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<a name="Gamma-and-Factorial-Functions"></a>
<div class="header">
<p>
Next: <a href="maxima_88.html#Exponential-Integrals" accesskey="n" rel="next">Exponential Integrals</a>, Previous: <a href="maxima_86.html#Airy-Functions" accesskey="p" rel="previous">Airy Functions</a>, Up: <a href="maxima_83.html#Special-Functions" accesskey="u" rel="up">Special Functions</a> &nbsp; [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Gamma-and-Factorial-Functions-1"></a>
<h3 class="section">15.4 Gamma and Factorial Functions</h3>

<p>The gamma function and the related beta, psi and incomplete gamma 
functions are defined in Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, Chapter 6.
</p>



<a name="bffac"></a><a name="Item_003a-Special_002fdeffn_002fbffac"></a><dl>
<dt><a name="index-bffac"></a>Function: <strong>bffac</strong> <em>(<var>expr</var>, <var>n</var>)</em></dt>
<dd>
<p>Bigfloat version of the factorial (shifted gamma)
function.  The second argument is how many digits to retain and return,
it&rsquo;s a good idea to request a couple of extra.
</p>
<div class="example">
<pre class="example">(%i1) bffac(1/2,16);
(%o1)                        8.862269254527584b-1
(%i2) (1/2)!,numer;
(%o2)                          0.886226925452758
(%i3) bffac(1/2,32);
(%o3)                 8.862269254527580136490837416707b-1
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Numerical-evaluation">Numerical evaluation</a>
&middot;</div></dd></dl>

<a name="bfpsi"></a><a name="Item_003a-Special_002fdeffn_002fbfpsi"></a><dl>
<dt><a name="index-bfpsi"></a>Function: <strong>bfpsi</strong> <em>(<var>n</var>, <var>z</var>, <var>fpprec</var>)</em></dt>
<dd><a name="Item_003a-Special_002fdeffn_002fbfpsi0"></a></dd><dt><a name="index-bfpsi0"></a>Function: <strong>bfpsi0</strong> <em>(<var>z</var>, <var>fpprec</var>)</em></dt>
<dd>
<p><code>bfpsi</code> is the polygamma function of real argument <var>z</var> and
integer order <var>n</var>.  See <a href="#polygamma">psi</a> for further
information.  <code>bfpsi0</code> is the digamma function.
<code>bfpsi0(<var>z</var>, <var>fpprec</var>)</code> is equivalent to <code>bfpsi(0,
<var>z</var>, <var>fpprec</var>)</code>.
</p>
<p>These functions return bigfloat values.
<var>fpprec</var> is the bigfloat precision of the return value.
</p>

<div class="example">
<pre class="example">(%i1) bfpsi0(1/3, 15);
(%o1)                        - 3.13203378002081b0
(%i2) bfpsi0(1/3, 32);
(%o2)                - 3.1320337800208063229964190742873b0
(%i3) bfpsi(0,1/3,32);
(%o3)                - 3.1320337800208063229964190742873b0
(%i4) psi[0](1/3);
                          3 log(3)       %pi
(%o4)                  (- --------) - --------- - %gamma
                             2        2 sqrt(3)
(%i5) float(%);
(%o5)                         - 3.132033780020806
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Numerical-evaluation">Numerical evaluation</a>
&middot;</div></dd></dl>

<a name="cbffac"></a><a name="Item_003a-Special_002fdeffn_002fcbffac"></a><dl>
<dt><a name="index-cbffac"></a>Function: <strong>cbffac</strong> <em>(<var>z</var>, <var>fpprec</var>)</em></dt>
<dd><p>Complex bigfloat factorial.
</p>
<p><code>load (&quot;bffac&quot;)</code> loads this function.
</p>
<div class="example">
<pre class="example">(%i1) cbffac(1+%i,16);
(%o1)           3.430658398165453b-1 %i + 6.529654964201666b-1
(%i2) (1+%i)!,numer;
(%o2)             0.3430658398165453 %i + 0.6529654964201667
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Complex-variables">Complex variables</a>
&middot;<a href="maxima_424.html#Category_003a-Numerical-evaluation">Numerical evaluation</a>
&middot;</div></dd></dl>

<a name="gamma"></a><a name="Item_003a-Special_002fdeffn_002fgamma"></a><dl>
<dt><a name="index-gamma"></a>Function: <strong>gamma</strong> <em>(<var>z</var>)</em></dt>
<dd>
<p>The basic definition of the gamma function (<a href="https://dlmf.nist.gov/5.2.E1">DLMF 5.2.E1</a> and <a href="https://personal.math.ubc.ca/~cbm/aands/page_255.htm">A&amp;S eqn 6.1.1</a>) is
</p>
$$
\Gamma\left(z\right)=\int_{0}^{\infty }{t^{z-1}\,e^ {- t }\;dt}
$$

<p>Maxima simplifies <code>gamma</code> for positive integer and positive and negative 
rational numbers. For half integral values the result is a rational number
times 
\(\sqrt{\pi}\). The simplification for integer values is controlled by 
<code>factlim</code>. For integers greater than <code>factlim</code> the numerical result of 
the factorial function, which is used to calculate <code>gamma</code>, will overflow. 
The simplification for rational numbers is controlled by <code>gammalim</code> to 
avoid internal overflow. See <code>factlim</code> and <code>gammalim</code>.
</p>
<p>For negative integers <code>gamma</code> is not defined.
</p>
<p>Maxima can evaluate <code>gamma</code> numerically for real and complex values in float
and bigfloat precision.
</p>
<p><code>gamma</code> has mirror symmetry.
</p>
<p>When <code><a href="#gamma_005fexpand">gamma_expand</a></code> is <code>true</code>, Maxima expands <code>gamma</code> for 
arguments <code>z+n</code> and <code>z-n</code> where <code>n</code> is an integer.
</p>
<p>Maxima knows the derivative of <code>gamma</code>.
</p>
<p>Examples:
</p>
<p>Simplification for integer, half integral, and rational numbers:
</p>
<div class="example">
<pre class="example">(%i1) map('gamma,[1,2,3,4,5,6,7,8,9]);
(%o1)        [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
(%i2) map('gamma,[1/2,3/2,5/2,7/2]);
                    sqrt(%pi)  3 sqrt(%pi)  15 sqrt(%pi)
(%o2)   [sqrt(%pi), ---------, -----------, ------------]
                        2           4            8
(%i3) map('gamma,[2/3,5/3,7/3]);
                                  2           1
                          2 gamma(-)  4 gamma(-)
                      2           3           3
(%o3)          [gamma(-), ----------, ----------]
                      3       3           9
</pre></div>

<p>Numerical evaluation for real and complex values:
</p>
<div class="example">
<pre class="example">(%i4) map('gamma,[2.5,2.5b0]);
(%o4)     [1.329340388179137, 1.3293403881791370205b0]
(%i5) map('gamma,[1.0+%i,1.0b0+%i]);
(%o5) [0.498015668118356 - .1549498283018107 %i, 
          4.9801566811835604272b-1 - 1.5494982830181068513b-1 %i]
</pre></div>

<p><code>gamma</code> has mirror symmetry:
</p>
<div class="example">
<pre class="example">(%i6) declare(z,complex)$
(%i7) conjugate(gamma(z));
(%o7)                  gamma(conjugate(z))
</pre></div>

<p>Maxima expands <code>gamma(z+n)</code> and <code>gamma(z-n)</code>, when <code><a href="#gamma_005fexpand">gamma_expand</a></code> 
is <code>true</code>:
</p>
<div class="example">
<pre class="example">(%i8) gamma_expand:true$

(%i9) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)];
                               gamma(z)
(%o9)             [z gamma(z), --------, z + 1]
                                z - 1
</pre></div>

<p>The derivative of <code>gamma</code>:
</p>
<div class="example">
<pre class="example">(%i10) diff(gamma(z),z);
(%o10)                  psi (z) gamma(z)
                           0
</pre></div>

<p>See also <code><a href="#makegamma">makegamma</a></code>.
</p>
<p>The Euler-Mascheroni constant is <code>%gamma</code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
&middot;</div></dd></dl>

<a name="Item_003a-Special_002fdeffn_002flog_005fgamma"></a><dl>
<dt><a name="index-log_005fgamma"></a>Function: <strong>log_gamma</strong> <em>(<var>z</var>)</em></dt>
<dd>
<p>The natural logarithm of the gamma function.
</p>
<div class="example">
<pre class="example">(%i1) gamma(6);
(%o1)                                 120
(%i2) log_gamma(6);
(%o2)                              log(120)
(%i3) log_gamma(0.5);
(%o3)                         0.5723649429247004
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
&middot;</div></dd></dl>

<a name="Item_003a-Special_002fdeffn_002fgamma_005fincomplete_005flower"></a><dl>
<dt><a name="index-gamma_005fincomplete_005flower"></a>Function: <strong>gamma_incomplete_lower</strong> <em>(<var>a</var>, <var>z</var>)</em></dt>
<dd>
<p>The lower incomplete gamma function (<a href="https://dlmf.nist.gov/8.2.E1">DLMF 8.2.E1</a> and <a href="https://personal.math.ubc.ca/~cbm/aands/page_260.htm">A&amp;S eqn 6.5.2</a>):
</p>
$$
\gamma\left(a , z\right)=\int_{0}^{z}{t^{a-1}\,e^ {- t }\;dt}
$$

<p>See also <code><a href="#gamma_005fincomplete">gamma_incomplete</a></code> (upper incomplete gamma function).
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
&middot;</div></dd></dl>

<a name="gamma_005fincomplete"></a><a name="Item_003a-Special_002fdeffn_002fgamma_005fincomplete"></a><dl>
<dt><a name="index-gamma_005fincomplete"></a>Function: <strong>gamma_incomplete</strong> <em>(<var>a</var>, <var>z</var>)</em></dt>
<dd>
<p>The incomplete upper gamma function (<a href="https://dlmf.nist.gov/8.2.E2">DLMF 8.2.E2</a> and <a href="https://personal.math.ubc.ca/~cbm/aands/page_260.htm">A&amp;S eqn 6.5.3</a>):
</p>
$$
\Gamma\left(a , z\right)=\int_{z}^{\infty }{t^{a-1}\,e^ {- t }\;dt}
$$

<p>See also <code><a href="#gamma_005fexpand">gamma_expand</a></code> for controlling how
<code>gamma_incomplete</code> is expressed in terms of elementary functions
and <code>erfc</code>.
</p>
<p>Also see the related functions <code><a href="#gamma_005fincomplete_005fregularized">gamma_incomplete_regularized</a></code> and
<code><a href="#gamma_005fincomplete_005fgeneralized">gamma_incomplete_generalized</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
&middot;</div></dd></dl>

<a name="gamma_005fincomplete_005fregularized"></a><a name="Item_003a-Special_002fdeffn_002fgamma_005fincomplete_005fregularized"></a><dl>
<dt><a name="index-gamma_005fincomplete_005fregularized"></a>Function: <strong>gamma_incomplete_regularized</strong> <em>(<var>a</var>, <var>z</var>)</em></dt>
<dd>
<p>The regularized incomplete upper gamma function (<a href="https://dlmf.nist.gov/8.2.E4">DLMF 8.2.E4</a>):
</p>
$$
Q\left(a , z\right)={{\Gamma\left(a , z\right)}\over{\Gamma\left(a\right)}}
$$

<p>See also <code><a href="#gamma_005fexpand">gamma_expand</a></code> for controlling how
<code><a href="#gamma_005fincomplete">gamma_incomplete</a></code> is expressed in terms of elementary functions
and <code><a href="maxima_89.html#erfc">erfc</a></code>.
</p>
<p>Also see <code><a href="#gamma_005fincomplete">gamma_incomplete</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
&middot;</div></dd></dl>

<a name="gamma_005fincomplete_005fgeneralized"></a><a name="Item_003a-Special_002fdeffn_002fgamma_005fincomplete_005fgeneralized"></a><dl>
<dt><a name="index-gamma_005fincomplete_005fgeneralized"></a>Function: <strong>gamma_incomplete_generalized</strong> <em>(<var>a</var>, <var>z1</var>, <var>z1</var>)</em></dt>
<dd>
<p>The generalized incomplete gamma function.
</p>
$$
\Gamma\left(a , z_{1}, z_{2}\right)=\int_{z_{1}}^{z_{2}}{t^{a-1}\,e^ {- t }\;dt}
$$

<p>Also see <code><a href="#gamma_005fincomplete">gamma_incomplete</a></code> and <code><a href="#gamma_005fincomplete_005fregularized">gamma_incomplete_regularized</a></code>.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Special-functions">Special functions</a>
&middot;</div></dd></dl>


<a name="gamma_005fexpand"></a><a name="Item_003a-Special_002fdefvr_002fgamma_005fexpand"></a><dl>
<dt><a name="index-gamma_005fexpand"></a>Option variable: <strong>gamma_expand</strong></dt>
<dd><p>Default value: <code>false</code>
</p>
<p><code>gamma_expand</code> controls expansion of <code><a href="#gamma_005fincomplete">gamma_incomplete</a></code>.
When <code>gamma_expand</code> is <code>true</code>, <code>gamma_incomplete(v,z)</code>
is expanded in terms of
<code>z</code>, <code>exp(z)</code>, and <code><a href="#gamma_005fincomplete">gamma_incomplete</a></code> or <code><a href="maxima_89.html#erfc">erfc</a></code> when possible.
</p>
<div class="example">
<pre class="example">(%i1) gamma_incomplete(2,z);
(%o1)                       gamma_incomplete(2, z)
(%i2) gamma_expand:true;
(%o2)                                true
(%i3) gamma_incomplete(2,z);
                                           - z
(%o3)                            (z + 1) %e
</pre><pre class="example">(%i4) gamma_incomplete(3/2,z);
                              - z   sqrt(%pi) erfc(sqrt(z))
(%o4)               sqrt(z) %e    + -----------------------
                                               2
</pre><pre class="example">(%i5) gamma_incomplete(4/3,z);
                                                    1
                                   gamma_incomplete(-, z)
                       1/3   - z                    3
(%o5)                 z    %e    + ----------------------
                                             3
</pre><pre class="example">(%i6) gamma_incomplete(a+2,z);
             a               - z
(%o6)       z  (z + a + 1) %e    + a (a + 1) gamma_incomplete(a, z)
(%i7) gamma_incomplete(a-2, z);
        gamma_incomplete(a, z)    a - 2         z            1      - z
(%o7)   ---------------------- - z      (--------------- + -----) %e
           (1 - a) (2 - a)               (a - 2) (a - 1)   a - 2

</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
&middot;</div>
</dd></dl>
<a name="gammalim"></a><a name="Item_003a-Special_002fdefvr_002fgammalim"></a><dl>
<dt><a name="index-gammalim"></a>Option variable: <strong>gammalim</strong></dt>
<dd><p>Default value: 10000
</p>
<p><code>gammalim</code> controls simplification of the gamma
function for integral and rational number arguments.  If the absolute
value of the argument is not greater than <code>gammalim</code>, then
simplification will occur.  Note that the <code>factlim</code> switch controls
simplification of the result of <code>gamma</code> of an integer argument as well.
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;<a href="maxima_424.html#Category_003a-Simplification-flags-and-variables">Simplification flags and variables</a>
&middot;</div>
</dd></dl>


<a name="makegamma"></a><a name="Item_003a-Special_002fdeffn_002fmakegamma"></a><dl>
<dt><a name="index-makegamma"></a>Function: <strong>makegamma</strong> <em>(<var>expr</var>)</em></dt>
<dd><p>Transforms instances of binomial, factorial, and beta
functions in <var>expr</var> into gamma functions.
</p>
<p>See also <code><a href="#makefact">makefact</a></code>.
</p>
<div class="example">
<pre class="example">(%i1) makegamma(binomial(n,k));
                                 gamma(n + 1)
(%o1)                    -----------------------------
                         gamma(k + 1) gamma(n - k + 1)
(%i2) makegamma(x!);
(%o2)                            gamma(x + 1)
(%i3) makegamma(beta(a,b));
                               gamma(a) gamma(b)
(%o3)                          -----------------
                                 gamma(b + a)
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;</div></dd></dl>

<a name="beta"></a><a name="Item_003a-Special_002fdeffn_002fbeta"></a><dl>
<dt><a name="index-beta"></a>Function: <strong>beta</strong> <em>(<var>a</var>, <var>b</var>)</em></dt>
<dd><p>The beta function is defined as
$$
{\rm B}(a, b) = {{\Gamma(a) \Gamma(b)}\over{\Gamma(a+b)}}
$$</p>
<p>(<a href="https://dlmf.nist.gov/5.12.E1">DLMF 5.12.E1</a> and <a href="https://personal.math.ubc.ca/~cbm/aands/page_258.htm">A&amp;S eqn 6.2.1</a>).
</p>
<p>Maxima simplifies the beta function for positive integers and rational 
numbers, which sum to an integer. When <code>beta_args_sum_to_integer</code> is 
<code>true</code>, Maxima simplifies also general expressions which sum to an integer. 
</p>
<p>For <var>a</var> or <var>b</var> equal to zero the beta function is not defined.
</p>
<p>In general the beta function is not defined for negative integers as an 
argument. The exception is for <var>a=-n</var>, <var>n</var> a positive integer 
and <var>b</var> a positive integer with <code>b&lt;=n</code>, it is possible to define an
analytic continuation. Maxima gives for this case a result.
</p>
<p>When <code><a href="#beta_005fexpand">beta_expand</a></code> is <code>true</code>, expressions like <code>beta(a+n,b)</code> and 
<code>beta(a-n,b)</code> or <code>beta(a,b+n)</code> and <code>beta(a,b-n)</code> with <code>n</code> 
an integer are simplified.
</p>
<p>Maxima can evaluate the beta function for real and complex values in float and 
bigfloat precision. For numerical evaluation Maxima uses <code>log_gamma</code>:
</p>
<div class="example">
<pre class="example">           - log_gamma(b + a) + log_gamma(b) + log_gamma(a)
         %e
</pre></div>

<p>Maxima knows that the beta function is symmetric and has mirror symmetry.
</p>
<p>Maxima knows the derivatives of the beta function with respect to <var>a</var> or 
<var>b</var>.
</p>
<p>To express the beta function as a ratio of gamma functions see <code>makegamma</code>. 
</p>
<p>Examples:
</p>
<p>Simplification, when one of the arguments is an integer:
</p>
<div class="example">
<pre class="example">(%i1) [beta(2,3),beta(2,1/3),beta(2,a)];
                               1   9      1
(%o1)                         [--, -, ---------]
                               12  4  a (a + 1)
</pre></div>

<p>Simplification for two rational numbers as arguments which sum to an integer:
</p>
<div class="example">
<pre class="example">(%i2) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)];
                          3 %pi   2 %pi
(%o2)                    [-----, -------, sqrt(2) %pi]
                            8    sqrt(3)
</pre></div>

<p>When setting <code>beta_args_sum_to_integer</code> to <code>true</code> more general 
expression are simplified, when the sum of the arguments is an integer:
</p>
<div class="example">
<pre class="example">(%i3) beta_args_sum_to_integer:true$
(%i4) beta(a+1,-a+2);
                                %pi (a - 1) a
(%o4)                         ------------------
                              2 sin(%pi (2 - a))
</pre></div>

<p>The possible results, when one of the arguments is a negative integer: 
</p>
<div class="example">
<pre class="example">(%i5) [beta(-3,1),beta(-3,2),beta(-3,3)];
                                    1  1    1
(%o5)                            [- -, -, - -]
                                    3  6    3
</pre></div>

<p><code>beta(a+n,b)</code> or <code>beta(a-n,b)</code> with <code>n</code> an integer simplifies when 
<code><a href="#beta_005fexpand">beta_expand</a></code> is <code>true</code>:
</p>
<div class="example">
<pre class="example">(%i6) beta_expand:true$
(%i7) [beta(a+1,b),beta(a-1,b),beta(a+1,b)/beta(a,b+1)];
                    a beta(a, b)  beta(a, b) (b + a - 1)  a
(%o7)              [------------, ----------------------, -]
                       b + a              a - 1           b

</pre></div>

<p>Beta is not defined, when one of the arguments is zero:
</p>
<div class="example">
<pre class="example">(%i7) beta(0,b);
beta: expected nonzero arguments; found 0, b
 -- an error.  To debug this try debugmode(true);
</pre></div>

<p>Numerical evaluation for real and complex arguments in float or bigfloat
precision:
</p>
<div class="example">
<pre class="example">(%i8) beta(2.5,2.3);
(%o8) .08694748611299981

(%i9) beta(2.5,1.4+%i);
(%o9) 0.0640144950796695 - .1502078053286415 %i

(%i10) beta(2.5b0,2.3b0);
(%o10) 8.694748611299969b-2

(%i11) beta(2.5b0,1.4b0+%i);
(%o11) 6.401449507966944b-2 - 1.502078053286415b-1 %i
</pre></div>

<p>Beta is symmetric and has mirror symmetry:
</p>
<div class="example">
<pre class="example">(%i14) beta(a,b)-beta(b,a);
(%o14)                                 0
(%i15) declare(a,complex,b,complex)$
(%i16) conjugate(beta(a,b));
(%o16)                 beta(conjugate(a), conjugate(b))
</pre></div>

<p>The derivative of the beta function wrt <code>a</code>:
</p>
<div class="example">
<pre class="example">(%i17) diff(beta(a,b),a);
(%o17)               - beta(a, b) (psi (b + a) - psi (a))
                                      0             0
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;</div></dd></dl>

<a name="Item_003a-Special_002fdeffn_002fbeta_005fincomplete"></a><dl>
<dt><a name="index-beta_005fincomplete"></a>Function: <strong>beta_incomplete</strong> <em>(<var>a</var>, <var>b</var>, <var>z</var>)</em></dt>
<dd>
<p>The basic definition of the incomplete beta function
(<a href="https://dlmf.nist.gov/8.17.E1">DLMF 8.17.E1</a> and <a href="https://personal.math.ubc.ca/~cbm/aands/page_263.htm">A&amp;S eqn 6.6.1</a>) is
</p>
$$
{\rm B}_z(a,b) = \int_0^z t^{a-1}(1-t)^{b-1}\; dt
$$

<p>This definition is possible for 
\({\rm Re}(a) > 0\) and 
\({\rm Re}(b) > 0\) and 
\(|z| < 1\).
For other values the incomplete beta function can be 
defined through a generalized hypergeometric function:
</p>
<div class="example">
<pre class="example">   gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z
</pre></div>

<p>(See <a href="https://functions.wolfram.com/GammaBetaErf/Beta3/">https://functions.wolfram.com/GammaBetaErf/Beta3/</a> for a complete definition of the incomplete beta
function.)
</p>
<p>For negative integers <em>a = -n</em> and positive integers <em>b=m</em> with 
\(m \le n\) the incomplete beta function is defined through
</p>
$$
z^{n-1}\sum_{k=0}^{m-1} {{(1-m)_k z^k} \over {k! (n-k)}}
$$

<p>Maxima uses this definition to simplify <code>beta_incomplete</code> for <var>a</var> a 
negative integer.
</p>
<p>For <var>a</var> a positive integer, <code>beta_incomplete</code> simplifies for any 
argument <var>b</var> and <var>z</var> and for <var>b</var> a positive integer for any 
argument <var>a</var> and <var>z</var>, with the exception of <var>a</var> a negative integer.
</p>
<p>For <em>z=0</em> and 
\({\rm Re}(a) > 0\), <code>beta_incomplete</code> has the 
specific value zero. For <em>z=1</em> and 
\({\rm Re}(b) > 0\), 
<code>beta_incomplete</code> simplifies to the beta function <code>beta(a,b)</code>.
</p>
<p>Maxima evaluates <code>beta_incomplete</code> numerically for real and complex values 
in float or bigfloat precision. For the numerical evaluation an expansion of the 
incomplete beta function in continued fractions is used.
</p>
<p>When the option variable <code><a href="#beta_005fexpand">beta_expand</a></code> is <code>true</code>, Maxima expands
expressions like <code>beta_incomplete(a+n,b,z)</code> and
<code>beta_incomplete(a-n,b,z)</code> where n is a positive integer.
</p>
<p>Maxima knows the derivatives of <code>beta_incomplete</code> with respect to the 
variables <var>a</var>, <var>b</var> and <var>z</var> and the integral with respect to the 
variable <var>z</var>.
</p>
<p>Examples:
</p>
<p>Simplification for <var>a</var> a positive integer:
</p>
<div class="example">
<pre class="example">(%i1) beta_incomplete(2,b,z);
                                       b
                            1 - (1 - z)  (b z + 1)
(%o1)                       ----------------------
                                  b (b + 1)
</pre></div>

<p>Simplification for <var>b</var> a positive integer:
</p>
<div class="example">
<pre class="example">(%i2) beta_incomplete(a,2,z);
                                               a
                              (a (1 - z) + 1) z
(%o2)                         ------------------
                                  a (a + 1)
</pre></div>

<p>Simplification for <var>a</var> and <var>b</var> a positive integer:
</p>
<div class="example">
<pre class="example">(%i3) beta_incomplete(3,2,z);
</pre><pre class="example">                                               3
                              (3 (1 - z) + 1) z
(%o3)                         ------------------
                                      12
</pre></div>

<p><var>a</var> is a negative integer and <em>b&lt;=(-a)</em>, Maxima simplifies:
</p>
<div class="example">
<pre class="example">(%i4) beta_incomplete(-3,1,z);
                                       1
(%o4)                              - ----
                                        3
                                     3 z
</pre></div>

<p>For the specific values <em>z=0</em> and <em>z=1</em>, Maxima simplifies:
</p>
<div class="example">
<pre class="example">(%i5) assume(a&gt;0,b&gt;0)$
(%i6) beta_incomplete(a,b,0);
(%o6)                                 0
(%i7) beta_incomplete(a,b,1);
(%o7)                            beta(a, b)
</pre></div>

<p>Numerical evaluation in float or bigfloat precision:
</p>
<div class="example">
<pre class="example">(%i8) beta_incomplete(0.25,0.50,0.9);
(%o8)                          4.594959440269333
(%i9)  fpprec:25$
(%i10) beta_incomplete(0.25,0.50,0.9b0);
(%o10)                    4.594959440269324086971203b0
</pre></div>

<p>For <em>abs(z)&gt;1</em> <code>beta_incomplete</code> returns a complex result:
</p>
<div class="example">
<pre class="example">(%i11) beta_incomplete(0.25,0.50,1.7);
(%o11)              5.244115108584249 - 1.45518047787844 %i
</pre></div>

<p>Results for more general complex arguments:
</p>
<div class="example">
<pre class="example">(%i14) beta_incomplete(0.25+%i,1.0+%i,1.7+%i);
(%o14)             2.726960675662536 - .3831175704269199 %i
(%i15) beta_incomplete(1/2,5/4*%i,2.8+%i);
(%o15)             13.04649635168716 %i - 5.802067956270001
(%i16) 
</pre></div>

<p>Expansion, when <code><a href="#beta_005fexpand">beta_expand</a></code> is <code>true</code>:
</p>
<div class="example">
<pre class="example">(%i23) beta_incomplete(a+1,b,z),beta_expand:true;
                                                       b  a
                   a beta_incomplete(a, b, z)   (1 - z)  z
(%o23)             -------------------------- - -----------
                             b + a                 b + a

(%i24) beta_incomplete(a-1,b,z),beta_expand:true;
                                                       b  a - 1
       beta_incomplete(a, b, z) (- b - a + 1)   (1 - z)  z
(%o24) -------------------------------------- - ---------------
                       1 - a                         1 - a
</pre></div>
 
<p>Derivative and integral for <code>beta_incomplete</code>:
</p>
<div class="example">
<pre class="example">(%i34) diff(beta_incomplete(a, b, z), z);
</pre><pre class="example">                              b - 1  a - 1
(%o34)                 (1 - z)      z
</pre><pre class="example">(%i35) integrate(beta_incomplete(a, b, z), z);
              b  a
       (1 - z)  z
(%o35) ----------- + beta_incomplete(a, b, z) z
          b + a
                                       a beta_incomplete(a, b, z)
                                     - --------------------------
                                                 b + a
(%i36) factor(diff(%, z));
(%o36)              beta_incomplete(a, b, z)
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;</div></dd></dl>

<a name="beta_005fincomplete_005fregularized"></a><a name="Item_003a-Special_002fdeffn_002fbeta_005fincomplete_005fregularized"></a><dl>
<dt><a name="index-beta_005fincomplete_005fregularized"></a>Function: <strong>beta_incomplete_regularized</strong> <em>(<var>a</var>, <var>b</var>, <var>z</var>)</em></dt>
<dd>
<p>The regularized incomplete beta function (<a href="https://dlmf.nist.gov/8.17.E2">DLMF 8.17.E2</a> and
<a href="https://personal.math.ubc.ca/~cbm/aands/page_263.htm">A&amp;S eqn 6.6.2</a>), defined as 
</p>
$$
I_z(a,b) = {{\rm B}_z(a,b)\over {\rm B}(a,b)}
$$

<p>As for <code>beta_incomplete</code> this definition is not complete. See 
<a href="https://functions.wolfram.com/GammaBetaErf/BetaRegularized/">https://functions.wolfram.com/GammaBetaErf/BetaRegularized/</a> for a complete definition of
<code>beta_incomplete_regularized</code>.
</p>
<p><code>beta_incomplete_regularized</code> simplifies <var>a</var> or <var>b</var> a positive 
integer.
</p>
<p>For <em>z=0</em> and 
\({\rm Re}(a)>0\),
<code>beta_incomplete_regularized</code> has 
the specific value 0. For <em>z=1</em> and 
\({\rm Re}(b) > 0\), 
<code>beta_incomplete_regularized</code> simplifies to 1.
</p>
<p>Maxima can evaluate <code>beta_incomplete_regularized</code> for real and complex 
arguments in float and bigfloat precision.
</p>
<p>When <code><a href="#beta_005fexpand">beta_expand</a></code> is <code>true</code>, Maxima expands 
<code>beta_incomplete_regularized</code> for arguments <em>a+n</em> or <em>a-n</em>, 
where n is an integer.
</p>
<p>Maxima knows the derivatives of <code>beta_incomplete_regularized</code> with respect 
to the variables <var>a</var>, <var>b</var>, and <var>z</var> and the integral with respect to 
the variable <var>z</var>.
</p>
<p>Examples:
</p>
<p>Simplification for <var>a</var> or <var>b</var> a positive integer:
</p>
<div class="example">
<pre class="example">(%i1) beta_incomplete_regularized(2,b,z);
                                       b
(%o1)                       1 - (1 - z)  (b z + 1)

(%i2) beta_incomplete_regularized(a,2,z);
                                               a
(%o2)                         (a (1 - z) + 1) z

(%i3) beta_incomplete_regularized(3,2,z);
                                               3
(%o3)                         (3 (1 - z) + 1) z
</pre></div>

<p>For the specific values <em>z=0</em> and <em>z=1</em>, Maxima simplifies:
</p>
<div class="example">
<pre class="example">(%i4) assume(a&gt;0,b&gt;0)$
(%i5) beta_incomplete_regularized(a,b,0);
(%o5)                                 0
(%i6) beta_incomplete_regularized(a,b,1);
(%o6)                                 1
</pre></div>

<p>Numerical evaluation for real and complex arguments in float and bigfloat 
precision:
</p>
<div class="example">
<pre class="example">(%i7) beta_incomplete_regularized(0.12,0.43,0.9);
(%o7)                         .9114011367359802
(%i8) fpprec:32$
(%i9) beta_incomplete_regularized(0.12,0.43,0.9b0);
(%o9)               9.1140113673598075519946998779975b-1
(%i10) beta_incomplete_regularized(1+%i,3/3,1.5*%i);
(%o10)             .2865367499935403 %i - 0.122995963334684
(%i11) fpprec:20$
(%i12) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i);
(%o12)      2.8653674999354036142b-1 %i - 1.2299596333468400163b-1
</pre></div>

<p>Expansion, when <code><a href="#beta_005fexpand">beta_expand</a></code> is <code>true</code>:
</p>
<div class="example">
<pre class="example">(%i13) beta_incomplete_regularized(a+1,b,z);
                                                     b  a
                                              (1 - z)  z
(%o13) beta_incomplete_regularized(a, b, z) - ------------
                                              a beta(a, b)
(%i14) beta_incomplete_regularized(a-1,b,z);
(%o14) beta_incomplete_regularized(a, b, z)
                                                     b  a - 1
                                              (1 - z)  z
                                         - ----------------------
                                           beta(a, b) (b + a - 1)
</pre></div>

<p>The derivative and the integral wrt <var>z</var>:
</p>
<div class="example">
<pre class="example">(%i15) diff(beta_incomplete_regularized(a,b,z),z);
                              b - 1  a - 1
                       (1 - z)      z
(%o15)                 -------------------
                           beta(a, b)
(%i16) integrate(beta_incomplete_regularized(a,b,z),z);
(%o16) beta_incomplete_regularized(a, b, z) z
                                                           b  a
                                                    (1 - z)  z
          a (beta_incomplete_regularized(a, b, z) - ------------)
                                                    a beta(a, b)
        - -------------------------------------------------------
                                   b + a
</pre></div>

<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
&middot;</div></dd></dl>

<a name="Item_003a-Special_002fdeffn_002fbeta_005fincomplete_005fgeneralized"></a><dl>
<dt><a name="index-beta_005fincomplete_005fgeneralized"></a>Function: <strong>beta_incomplete_generalized</strong> <em>(<var>a</var>, <var>b</var>, <var>z1</var>, <var>z2</var>)</em></dt>
<dd>
<p>The basic definition of the generalized incomplete beta function is
</p>
$$
\int_{z_1}^{z_2} t^{a-1}(1-t)^{b-1}\; dt
$$

<p>Maxima simplifies <code>beta_incomplete_regularized</code> for <var>a</var> and <var>b</var> 
a positive integer.
</p>
<p>For 
\({\rm Re}(a) > 0\) and 
\(z_1 = 0\) or 
\(z_2 = 0\), Maxima simplifies
<code>beta_incomplete_generalized</code> to <code>beta_incomplete</code>.
For 
\({\rm Re}(b) > 0\) and 
\(z_1 = 1\) or 
\(z_2 = 1\), Maxima simplifies to an 
expression with <code>beta</code> and <code>beta_incomplete</code>.
</p>
<p>Maxima evaluates <code>beta_incomplete_regularized</code> for real and complex values 
in float and bigfloat precision.
</p>
<p>When <code><a href="#beta_005fexpand">beta_expand</a></code> is <code>true</code>, Maxima expands 
<code>beta_incomplete_generalized</code> for <em>a+n</em> and <em>a-n</em>, <var>n</var> a 
positive integer.
</p>
<p>Maxima knows the derivative of <code>beta_incomplete_generalized</code> with respect 
to the variables <var>a</var>, <var>b</var>, <var>z1</var>, and <var>z2</var> and the integrals with
respect to the variables <var>z1</var> and <var>z2</var>.
</p>
<p>Examples:
</p>
<p>Maxima simplifies <code>beta_incomplete_generalized</code> for <var>a</var> and <var>b</var> a 
positive integer:
</p>
<div class="example">
<pre class="example">(%i1) beta_incomplete_generalized(2,b,z1,z2);
                   b                      b
           (1 - z1)  (b z1 + 1) - (1 - z2)  (b z2 + 1)
(%o1)      -------------------------------------------
                            b (b + 1)
(%i2) beta_incomplete_generalized(a,2,z1,z2);
</pre><pre class="example">                              a                      a
           (a (1 - z2) + 1) z2  - (a (1 - z1) + 1) z1
(%o2)      -------------------------------------------
                            a (a + 1)
</pre><pre class="example">(%i3) beta_incomplete_generalized(3,2,z1,z2);
              2      2                       2      2
      (1 - z1)  (3 z1  + 2 z1 + 1) - (1 - z2)  (3 z2  + 2 z2 + 1)
(%o3) -----------------------------------------------------------
                                  12
</pre></div>

<p>Simplification for specific values <em>z1=0</em>, <em>z2=0</em>, <em>z1=1</em>, or 
<em>z2=1</em>:
</p>
<div class="example">
<pre class="example">(%i4) assume(a &gt; 0, b &gt; 0)$
(%i5) beta_incomplete_generalized(a,b,z1,0);
(%o5)                    - beta_incomplete(a, b, z1)

(%i6) beta_incomplete_generalized(a,b,0,z2);
(%o6)                    - beta_incomplete(a, b, z2)

(%i7) beta_incomplete_generalized(a,b,z1,1);
(%o7)              beta(a, b) - beta_incomplete(a, b, z1)

(%i8) beta_incomplete_generalized(a,b,1,z2);
(%o8)              beta_incomplete(a, b, z2) - beta(a, b)
</pre></div>

<p>Numerical evaluation for real arguments in float or bigfloat precision:
</p>
<div class="example">
<pre class="example">(%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31);
(%o9)                        .09638178086368676

(%i10) fpprec:32$
(%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0);
(%o10)               9.6381780863686935309170054689964b-2
</pre></div>

<p>Numerical evaluation for complex arguments in float or bigfloat precision:
</p>
<div class="example">
<pre class="example">(%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31);
(%o11)           - .09625463003205376 %i - .003323847735353769
(%i12) fpprec:20$
(%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0);
(%o13)     - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3
</pre></div>

<p>Expansion for <em>a+n</em> or <em>a-n</em>, <var>n</var> a positive integer, when 
<code><a href="#beta_005fexpand">beta_expand</a></code> is <code>true</code>: 
</p>
<div class="example">
<pre class="example">(%i14) beta_expand:true$

(%i15) beta_incomplete_generalized(a+1,b,z1,z2);

               b   a           b   a
       (1 - z1)  z1  - (1 - z2)  z2
(%o15) -----------------------------
                   b + a
                      a beta_incomplete_generalized(a, b, z1, z2)
                    + -------------------------------------------
                                         b + a
(%i16) beta_incomplete_generalized(a-1,b,z1,z2);

       beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
(%o16) -------------------------------------------------------
                                1 - a
                                    b   a - 1           b   a - 1
                            (1 - z2)  z2      - (1 - z1)  z1
                          - -------------------------------------
                                            1 - a
</pre></div>

<p>Derivative wrt the variable <var>z1</var> and integrals wrt <var>z1</var> and <var>z2</var>:
</p>
<div class="example">
<pre class="example">(%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
                               b - 1   a - 1
(%o17)               - (1 - z1)      z1
(%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
(%o18) beta_incomplete_generalized(a, b, z1, z2) z1
                                  + beta_incomplete(a + 1, b, z1)
(%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
(%o19) beta_incomplete_generalized(a, b, z1, z2) z2
                                  - beta_incomplete(a + 1, b, z2)
</pre></div>

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<a name="beta_005fexpand"></a><a name="Item_003a-Special_002fdefvr_002fbeta_005fexpand"></a><dl>
<dt><a name="index-beta_005fexpand"></a>Option variable: <strong>beta_expand</strong></dt>
<dd><p>Default value: false
</p>
<p>When <code>beta_expand</code> is <code>true</code>, <code>beta(a,b)</code> and related 
functions are expanded for arguments like <em>a+n</em> or <em>a-n</em>, 
where <em>n</em> is an integer.
</p>
<p>See <a href="#beta">beta</a> for examples.
</p>
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&middot;</div></dd></dl>

<a name="Item_003a-Special_002fdefvr_002fbeta_005fargs_005fsum_005fto_005finteger"></a><dl>
<dt><a name="index-beta_005fargs_005fsum_005fto_005finteger"></a>Option variable: <strong>beta_args_sum_to_integer</strong></dt>
<dd><p>Default value: false
</p>
<p>When <code>beta_args_sum_to_integer</code> is <code>true</code>, Maxima simplifies 
<code>beta(a,b)</code>, when the arguments <var>a</var> and <var>b</var> sum to an integer.
</p>
<p>See <a href="#beta">beta</a> for examples.
</p>
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<a name="polygamma"></a><a name="Item_003a-Special_002fdeffn_002fpsi"></a><dl>
<dt><a name="index-psi"></a>Function: <strong>psi</strong> <em>[<var>n</var>](<var>x</var>)</em></dt>
<dd>
<p><code>psi[n](x)</code> is the polygamma function (<a href="https://dlmf.nist.gov/5.2E2">DLMF 5.2E2</a>,
<a href="https://dlmf.nist.gov/5.15">DLMF 5.15</a>, <a href="https://personal.math.ubc.ca/~cbm/aands/page_258.htm">A&amp;S eqn 6.3.1</a> and <a href="https://personal.math.ubc.ca/~cbm/aands/page_260.htm">A&amp;S eqn 6.4.1</a>) defined by
$$
\psi^{(n)}(x) = {d^{n+1}\over{dx^{n+1}}} \log\Gamma(x)
$$</p>
<p>Thus, <code>psi[0](<var>x</var>)</code> is the first derivative,
<code>psi[1](<var>x</var>)</code> is the second derivative, etc.
</p>
<p>Maxima can compute some exact values for rational args as well for
float and bfloat args.  Several variables control what range of
rational args 
\(\psi^{(n)}(x)\)) will return an
exact value, if possible.  See <code><a href="#maxpsiposint">maxpsiposint</a></code>,
<code><a href="#maxpsinegint">maxpsinegint</a></code>, <code><a href="#maxpsifracnum">maxpsifracnum</a></code>, and
<code><a href="#maxpsifracdenom">maxpsifracdenom</a></code>. That is, <em>x</em> must lie between
<code>maxpsinegint</code> and <code>maxpsiposint</code>.  If the absolute value of
the fractional part of <em>x</em> is rational and has a numerator less
than <code>maxpsifracnum</code> and has a denominator less than
<code>maxpsifracdenom</code>, 
\(\psi^{(0)}(x)\) will
return an exact value.
</p>
<p>The function <code><a href="#bfpsi">bfpsi</a></code> in the <code>bffac</code> package can compute
numerical values.
</p>
<div class="example">
<pre class="example">(%i1) psi[0](.25);
(%o1)                        - 4.227453533376265
(%i2) psi[0](1/4);
                                        %pi
(%o2)                    (- 3 log(2)) - --- - %gamma
                                         2
(%i3) float(%);
(%o3)                        - 4.227453533376265
(%i4) psi[2](0.75);
(%o4)                        - 5.30263321633764
(%i5) psi[2](3/4);
                                   1         3
(%o5)                         psi (-) + 4 %pi
                                 2 4
(%i6) float(%);
(%o6)                        - 5.30263321633764
</pre></div>

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<a name="maxpsiposint"></a><a name="Item_003a-Special_002fdefvr_002fmaxpsiposint"></a><dl>
<dt><a name="index-maxpsiposint"></a>Option variable: <strong>maxpsiposint</strong></dt>
<dd><p>Default value: 20
</p>
<p><code>maxpsiposint</code> is the largest positive integer value for
which 
\(\psi^{(n)}(m)\) gives an exact value for
rational <em>x</em>.
</p>
<div class="example">
<pre class="example">(%i1) psi[0](20);
                             275295799
(%o1)                        --------- - %gamma
                             77597520
(%i2) psi[0](21);
(%o2)                             psi (21)
                                     0
(%i3) psi[2](20);
                      1683118856778495358491487
(%o3)              2 (------------------------- - zeta(3))
                      1401731326612193601024000
(%i4) psi[2](21);
(%o4)                            psi (21)
                                      2

</pre></div>

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</dd></dl>

<a name="maxpsinegint"></a><a name="Item_003a-Special_002fdefvr_002fmaxpsinegint"></a><dl>
<dt><a name="index-maxpsinegint"></a>Option variable: <strong>maxpsinegint</strong></dt>
<dd><p>Default value: -10
</p>
<p><code>maxpsinegint</code> is the most negative value for
which 
\(\psi^{(0)}(x)\) will try to compute an exact
value for rational <em>x</em>.  That is if <em>x</em> is less than
<code>maxpsinegint</code>, 
\(\psi^{(n)}(x)\) will not
return simplified answer, even if it could.
</p>
<div class="example">
<pre class="example">(%i1) psi[0](-100/9);
                                        100
(%o1)                            psi (- ---)
                                    0    9
(%i2) psi[0](-100/11);
                         100 %pi         1     5231385863539
(%o2)            %pi cot(-------) + psi (--) + -------------
                           11          0 11    381905105400

(%i3) psi[2](-100/9);
                                        100
(%o3)                            psi (- ---)
                                    2    9
(%i4) psi[2](-100/11);
           3     100 %pi     2 100 %pi         1
(%o4) 2 %pi  cot(-------) csc (-------) + psi (--)
                   11            11          2 11
                                         74191313259470963498957651385614962459
                                       + --------------------------------------
                                          27850718060013605318710152732000000
</pre></div>
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</dd></dl>

<a name="maxpsifracnum"></a><a name="Item_003a-Special_002fdefvr_002fmaxpsifracnum"></a><dl>
<dt><a name="index-maxpsifracnum"></a>Option variable: <strong>maxpsifracnum</strong></dt>
<dd><p>Default value: 6
</p>
<p>Let <em>x</em> be a rational number of the form <em>p/q</em>.
If <em>p</em> is greater than <code>maxpsifracnum</code>,
then 
\(\psi^{(0)}(x)\) will not try to
return a simplified value.
</p>
<div class="example">
<pre class="example">(%i1) psi[0](3/4);
                                        %pi
(%o1)                    (- 3 log(2)) + --- - %gamma
                                         2
(%i2) psi[2](3/4);
                                   1         3
(%o2)                         psi (-) + 4 %pi
                                 2 4
(%i3) maxpsifracnum:2;
(%o3)                                 2
(%i4) psi[0](3/4);
                                        3
(%o4)                              psi (-)
                                      0 4
(%i5) psi[2](3/4);
                                   1         3
(%o5)                         psi (-) + 4 %pi
                                 2 4
</pre></div>

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&middot;</div>
</dd></dl>

<a name="maxpsifracdenom"></a><a name="Item_003a-Special_002fdefvr_002fmaxpsifracdenom"></a><dl>
<dt><a name="index-maxpsifracdenom"></a>Option variable: <strong>maxpsifracdenom</strong></dt>
<dd><p>Default value: 6
</p>
<p>Let <em>x</em> be a rational number of the form <em>p/q</em>.
If <em>q</em> is greater than <code>maxpsifracdenom</code>,
then 
\(\psi^{(0)}(x)\) will
not try to return a simplified value.
</p>
<div class="example">
<pre class="example">(%i1) psi[0](3/4);
                                        %pi
(%o1)                    (- 3 log(2)) + --- - %gamma
                                         2
(%i2) psi[2](3/4);
                                   1         3
(%o2)                         psi (-) + 4 %pi
                                 2 4
(%i3) maxpsifracdenom:2;
(%o3)                                 2
(%i4) psi[0](3/4);
                                        3
(%o4)                              psi (-)
                                      0 4
(%i5) psi[2](3/4);
                                   1         3
(%o5)                         psi (-) + 4 %pi
                                 2 4
</pre></div>

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&middot;</div>
</dd></dl>

<a name="makefact"></a><a name="Item_003a-Special_002fdeffn_002fmakefact"></a><dl>
<dt><a name="index-makefact"></a>Function: <strong>makefact</strong> <em>(<var>expr</var>)</em></dt>
<dd><p>Transforms instances of binomial, gamma, and beta
functions in <var>expr</var> into factorials.
</p>
<p>See also <code><a href="#makegamma">makegamma</a></code>.
</p>
<div class="example">
<pre class="example">(%i1) makefact(binomial(n,k));
                                      n!
(%o1)                             -----------
                                  k! (n - k)!
(%i2) makefact(gamma(x));
(%o2)                              (x - 1)!
(%i3) makefact(beta(a,b));
                               (a - 1)! (b - 1)!
(%o3)                          -----------------
                                 (b + a - 1)!
</pre></div>

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<a name="Item_003a-Special_002fdeffn_002fnumfactor"></a><dl>
<dt><a name="index-numfactor"></a>Function: <strong>numfactor</strong> <em>(<var>expr</var>)</em></dt>
<dd><p>Returns the numerical factor multiplying the expression
<var>expr</var>, which should be a single term.
</p>
<p><code><a href="maxima_80.html#content">content</a></code> returns the greatest common divisor (gcd) of all terms in a sum.
</p>
<div class="example">
<pre class="example">(%i1) gamma (7/2);
                          15 sqrt(%pi)
(%o1)                     ------------
                               8
(%i2) numfactor (%);
                               15
(%o2)                          --
                               8
</pre></div>

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