + . . .) x
6 2
3 2
y y 2 1 y 3
+ (- - + -- + . . .) x + (- - + -- + . . .) x + . . .
2 12 6 12
(%i14) taylor (sin (y + x), [x, y], 0, 3);
3 2 2 3
x + 3 y x + 3 y x + y
(%o14)/T/ y + x - ------------------------- + . . .
6
(%i15) taylor (1/sin (y + x), x, 0, 3, y, 0, 3);
1 y 1 1 1 2
(%o15)/T/ - + - + . . . + (- -- + - + . . .) x + (-- + . . .) x
y 6 2 6 3
y y
1 3
+ (- -- + . . .) x + . . .
4
y
(%i16) taylor (1/sin (y + x), [x, y], 0, 3);
3 2 2 3
1 x + y 7 x + 21 y x + 21 y x + 7 y
(%o16)/T/ ----- + ----- + ------------------------------- + . . .
x + y 6 360
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-taylordepth"></a>
</p><dl>
<dt><u>Option variable:</u> <b>taylordepth</b>
<a name="IDX1123"></a>
</dt>
<dd><p>Default value: 3
</p>
<p>If there are still no nonzero terms, <code>taylor</code> doubles the
degree of the expansion of <code><var>g</var>(<var>x</var>)</code>
so long as the degree of the expansion is less than or equal to <code><var>n</var> 2^taylordepth</code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-taylorinfo"></a>
</p><dl>
<dt><u>Function:</u> <b>taylorinfo</b><i> (<var>expr</var>)</i>
<a name="IDX1124"></a>
</dt>
<dd><p>Returns information about the Taylor series <var>expr</var>.
The return value is a list of lists.
Each list comprises the name of a variable,
the point of expansion, and the degree of the expansion.
</p>
<p><code>taylorinfo</code> returns <code>false</code> if <var>expr</var> is not a Taylor series.
</p>
<p>Example:
</p>
<pre class="example">(%i1) taylor ((1 - y^2)/(1 - x), x, 0, 3, [y, a, inf]);
2 2
(%o1)/T/ - (y - a) - 2 a (y - a) + (1 - a )
2 2
+ (1 - a - 2 a (y - a) - (y - a) ) x
2 2 2
+ (1 - a - 2 a (y - a) - (y - a) ) x
2 2 3
+ (1 - a - 2 a (y - a) - (y - a) ) x + . . .
(%i2) taylorinfo(%);
(%o2) [[y, a, inf], [x, 0, 3]]
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-taylorp"></a>
</p><dl>
<dt><u>Function:</u> <b>taylorp</b><i> (<var>expr</var>)</i>
<a name="IDX1125"></a>
</dt>
<dd><p>Returns <code>true</code> if <var>expr</var> is a Taylor series,
and <code>false</code> otherwise.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
·
<a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-taylor_005flogexpand"></a>
</p><dl>
<dt><u>Option variable:</u> <b>taylor_logexpand</b>
<a name="IDX1126"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>taylor_logexpand</code> controls expansions of logarithms in
<code>taylor</code> series.
</p>
<p>When <code>taylor_logexpand</code> is <code>true</code>, all logarithms are expanded fully so that
zero-recognition problems involving logarithmic identities do not
disturb the expansion process. However, this scheme is not always
mathematically correct since it ignores branch information.
</p>
<p>When <code>taylor_logexpand</code> is set to <code>false</code>, then the only expansion of logarithms
that occur is that necessary to obtain a formal power series.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
·
<a href="maxima_95.html#Category_003a-Exponential-and-logarithm-functions">Exponential and logarithm functions</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-taylor_005forder_005fcoefficients"></a>
</p><dl>
<dt><u>Option variable:</u> <b>taylor_order_coefficients</b>
<a name="IDX1127"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p><code>taylor_order_coefficients</code> controls the ordering of
coefficients in a Taylor series.
</p>
<p>When <code>taylor_order_coefficients</code> is <code>true</code>,
coefficients of taylor series are ordered canonically.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-taylor_005fsimplifier"></a>
</p><dl>
<dt><u>Function:</u> <b>taylor_simplifier</b><i> (<var>expr</var>)</i>
<a name="IDX1128"></a>
</dt>
<dd><p>Simplifies coefficients of the power series <var>expr</var>.
<code>taylor</code> calls this function.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-taylor_005ftruncate_005fpolynomials"></a>
</p><dl>
<dt><u>Option variable:</u> <b>taylor_truncate_polynomials</b>
<a name="IDX1129"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>taylor_truncate_polynomials</code> is <code>true</code>,
polynomials are truncated based upon the input truncation levels.
</p>
<p>Otherwise,
polynomials input to <code>taylor</code> are considered to have infinite precison.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-taytorat"></a>
</p><dl>
<dt><u>Function:</u> <b>taytorat</b><i> (<var>expr</var>)</i>
<a name="IDX1130"></a>
</dt>
<dd><p>Converts <var>expr</var> from <code>taylor</code> form to canonical rational expression (CRE) form.
The effect is the same as <code>rat (ratdisrep (<var>expr</var>))</code>, but faster.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
·
<a href="maxima_95.html#Category_003a-Rational-expressions">Rational expressions</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-trunc"></a>
</p><dl>
<dt><u>Function:</u> <b>trunc</b><i> (<var>expr</var>)</i>
<a name="IDX1131"></a>
</dt>
<dd><p>Annotates the internal representation of the general expression <var>expr</var>
so that it is displayed as if its sums were truncated Taylor series.
<var>expr</var> is not otherwise modified.
</p>
<p>Example:
</p>
<pre class="example">(%i1) expr: x^2 + x + 1;
2
(%o1) x + x + 1
(%i2) trunc (expr);
2
(%o2) 1 + x + x + . . .
(%i3) is (expr = trunc (expr));
(%o3) true
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-unsum"></a>
</p><dl>
<dt><u>Function:</u> <b>unsum</b><i> (<var>f</var>, <var>n</var>)</i>
<a name="IDX1132"></a>
</dt>
<dd><p>Returns the first backward difference <code><var>f</var>(<var>n</var>) - <var>f</var>(<var>n</var> - 1)</code>.
Thus <code>unsum</code> in a sense is the inverse of <code>sum</code>.
</p>
<p>See also <code>nusum</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) g(p) := p*4^n/binomial(2*n,n);
n
p 4
(%o1) g(p) := ----------------
binomial(2 n, n)
(%i2) g(n^4);
4 n
n 4
(%o2) ----------------
binomial(2 n, n)
(%i3) nusum (%, n, 0, n);
4 3 2 n
2 (n + 1) (63 n + 112 n + 18 n - 22 n + 3) 4 2
(%o3) ------------------------------------------------ - ------
693 binomial(2 n, n) 3 11 7
(%i4) unsum (%, n);
4 n
n 4
(%o4) ----------------
binomial(2 n, n)
</pre>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Sums-and-products">Sums and products</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-verbose"></a>
</p><dl>
<dt><u>Option variable:</u> <b>verbose</b>
<a name="IDX1133"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>verbose</code> is <code>true</code>,
<code>powerseries</code> prints progress messages.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Power-series">Power series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-Poisson-series"></a>
</p><hr size="6">
<a name="Poisson-series"></a>
<a name="SEC137"></a>
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</tr></table>
<h2 class="section"> 30.3 Poisson series </h2>
<p><a name="Item_003a-intopois"></a>
</p><dl>
<dt><u>Function:</u> <b>intopois</b><i> (<var>a</var>)</i>
<a name="IDX1134"></a>
</dt>
<dd><p>Converts <var>a</var> into a Poisson encoding.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-outofpois"></a>
</p><dl>
<dt><u>Function:</u> <b>outofpois</b><i> (<var>a</var>)</i>
<a name="IDX1135"></a>
</dt>
<dd><p>Converts <var>a</var> from Poisson encoding to general
representation. If <var>a</var> is not in Poisson form, <code>outofpois</code> carries out the conversion,
i.e., the return value is <code>outofpois (intopois (<var>a</var>))</code>.
This function is thus a canonical simplifier
for sums of powers of sine and cosine terms of a particular type.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poisdiff"></a>
</p><dl>
<dt><u>Function:</u> <b>poisdiff</b><i> (<var>a</var>, <var>b</var>)</i>
<a name="IDX1136"></a>
</dt>
<dd><p>Differentiates <var>a</var> with respect to <var>b</var>. <var>b</var> must occur only
in the trig arguments or only in the coefficients.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poisexpt"></a>
</p><dl>
<dt><u>Function:</u> <b>poisexpt</b><i> (<var>a</var>, <var>b</var>)</i>
<a name="IDX1137"></a>
</dt>
<dd><p>Functionally identical to <code>intopois (<var>a</var>^<var>b</var>)</code>.
<var>b</var> must be a positive integer.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poisint"></a>
</p><dl>
<dt><u>Function:</u> <b>poisint</b><i> (<var>a</var>, <var>b</var>)</i>
<a name="IDX1138"></a>
</dt>
<dd><p>Integrates in a similarly restricted sense (to
<code>poisdiff</code>). Non-periodic terms in <var>b</var> are dropped if <var>b</var> is in the trig
arguments.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poislim"></a>
</p><dl>
<dt><u>Option variable:</u> <b>poislim</b>
<a name="IDX1139"></a>
</dt>
<dd><p>Default value: 5
</p>
<p><code>poislim</code> determines the domain of the coefficients in
the arguments of the trig functions. The initial value of 5
corresponds to the interval [-2^(5-1)+1,2^(5-1)], or [-15,16], but it
can be set to [-2^(n-1)+1, 2^(n-1)].
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poismap"></a>
</p><dl>
<dt><u>Function:</u> <b>poismap</b><i> (<var>series</var>, <var>sinfn</var>, <var>cosfn</var>)</i>
<a name="IDX1140"></a>
</dt>
<dd><p>will map the functions <var>sinfn</var> on the
sine terms and <var>cosfn</var> on the cosine terms of the Poisson series given.
<var>sinfn</var> and <var>cosfn</var> are functions of two arguments which are a coefficient
and a trigonometric part of a term in series respectively.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poisplus"></a>
</p><dl>
<dt><u>Function:</u> <b>poisplus</b><i> (<var>a</var>, <var>b</var>)</i>
<a name="IDX1141"></a>
</dt>
<dd><p>Is functionally identical to <code>intopois (a + b)</code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poissimp"></a>
</p><dl>
<dt><u>Function:</u> <b>poissimp</b><i> (<var>a</var>)</i>
<a name="IDX1142"></a>
</dt>
<dd><p>Converts <var>a</var> into a Poisson series for <var>a</var> in general
representation.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poisson"></a>
</p><dl>
<dt><u>Special symbol:</u> <b>poisson</b>
<a name="IDX1143"></a>
</dt>
<dd><p>The symbol <code>/P/</code> follows the line label of Poisson series
expressions.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poissubst"></a>
</p><dl>
<dt><u>Function:</u> <b>poissubst</b><i> (<var>a</var>, <var>b</var>, <var>c</var>)</i>
<a name="IDX1144"></a>
</dt>
<dd><p>Substitutes <var>a</var> for <var>b</var> in <var>c</var>. <var>c</var> is a Poisson series.
</p>
<p>(1) Where <var>B</var> is a variable <var>u</var>, <var>v</var>, <var>w</var>, <var>x</var>, <var>y</var>, or <var>z</var>,
then <var>a</var> must be an
expression linear in those variables (e.g., <code>6*u + 4*v</code>).
</p>
<p>(2) Where <var>b</var> is other than those variables, then <var>a</var> must also be
free of those variables, and furthermore, free of sines or cosines.
</p>
<p><code>poissubst (<var>a</var>, <var>b</var>, <var>c</var>, <var>d</var>, <var>n</var>)</code> is a special type of substitution which
operates on <var>a</var> and <var>b</var> as in type (1) above, but where <var>d</var> is a Poisson
series, expands <code>cos(<var>d</var>)</code> and <code>sin(<var>d</var>)</code> to order <var>n</var> so as to provide the
result of substituting <code><var>a</var> + <var>d</var></code> for <var>b</var> in <var>c</var>. The idea is that <var>d</var> is an
expansion in terms of a small parameter. For example,
<code>poissubst (u, v, cos(v), %e, 3)</code> yields <code>cos(u)*(1 - %e^2/2) - sin(u)*(%e - %e^3/6)</code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poistimes"></a>
</p><dl>
<dt><u>Function:</u> <b>poistimes</b><i> (<var>a</var>, <var>b</var>)</i>
<a name="IDX1145"></a>
</dt>
<dd><p>Is functionally identical to <code>intopois (<var>a</var>*<var>b</var>)</code>.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-poistrim"></a>
</p><dl>
<dt><u>Function:</u> <b>poistrim</b><i> ()</i>
<a name="IDX1146"></a>
</dt>
<dd><p>is a reserved function name which (if the user has defined
it) gets applied during Poisson multiplication. It is a predicate
function of 6 arguments which are the coefficients of the <var>u</var>, <var>v</var>, ..., <var>z</var>
in a term. Terms for which <code>poistrim</code> is <code>true</code> (for the coefficients of
that term) are eliminated during multiplication.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-printpois"></a>
</p><dl>
<dt><u>Function:</u> <b>printpois</b><i> (<var>a</var>)</i>
<a name="IDX1147"></a>
</dt>
<dd><p>Prints a Poisson series in a readable format. In common
with <code>outofpois</code>, it will convert <var>a</var> into a Poisson encoding first, if
necessary.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Poisson-series">Poisson series</a>
·
<a href="maxima_95.html#Category_003a-Display-functions">Display functions</a>
</p>
</div>
</dd></dl>
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