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<h1 class="chapter"> 15. Special Functions </h1>
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<tr><td align="left" valign="top"><a href="#SEC78">15.1 Introduction to Special Functions</a></td><td> </td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC79">15.2 Bessel Functions</a></td><td> </td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC80">15.3 Airy Functions</a></td><td> </td><td align="left" valign="top">
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<tr><td align="left" valign="top"><a href="#SEC81">15.4 Gamma and factorial Functions</a></td><td> </td><td align="left" valign="top">
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<tr><td align="left" valign="top"><a href="#SEC82">15.5 Exponential Integrals</a></td><td> </td><td align="left" valign="top">
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<tr><td align="left" valign="top"><a href="#SEC83">15.6 Error Function</a></td><td> </td><td align="left" valign="top">
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<tr><td align="left" valign="top"><a href="#SEC84">15.7 Struve Functions</a></td><td> </td><td align="left" valign="top">
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<tr><td align="left" valign="top"><a href="#SEC85">15.8 Hypergeometric Functions</a></td><td> </td><td align="left" valign="top">
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<tr><td align="left" valign="top"><a href="#SEC86">15.9 Parabolic Cylinder Functions</a></td><td> </td><td align="left" valign="top">
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<tr><td align="left" valign="top"><a href="#SEC87">15.10 Functions and Variables for Special Functions</a></td><td> </td><td align="left" valign="top">
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<h2 class="section"> 15.1 Introduction to Special Functions </h2>
<p>Special function notation follows:
</p>
<pre class="example">bessel_j (index, expr) Bessel function, 1st kind
bessel_y (index, expr) Bessel function, 2nd kind
bessel_i (index, expr) Modified Bessel function, 1st kind
bessel_k (index, expr) Modified Bessel function, 2nd kind
hankel_1 (v,z) Hankel function of the 1st kind
hankel_2 (v,z) Hankel function of the 2nd kind
struve_h (v,z) Struve H function
struve_l (v,z) Struve L function
assoc_legendre_p[v,u] (z) Legendre function of degree v and order u
assoc_legendre_q[v,u] (z) Legendre function, 2nd kind
%f[p,q] ([], [], expr) Generalized Hypergeometric function
gamma (z) Gamma function
gamma_greek (a,z) Incomplete gamma function
gamma_incomplete (a,z) Tail of incomplete gamma function
hypergeometric (l1, l2, z) Hypergeometric function
slommel
%m[u,k] (z) Whittaker function, 1st kind
%w[u,k] (z) Whittaker function, 2nd kind
erfc (z) Complement of the erf function
expintegral_e (v,z) Exponential integral E
expintegral_e1 (z) Exponential integral E1
expintegral_ei (z) Exponential integral Ei
expintegral_li (z) Logarithmic integral Li
expintegral_si (z) Exponential integral Si
expintegral_ci (z) Exponential integral Ci
expintegral_shi (z) Exponential integral Shi
expintegral_chi (z) Exponential integral Chi
kelliptic (z) Complete elliptic integral of the first
kind (K)
parabolic_cylinder_d (v,z) Parabolic cylinder D function
</pre>
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·
<p>@ref{Category: Bessel functions} ·
@ref{Category: Airy functions} ·
@ref{Category: Special functions}
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<h2 class="section"> 15.2 Bessel Functions </h2>
<p><a name="bessel_005fj"></a>
<a name="Item_003a-bessel_005fj"></a>
</p><dl>
<dt><u>Function:</u> <b>bessel_j</b><i> (<var>v</var>, <var>z</var>)</i>
<a name="IDX708"></a>
</dt>
<dd><p>The Bessel function of the first kind of order <em>v</em> and argument <em>z</em>.
</p>
<p><code>bessel_j</code> is defined as
</p>
<pre class="example"> inf
==== k - v - 2 k v + 2 k
\ (- 1) 2 z
> --------------------------
/ k! gamma(v + k + 1)
====
k = 0
</pre>
<p>although the infinite series is not used for computations.
</p>
<div class=categorybox>
·
<p>@ref{Category: Bessel functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="bessel_005fy"></a>
<a name="Item_003a-bessel_005fy"></a>
</p><dl>
<dt><u>Function:</u> <b>bessel_y</b><i> (<var>v</var>, <var>z</var>)</i>
<a name="IDX709"></a>
</dt>
<dd><p>The Bessel function of the second kind of order <em>v</em> and argument <em>z</em>.
</p>
<p><code>bessel_y</code> is defined as
</p><pre class="example"> cos(%pi v) bessel_j(v, z) - bessel_j(-v, z)
-------------------------------------------
sin(%pi v)
</pre>
<p>when <em>v</em> is not an integer. When <em>v</em> is an integer <em>n</em>,
the limit as <em>v</em> approaches <em>n</em> is taken.
</p>
<div class=categorybox>
·
<p>@ref{Category: Bessel functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="bessel_005fi"></a>
<a name="Item_003a-bessel_005fi"></a>
</p><dl>
<dt><u>Function:</u> <b>bessel_i</b><i> (<var>v</var>, <var>z</var>)</i>
<a name="IDX710"></a>
</dt>
<dd><p>The modified Bessel function of the first kind of order <em>v</em> and argument
<em>z</em>.
</p>
<p><code>bessel_i</code> is defined as
</p><pre class="example"> inf
==== - v - 2 k v + 2 k
\ 2 z
> -------------------
/ k! gamma(v + k + 1)
====
k = 0
</pre>
<p>although the infinite series is not used for computations.
</p>
<div class=categorybox>
·
<p>@ref{Category: Bessel functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="bessel_005fk"></a>
<a name="Item_003a-bessel_005fk"></a>
</p><dl>
<dt><u>Function:</u> <b>bessel_k</b><i> (<var>v</var>, <var>z</var>)</i>
<a name="IDX711"></a>
</dt>
<dd><p>The modified Bessel function of the second kind of order <em>v</em> and argument
<em>z</em>.
</p>
<p><code>bessel_k</code> is defined as
</p><pre class="example"> %pi csc(%pi v) (bessel_i(-v, z) - bessel_i(v, z))
-------------------------------------------------
2
</pre>
<p>when <em>v</em> is not an integer. If <em>v</em> is an integer <em>n</em>,
then the limit as <em>v</em> approaches <em>n</em> is taken.
</p>
<div class=categorybox>
·
<p>@ref{Category: Bessel functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="hankel_005f1"></a>
<a name="Item_003a-hankel_005f1"></a>
</p><dl>
<dt><u>Function:</u> <b>hankel_1</b><i> (<var>v</var>, <var>z</var>)</i>
<a name="IDX712"></a>
</dt>
<dd><p>The Hankel function of the first kind of order <em>v</em> and argument <em>z</em>
(A&S 9.1.3). <code>hankel_1</code> is defined as
</p>
<pre class="example"> bessel_j(v,z) + %i * bessel_y(v,z)
</pre>
<p>Maxima evaluates <code>hankel_1</code> numerically for a real order <em>v</em> and
complex argument <em>z</em> in float precision. The numerical evaluation in
bigfloat precision and for a complex order <em>v</em> is not supported.
</p>
<p>When <code>besselexpand</code> is <code>true</code>, <code>hankel_1</code> is expanded in terms
of elementary functions when the order <em>v</em> is half of an odd integer.
See <code>besselexpand</code>.
</p>
<p>Maxima knows the derivative of <code>hankel_1</code> wrt the argument <em>z</em>.
</p>
<p>Examples:
</p>
<p>Numerical evaluation:
</p>
<pre class="example">(%i1) hankel_1(1,0.5);
(%o1) .2422684576748738 - 1.471472392670243 %i
(%i2) hankel_1(1,0.5+%i);
(%o2) - .2558287994862166 %i - 0.239575601883016
</pre>
<p>A complex order <em>v</em> is not supported. Maxima returns a noun form:
</p>
<pre class="example">(%i3) hankel_1(%i,0.5+%i);
(%o3) hankel_1(%i, %i + 0.5)
</pre>
<p>Expansion of <code>hankel_1</code> when <code>besselexpand</code> is <code>true</code>:
</p>
<pre class="example">(%i4) hankel_1(1/2,z),besselexpand:true;
sqrt(2) sin(z) - sqrt(2) %i cos(z)
(%o4) ----------------------------------
sqrt(%pi) sqrt(z)
</pre>
<p>Derivative of <code>hankel_1</code> wrt the argument <em>z</em>. The derivative wrt the
order <em>v</em> is not supported. Maxima returns a noun form:
</p>
<pre class="example">(%i5) diff(hankel_1(v,z),z);
hankel_1(v - 1, z) - hankel_1(v + 1, z)
(%o5) ---------------------------------------
2
(%i6) diff(hankel_1(v,z),v);
d
(%o6) -- (hankel_1(v, z))
dv
</pre>
<div class=categorybox>
·
<p>@ref{Category: Bessel functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="hankel_005f2"></a>
<a name="Item_003a-hankel_005f2"></a>
</p><dl>
<dt><u>Function:</u> <b>hankel_2</b><i> (<var>v</var>, <var>z</var>)</i>
<a name="IDX713"></a>
</dt>
<dd><p>The Hankel function of the second kind of order <em>v</em> and argument <em>z</em>
(A&S 9.1.4). <code>hankel_2</code> is defined as
</p>
<pre class="example"> bessel_j(v,z) - %i * bessel_y(v,z)
</pre>
<p>Maxima evaluates <code>hankel_2</code> numerically for a real order <em>v</em> and
complex argument <em>z</em> in float precision. The numerical evaluation in
bigfloat precision and for a complex order <em>v</em> is not supported.
</p>
<p>When <code>besselexpand</code> is <code>true</code>, <code>hankel_2</code> is expanded in terms
of elementary functions when the order <em>v</em> is half of an odd integer.
See <code>besselexpand</code>.
</p>
<p>Maxima knows the derivative of <code>hankel_2</code> wrt the argument <em>z</em>.
</p>
<p>For examples see <code>hankel_1</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Bessel functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="besselexpand"></a>
<a name="Item_003a-besselexpand"></a>
</p><dl>
<dt><u>Option variable:</u> <b>besselexpand</b>
<a name="IDX714"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>Controls expansion of the Bessel functions when the order is half of
an odd integer. In this case, the Bessel functions can be expanded
in terms of other elementary functions. When <code>besselexpand</code> is <code>true</code>,
the Bessel function is expanded.
</p>
<pre class="example">(%i1) besselexpand: false$
(%i2) bessel_j (3/2, z);
3
(%o2) bessel_j(-, z)
2
(%i3) besselexpand: true$
(%i4) bessel_j (3/2, z);
sin(z) cos(z)
sqrt(2) sqrt(z) (------ - ------)
2 z
z
(%o4) ---------------------------------
sqrt(%pi)
</pre>
<div class=categorybox>
·
<p>@ref{Category: Bessel functions} ·
@ref{Category: Simplification flags and variables}
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-scaled_005fbessel_005fi"></a>
</p><dl>
<dt><u>Function:</u> <b>scaled_bessel_i</b><i> (<var>v</var>, <var>z</var>) </i>
<a name="IDX715"></a>
</dt>
<dd><p>The scaled modified Bessel function of the first kind of order
<em>v</em> and argument <em>z</em>. That is, <em>scaled_bessel_i(v,z) =
exp(-abs(z))*bessel_i(v, z)</em>. This function is particularly useful
for calculating <em>bessel_i</em> for large <em>z</em>, which is large.
However, maxima does not otherwise know much about this function. For
symbolic work, it is probably preferable to work with the expression
<code>exp(-abs(z))*bessel_i(v, z)</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Bessel functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-scaled_005fbessel_005fi0"></a>
</p><dl>
<dt><u>Function:</u> <b>scaled_bessel_i0</b><i> (<var>z</var>) </i>
<a name="IDX716"></a>
</dt>
<dd><p>Identical to <code>scaled_bessel_i(0,z)</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Bessel functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-scaled_005fbessel_005fi1"></a>
</p><dl>
<dt><u>Function:</u> <b>scaled_bessel_i1</b><i> (<var>z</var>)</i>
<a name="IDX717"></a>
</dt>
<dd><p>Identical to <code>scaled_bessel_i(1,z)</code>.
<div class=categorybox>
·
@ref{Category: Bessel functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-_0025s"></a>
</p><dl>
<dt><u>Function:</u> <b>%s</b><i> [<var>u</var>,<var>v</var>] (<var>z</var>) </i>
<a name="IDX718"></a>
</dt>
<dd><p>Lommel's little s[u,v](z) function.
Probably Gradshteyn & Ryzhik 8.570.1.
<div class=categorybox>
·
@ref{Category: Bessel functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-Airy-Functions"></a>
</p><hr size="6">
<a name="Airy-Functions"></a>
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<h2 class="section"> 15.3 Airy Functions </h2>
<p>The Airy functions Ai(x) and Bi(x) are defined in Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, Section 10.4.
</p>
<p><code>y = Ai(x)</code> and <code>y = Bi(x)</code> are two linearly independent solutions
of the Airy differential equation <code>diff (y(x), x, 2) - x y(x) = 0</code>.
</p>
<p>If the argument <code>x</code> is a real or complex floating point
number, the numerical value of the function is returned.
</p>
<p><a name="Item_003a-airy_005fai"></a>
</p><dl>
<dt><u>Function:</u> <b>airy_ai</b><i> (<var>x</var>)</i>
<a name="IDX719"></a>
</dt>
<dd><p>The Airy function Ai(x). (A&S 10.4.2)
</p>
<p>The derivative <code>diff (airy_ai(x), x)</code> is <code>airy_dai(x)</code>.
</p>
<p>See also <code>airy_bi</code>, <code>airy_dai</code>, <code>airy_dbi</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Airy functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-airy_005fdai"></a>
</p><dl>
<dt><u>Function:</u> <b>airy_dai</b><i> (<var>x</var>)</i>
<a name="IDX720"></a>
</dt>
<dd><p>The derivative of the Airy function Ai <code>airy_ai(x)</code>.
</p>
<p>See <code>airy_ai</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Airy functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-airy_005fbi"></a>
</p><dl>
<dt><u>Function:</u> <b>airy_bi</b><i> (<var>x</var>)</i>
<a name="IDX721"></a>
</dt>
<dd><p>The Airy function Bi(x). (A&S 10.4.3)
</p>
<p>The derivative <code>diff (airy_bi(x), x)</code> is <code>airy_dbi(x)</code>.
</p>
<p>See <code>airy_ai</code>, <code>airy_dbi</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Airy functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-airy_005fdbi"></a>
</p><dl>
<dt><u>Function:</u> <b>airy_dbi</b><i> (<var>x</var>)</i>
<a name="IDX722"></a>
</dt>
<dd><p>The derivative of the Airy Bi function <code>airy_bi(x)</code>.
</p>
<p>See <code>airy_ai</code> and <code>airy_bi</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Airy functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-Gamma-and-factorial-Functions"></a>
</p><hr size="6">
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<h2 class="section"> 15.4 Gamma and factorial Functions </h2>
<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>
<p><a name="bffac"></a>
<a name="Item_003a-bffac"></a>
</p><dl>
<dt><u>Function:</u> <b>bffac</b><i> (<var>expr</var>, <var>n</var>)</i>
<a name="IDX723"></a>
</dt>
<dd><p>Bigfloat version of the factorial (shifted gamma)
function. The second argument is how many digits to retain and return,
it's a good idea to request a couple of extra.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Numerical evaluation}
</div>
</p></dd></dl>
<p><a name="bfpsi"></a>
<a name="Item_003a-bfpsi"></a>
</p><dl>
<dt><u>Function:</u> <b>bfpsi</b><i> (<var>n</var>, <var>z</var>, <var>fpprec</var>)</i>
<a name="IDX724"></a>
</dt>
<dt><u>Function:</u> <b>bfpsi0</b><i> (<var>z</var>, <var>fpprec</var>)</i>
<a name="IDX725"></a>
</dt>
<dd><p><code>bfpsi</code> is the polygamma function of real argument <var>z</var> and integer
order <var>n</var>. <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=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Numerical evaluation}
</div>
</p></dd></dl>
<p><a name="cbffac"></a>
<a name="Item_003a-cbffac"></a>
</p><dl>
<dt><u>Function:</u> <b>cbffac</b><i> (<var>z</var>, <var>fpprec</var>)</i>
<a name="IDX726"></a>
</dt>
<dd><p>Complex bigfloat factorial.
</p>
<p><code>load ("bffac")</code> loads this function.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Complex variables} ·
@ref{Category: Numerical evaluation}
</div>
</p></dd></dl>
<p><a name="gamma"></a>
<a name="Item_003a-gamma"></a>
</p><dl>
<dt><u>Function:</u> <b>gamma</b><i> (<var>z</var>)</i>
<a name="IDX727"></a>
</dt>
<dd><p>The basic definition of the gamma function (A&S 6.1.1) is
</p>
<pre class="example"> inf
/
[ z - 1 - t
gamma(z) = I t %e dt
]
/
0
</pre>
<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
<code>sqrt(%pi)</code>. 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 evalute <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>gamma_expand</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 derivate of <code>gamma</code>.
</p>
<p>Examples:
</p>
<p>Simplification for integer, half integral, and rational numbers:
</p>
<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>
<p>Numerical evaluation for real and complex values:
</p>
<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>
<p><code>gamma</code> has mirror symmetry:
</p>
<pre class="example">(%i6) declare(z,complex)$
(%i7) conjugate(gamma(z));
(%o7) gamma(conjugate(z))
</pre>
<p>Maxima expands <code>gamma(z+n)</code> and <code>gamma(z-n)</code>, when <code>gamma_expand</code>
is <code>true</code>:
</p>
<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>
<p>The deriviative of <code>gamma</code>:
</p>
<pre class="example">(%i10) diff(gamma(z),z);
(%o10) psi (z) gamma(z)
0
</pre>
<p>See also <code>makegamma</code>.
</p>
<p>The Euler-Mascheroni constant is <code>%gamma</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-log_005fgamma"></a>
</p><dl>
<dt><u>Function:</u> <b>log_gamma</b><i> (<var>z</var>)</i>
<a name="IDX728"></a>
</dt>
<dd><p>The natural logarithm of the gamma function.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-gamma_005fgreek"></a>
</p><dl>
<dt><u>Function:</u> <b>gamma_greek</b><i> (<var>a</var>, <var>z</var>)</i>
<a name="IDX729"></a>
</dt>
<dd><p>The lower incomplete gamma function (A&S 6.5.2):
</p>
<pre class="example"> z
/
[ a - 1 - t
gamma_greek(a, z) = I t %e dt
]
/
0
</pre>
<p>See also <code>gamma_incomplete</code> (upper incomplete gamma function).
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-gamma_005fincomplete"></a>
</p><dl>
<dt><u>Function:</u> <b>gamma_incomplete</b><i> (<var>a</var>, <var>z</var>)</i>
<a name="IDX730"></a>
</dt>
<dd><p>The incomplete upper gamma function A&S 6.5.3:
</p>
<pre class="example"> inf
/
[ a - 1 - t
gamma_incomplete(a, z) = I t %e dt
]
/
z
</pre>
<p>See also <code>gamma_expand</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>gamma_incomplete_regularized</code> and
<code>gamma_incomplete_generalized</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-gamma_005fincomplete_005fregularized"></a>
</p><dl>
<dt><u>Function:</u> <b>gamma_incomplete_regularized</b><i> (<var>a</var>, <var>z</var>)</i>
<a name="IDX731"></a>
</dt>
<dd><p>The regularized incomplete upper gamma function A&S 6.5.1:
</p>
<pre class="example">gamma_incomplete_regularized(a, z) =
gamma_incomplete(a, z)
----------------------
gamma(a)
</pre>
<p>See also <code>gamma_expand</code> for controlling how
<code>gamma_incomplete</code> is expressed in terms of elementary functions
and <code>erfc</code>.
</p>
<p>Also see <code>gamma_incomplete</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-gamma_005fincomplete_005fgeneralized"></a>
</p><dl>
<dt><u>Function:</u> <b>gamma_incomplete_generalized</b><i> (<var>a</var>, <var>z1</var>, <var>z1</var>)</i>
<a name="IDX732"></a>
</dt>
<dd><p>The generalized incomplete gamma function.
</p>
<pre class="example">gamma_incomplete_generalized(a, z1, z2) =
z2
/
[ a - 1 - t
I t %e dt
]
/
z1
</pre>
<p>Also see <code>gamma_incomplete</code> and <code>gamma_incomplete_regularized</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-gamma_005fexpand"></a>
</p><dl>
<dt><u>Option variable:</u> <b>gamma_expand</b>
<a name="IDX733"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p><code>gamma_expand</code> controls expansion of <code>gamma_incomplete</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>erfc(z)</code> when possible.
</p>
<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
(%i4) gamma_incomplete(3/2,z);
- z sqrt(%pi) erfc(sqrt(z))
(%o4) sqrt(z) %e + -----------------------
2
</pre>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Simplification flags and variables}
</div>
</p>
</dd></dl>
<p><a name="gammalim"></a>
<a name="Item_003a-gammalim"></a>
</p><dl>
<dt><u>Option variable:</u> <b>gammalim</b>
<a name="IDX734"></a>
</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>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Simplification flags and variables}
</div>
</p>
</dd></dl>
<p><a name="makegamma"></a>
<a name="Item_003a-makegamma"></a>
</p><dl>
<dt><u>Function:</u> <b>makegamma</b><i> (<var>expr</var>)</i>
<a name="IDX735"></a>
</dt>
<dd><p>Transforms instances of binomial, factorial, and beta
functions in <var>expr</var> into gamma functions.
</p>
<p>See also <code>makefact</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-beta"></a>
</p><dl>
<dt><u>Function:</u> <b>beta</b><i> (<var>a</var>, <var>b</var>)</i>
<a name="IDX736"></a>
</dt>
<dd><p>The beta function is defined as <code>gamma(a) gamma(b)/gamma(a+b)</code>
(A&S 6.2.1).
</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 <var>b<=n</var>, it is possible to define an
analytic continuation. Maxima gives for this case a result.
</p>
<p>When <code>beta_expand</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>
<pre class="example"> - log_gamma(b + a) + log_gamma(b) + log_gamma(a)
%e
</pre>
<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>
<pre class="example">(%i1) [beta(2,3),beta(2,1/3),beta(2,a)];
1 9 1
(%o1) [--, -, ---------]
12 4 a (a + 1)
</pre>
<p>Simplification for two rational numbers as arguments which sum to an integer:
</p>
<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>
<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>
<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>
<p>The possible results, when one of the arguments is a negative integer:
</p>
<pre class="example">(%i5) [beta(-3,1),beta(-3,2),beta(-3,3)];
1 1 1
(%o5) [- -, -, - -]
3 6 3
</pre>
<p><code>beta(a+n,b)</code> or <code>beta(a-n)</code> with <code>n</code> an integer simplifies when
<code>beta_expand</code> is <code>true</code>:
</p>
<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>
<p>Beta is not defined, when one of the arguments is zero:
</p>
<pre class="example">(%i7) beta(0,b);
beta: expected nonzero arguments; found 0, b
-- an error. To debug this try debugmode(true);
</pre>
<p>Numercial evaluation for real and complex arguments in float or bigfloat
precision:
</p>
<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>
<p>Beta is symmetric and has mirror symmetry:
</p>
<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>
<p>The derivative of the beta function wrt <code>a</code>:
</p><pre class="example">(%i17) diff(beta(a,b),a);
(%o17) - beta(a, b) (psi (b + a) - psi (a))
0 0
</pre>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-beta_005fincomplete"></a>
</p><dl>
<dt><u>Function:</u> <b>beta_incomplete</b><i> (<var>a</var>, <var>b</var>, <var>z</var>)</i>
<a name="IDX737"></a>
</dt>
<dd><p>The basic definition of the incomplete beta function (A&S 6.6.1) is
</p>
<pre class="example"> z
/
[ b - 1 a - 1
I (1 - t) t dt
]
/
0
</pre>
<p>This definition is possible for <em>realpart(a)>0</em> and <em>realpart(b)>0</em>
and <em>abs(z)<1</em>. For other values the incomplete beta function can be
defined through a generalized hypergeometric function:
</p>
<pre class="example"> gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z
</pre>
<p>(See functions.wolfram.com 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
<em>m<=n</em> the incomplete beta function is defined through
</p>
<pre class="example"> m - 1 k
==== (1 - m) z
n - 1 \ k
z > -----------
/ k! (n - k)
====
k = 0
</pre>
<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 <em>realpart(a)>0</em>, <code>beta_incomplete</code> has the
specific value zero. For <var>z=1</var> and <em>realpart(b)>0</em>,
<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>beta_expand</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>
<pre class="example">(%i1) beta_incomplete(2,b,z);
b
1 - (1 - z) (b z + 1)
(%o1) ----------------------
b (b + 1)
</pre>
<p>Simplification for <var>b</var> a positive integer:
</p>
<pre class="example">(%i2) beta_incomplete(a,2,z);
a
(a (1 - z) + 1) z
(%o2) ------------------
a (a + 1)
</pre>
<p>Simplification for <var>a</var> and <var>b</var> a positive integer:
</p>
<pre class="example">(%i3) beta_incomplete(3,2,z);
3
(3 (1 - z) + 1) z
(%o3) ------------------
12
</pre>
<p><var>a</var> is a negative integer and <em>b<=(-a)</em>, Maxima simplifies:
</p>
<pre class="example">(%i4) beta_incomplete(-3,1,z);
1
(%o4) - ----
3
3 z
</pre>
<p>For the specific values <em>z=0</em> and <em>z=1</em>, Maxima simplifies:
</p>
<pre class="example">(%i5) assume(a>0,b>0)$
(%i6) beta_incomplete(a,b,0);
(%o6) 0
(%i7) beta_incomplete(a,b,1);
(%o7) beta(a, b)
</pre>
<p>Numerical evaluation in float or bigfloat precision:
</p>
<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>
<p>For <em>abs(z)>1</em> <code>beta_incomplete</code> returns a complex result:
</p>
<pre class="example">(%i11) beta_incomplete(0.25,0.50,1.7);
(%o11) 5.244115108584249 - 1.45518047787844 %i
</pre>
<p>Results for more general complex arguments:
</p>
<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>
<p>Expansion, when <code>beta_expand</code> is <code>true</code>:
</p>
<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>
<p>Derivative and integral for <code>beta_incomplete</code>:
</p>
<pre class="example">(%i34) diff(beta_incomplete(a, b, z), z);
b - 1 a - 1
(%o34) (1 - z) z
(%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 class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-beta_005fincomplete_005fregularized"></a>
</p><dl>
<dt><u>Function:</u> <b>beta_incomplete_regularized</b><i> (<var>a</var>, <var>b</var>, <var>z</var>)</i>
<a name="IDX738"></a>
</dt>
<dd><p>The regularized incomplete beta function A&S 6.6.2, defined as
</p>
<pre class="example">beta_incomplete_regularized(a, b, z) =
beta_incomplete(a, b, z)
------------------------
beta(a, b)
</pre>
<p>As for <code>beta_incomplete</code> this definition is not complete. See
functions.wolfram.com 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 <em>realpart(a)>0</em>, <code>beta_incomplete_regularized</code> has
the specific value 0. For <var>z=1</var> and <em>realpart(b)>0</em>,
<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>beta_expand</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>
<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>
<p>For the specific values <em>z=0</em> and <em>z=1</em>, Maxima simplifies:
</p>
<pre class="example">(%i4) assume(a>0,b>0)$
(%i5) beta_incomplete_regularized(a,b,0);
(%o5) 0
(%i6) beta_incomplete_regularized(a,b,1);
(%o6) 1
</pre>
<p>Numerical evaluation for real and complex arguments in float and bigfloat
precision:
</p>
<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>
<p>Expansion, when <code>beta_expand</code> is <code>true</code>:
</p>
<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>
<p>The derivative and the integral wrt <var>z</var>:
</p>
<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 class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-beta_005fincomplete_005fgeneralized"></a>
</p><dl>
<dt><u>Function:</u> <b>beta_incomplete_generalized</b><i> (<var>a</var>, <var>b</var>, <var>z1</var>, <var>z2</var>)</i>
<a name="IDX739"></a>
</dt>
<dd><p>The basic definition of the generalized incomplete beta function is
</p>
<pre class="example"> z2
/
[ b - 1 a - 1
I (1 - t) t dt
]
/
z1
</pre>
<p>Maxima simplifies <code>beta_incomplete_regularized</code> for <var>a</var> and <var>b</var>
a positive integer.
</p>
<p>For <em>realpart(a)>0</em> and <em>z1=0</em> or <em>z2=0</em>, Maxima simplifies
<code>beta_incomplete_generalized</code> to <code>beta_incomplete</code>. For
<em>realpart(b)>0</em> and <em>z1=1</em> or <var>z2=1</var>, 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>beta_expand</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>
<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);
a a
(a (1 - z2) + 1) z2 - (a (1 - z1) + 1) z1
(%o2) -------------------------------------------
a (a + 1)
(%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>
<p>Simplification for specific values <em>z1=0</em>, <em>z2=0</em>, <em>z1=1</em>, or
<em>z2=1</em>:
</p>
<pre class="example">(%i4) assume(a > 0, b > 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>
<p>Numerical evaluation for real arguments in float or bigfloat precision:
</p>
<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>
<p>Numerical evaluation for complex arguments in float or bigfloat precision:
</p>
<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>
<p>Expansion for <em>a+n</em> or <em>a-n</em>, <var>n</var> a positive integer, when
<code>beta_expand</code> is <code>true</code>:
</p>
<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>
<p>Derivative wrt the variable <var>z1</var> and integrals wrt <var>z1</var> and <var>z2</var>:
</p>
<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 class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-beta_005fexpand"></a>
</p><dl>
<dt><u>Option variable:</u> <b>beta_expand</b>
<a name="IDX740"></a>
</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>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Simplification flags and variables}
</div>
</p></dd></dl>
<p><a name="Item_003a-beta_005fargs_005fsum_005fto_005finteger"></a>
</p><dl>
<dt><u>Option variable:</u> <b>beta_args_sum_to_integer</b>
<a name="IDX741"></a>
</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>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions} ·
@ref{Category: Simplification flags and variables}
</div>
</p></dd></dl>
<p><a name="Item_003a-psi"></a>
</p><dl>
<dt><u>Function:</u> <b>psi</b><i> [<var>n</var>](<var>x</var>)</i>
<a name="IDX742"></a>
</dt>
<dd><p>The derivative of <code>log (gamma (<var>x</var>))</code> of order <code><var>n</var>+1</code>.
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 does not know how, in general, to compute a numerical value of
<code>psi</code>, but it can compute some exact values for rational args.
Several variables control what range of rational args <code>psi</code> will
return an exact value, if possible. See <code>maxpsiposint</code>,
<code>maxpsinegint</code>, <code>maxpsifracnum</code>, and <code>maxpsifracdenom</code>.
That is, <var>x</var> must lie between <code>maxpsinegint</code> and
<code>maxpsiposint</code>. If the absolute value of the fractional part of
<var>x</var> is rational and has a numerator less than <code>maxpsifracnum</code>
and has a denominator less than <code>maxpsifracdenom</code>, <code>psi</code>
will return an exact value.
</p>
<p>The function <code>bfpsi</code> in the <code>bffac</code> package can compute
numerical values.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-maxpsiposint"></a>
</p><dl>
<dt><u>Option variable:</u> <b>maxpsiposint</b>
<a name="IDX743"></a>
</dt>
<dd><p>Default value: 20
</p>
<p><code>maxpsiposint</code> is the largest positive value for which
<code>psi[n](x)</code> will try to compute an exact value.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p>
</dd></dl>
<p><a name="Item_003a-maxpsinegint"></a>
</p><dl>
<dt><u>Option variable:</u> <b>maxpsinegint</b>
<a name="IDX744"></a>
</dt>
<dd><p>Default value: -10
</p>
<p><code>maxpsinegint</code> is the most negative value for which
<code>psi[n](x)</code> will try to compute an exact value. That is if
<var>x</var> is less than <code>maxnegint</code>, <code>psi[n](<var>x</var>)</code> will not
return simplified answer, even if it could.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p>
</dd></dl>
<p><a name="Item_003a-maxpsifracnum"></a>
</p><dl>
<dt><u>Option variable:</u> <b>maxpsifracnum</b>
<a name="IDX745"></a>
</dt>
<dd><p>Default value: 6
</p>
<p>Let <var>x</var> be a rational number less than one of the form <code>p/q</code>.
If <code>p</code> is greater than <code>maxpsifracnum</code>, then
<code>psi[<var>n</var>](<var>x</var>)</code> will not try to return a simplified
value.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p>
</dd></dl>
<p><a name="Item_003a-maxpsifracdenom"></a>
</p><dl>
<dt><u>Option variable:</u> <b>maxpsifracdenom</b>
<a name="IDX746"></a>
</dt>
<dd><p>Default value: 6
</p>
<p>Let <var>x</var> be a rational number less than one of the form <code>p/q</code>.
If <code>q</code> is greater than <code>maxpsifracdenom</code>, then
<code>psi[<var>n</var>](<var>x</var>)</code> will not try to return a simplified
value.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p>
</dd></dl>
<p><a name="Item_003a-makefact"></a>
</p><dl>
<dt><u>Function:</u> <b>makefact</b><i> (<var>expr</var>)</i>
<a name="IDX747"></a>
</dt>
<dd><p>Transforms instances of binomial, gamma, and beta
functions in <var>expr</var> into factorials.
</p>
<p>See also <code>makegamma</code>.
</p>
<div class=categorybox>
·
<p>@ref{Category: Gamma and factorial functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-numfactor"></a>
</p><dl>
<dt><u>Function:</u> <b>numfactor</b><i> (<var>expr</var>)</i>
<a name="IDX748"></a>
</dt>
<dd><p>Returns the numerical factor multiplying the expression
<var>expr</var>, which should be a single term.
</p>
<p><code>content</code> returns the greatest common divisor (gcd) of all terms in a sum.
</p>
<pre class="example">(%i1) gamma (7/2);
15 sqrt(%pi)
(%o1) ------------
8
(%i2) numfactor (%);
15
(%o2) --
8
</pre>
<div class=categorybox>
·
<p>@ref{Category: Expressions}
</div>
</p></dd></dl>
<p><a name="Item_003a-Exponential-Integrals"></a>
</p><hr size="6">
<a name="Exponential-Integrals"></a>
<a name="SEC82"></a>
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</tr></table>
<h2 class="section"> 15.5 Exponential Integrals </h2>
<p>The Exponential Integral and related funtions are defined in
Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, Chapter 5
</p>
<p><a name="Item_003a-expintegral_005fe1"></a>
</p><dl>
<dt><u>Function:</u> <b>expintegral_e1</b><i> (<var>z</var>)</i>
<a name="IDX749"></a>
</dt>
<dd><p>The Exponential Integral E1(z) (A&S 5.1.1)
<div class=categorybox>
·
@ref{Category: Exponential Integrals} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-expintegral_005fei"></a>
</p><dl>
<dt><u>Function:</u> <b>expintegral_ei</b><i> (<var>z</var>)</i>
<a name="IDX750"></a>
</dt>
<dd><p>The Exponential Integral Ei(z) (A&S 5.1.2)
<div class=categorybox>
·
@ref{Category: Exponential Integrals} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-expintegral_005fli"></a>
</p><dl>
<dt><u>Function:</u> <b>expintegral_li</b><i> (<var>z</var>)</i>
<a name="IDX751"></a>
</dt>
<dd><p>The Exponential Integral Li(z) (A&S 5.1.3)
<div class=categorybox>
·
@ref{Category: Exponential Integrals} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-expintegral_005fe"></a>
</p><dl>
<dt><u>Function:</u> <b>expintegral_e</b><i> (<var>n</var>,<var>z</var>)</i>
<a name="IDX752"></a>
</dt>
<dd><p>The Exponential Integral En(z) (A&S 5.1.4)
<div class=categorybox>
·
@ref{Category: Exponential Integrals} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-expintegral_005fsi"></a>
</p><dl>
<dt><u>Function:</u> <b>expintegral_si</b><i> (<var>z</var>)</i>
<a name="IDX753"></a>
</dt>
<dd><p>The Exponential Integral Si(z) (A&S 5.2.1)
<div class=categorybox>
·
@ref{Category: Exponential Integrals} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-expintegral_005fci"></a>
</p><dl>
<dt><u>Function:</u> <b>expintegral_ci</b><i> (<var>z</var>)</i>
<a name="IDX754"></a>
</dt>
<dd><p>The Exponential Integral Ci(z) (A&S 5.2.2)
<div class=categorybox>
·
@ref{Category: Exponential Integrals} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-expintegral_005fshi"></a>
</p><dl>
<dt><u>Function:</u> <b>expintegral_shi</b><i> (<var>z</var>)</i>
<a name="IDX755"></a>
</dt>
<dd><p>The Exponential Integral Shi(z) (A&S 5.2.3)
<div class=categorybox>
·
@ref{Category: Exponential Integrals} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-expintegral_005fchi"></a>
</p><dl>
<dt><u>Function:</u> <b>expintegral_chi</b><i> (<var>z</var>)</i>
<a name="IDX756"></a>
</dt>
<dd><p>The Exponential Integral Chi(z) (A&S 5.2.4)
<div class=categorybox>
·
@ref{Category: Exponential Integrals} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-expintrep"></a>
</p><dl>
<dt><u>Option variable:</u> <b>expintrep</b>
<a name="IDX757"></a>
</dt>
<dd><p>Default value: false
</p>
<p>Change the representation of the Exponential Integral to
gamma_incomplete, expintegral_e1, expintegral_ei,
expintegral_li, expintegral_trig, expintegral_hyp
<div class=categorybox>
·
@ref{Category: Exponential Integrals}
</div>
</p></dd></dl>
<p><a name="Item_003a-expintexpand"></a>
</p><dl>
<dt><u>Option variable:</u> <b>expintexpand</b>
<a name="IDX758"></a>
</dt>
<dd><p>Default value: false
</p>
<p>Expand the Exponential Integral E[n](z)
for half integral values in terms of Erfc or Erf and
for positive integers in terms of Ei
<div class=categorybox>
·
@ref{Category: Exponential Integrals}
</div>
</p></dd></dl>
<p><a name="Item_003a-Error-Function"></a>
</p><hr size="6">
<a name="Error-Function"></a>
<a name="SEC83"></a>
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</tr></table>
<h2 class="section"> 15.6 Error Function </h2>
<p>The Error function and related funtions are defined in
Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, Chapter 7
</p>
<p><a name="erf"></a>
<a name="Item_003a-erf"></a>
</p><dl>
<dt><u>Function:</u> <b>erf</b><i> (<var>z</var>)</i>
<a name="IDX759"></a>
</dt>
<dd><p>The Error Function erf(z) (A&S 7.1.1)
</p>
<p>See also flag <code>erfflag</code>.
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-erfc"></a>
</p><dl>
<dt><u>Function:</u> <b>erfc</b><i> (<var>z</var>)</i>
<a name="IDX760"></a>
</dt>
<dd><p>The Complementary Error Function erfc(z) (A&S 7.1.2)
</p>
<p><code>erfc(z) = 1-erf(z)</code>
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-erfi"></a>
</p><dl>
<dt><u>Function:</u> <b>erfi</b><i> (<var>z</var>)</i>
<a name="IDX761"></a>
</dt>
<dd><p>The Imaginary Error Function.
</p>
<p><code>erfi(z) = -%i*erf(%i*z)</code>
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-erf_005fgeneralized"></a>
</p><dl>
<dt><u>Function:</u> <b>erf_generalized</b><i> (<var>z1</var>,<var>z2</var>)</i>
<a name="IDX762"></a>
</dt>
<dd><p>Generalized Error function Erf(z1,z2)
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-fresnel_005fc"></a>
</p><dl>
<dt><u>Function:</u> <b>fresnel_c</b><i> (<var>z</var>)</i>
<a name="IDX763"></a>
</dt>
<dd><p>The Fresnel Integral C(z) = integrate(cos((%pi/2)*t^2),t,0,z). (A&S 7.3.1)
</p>
<p>The simplification fresnel_c(-x) = -fresnel_c(x) is applied when
flag <code>trigsign</code> is true.
</p>
<p>The simplification fresnel_c(%i*x) = %i*fresnel_c(x) is applied when
flag <code>%iargs</code> is true.
</p>
<p>See flags <code>erf_representation</code> and <code>hypergeometric_representation</code>.
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-fresnel_005fs"></a>
</p><dl>
<dt><u>Function:</u> <b>fresnel_s</b><i> (<var>z</var>)</i>
<a name="IDX764"></a>
</dt>
<dd><p>The Fresnel Integral S(z) = integrate(sin((%pi/2)*t^2),t,0,z). (A&S 7.3.2)
</p>
<p>The simplification fresnel_s(-x) = -fresnel_s(x) is applied when
flag <code>trigsign</code> is true.
</p>
<p>The simplification fresnel_s(%i*x) = -%i*fresnel_s(x) is applied when
flag <code>%iargs</code> is true.
</p>
<p>See flags <code>erf_representation</code> and <code>hypergeometric_representation</code>.
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-erf_005frepresentation"></a>
</p><dl>
<dt><u>Option variable:</u> <b>erf_representation</b>
<a name="IDX765"></a>
</dt>
<dd><p>Default value: false
</p>
<p>When T erfc, erfi, erf_generalized, fresnel_s
and fresnel_c are transformed to erf.
</p></dd></dl>
<p><a name="Item_003a-hypergeometric_005frepresentation"></a>
</p><dl>
<dt><u>Option variable:</u> <b>hypergeometric_representation</b>
<a name="IDX766"></a>
</dt>
<dd><p>Default value: false
</p>
<p>Enables transformation to a Hypergeometric
representation for fresnel_s and fresnel_c
</p></dd></dl>
<p><a name="Item_003a-Struve-Functions"></a>
</p><hr size="6">
<a name="Struve-Functions"></a>
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<h2 class="section"> 15.7 Struve Functions </h2>
<p>The Struve functions are defined in Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, Chapter 12.
</p>
<p><a name="Item_003a-struve_005fh"></a>
</p><dl>
<dt><u>Function:</u> <b>struve_h</b><i> (<var>v</var>, <var>z</var>)</i>
<a name="IDX767"></a>
</dt>
<dd><p>The Struve Function H of order v and argument z. (A&S 12.1.1)
</p>
<div class=categorybox>
·
<p>@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-struve_005fl"></a>
</p><dl>
<dt><u>Function:</u> <b>struve_l</b><i> (<var>v</var>, <var>z</var>)</i>
<a name="IDX768"></a>
</dt>
<dd><p>The Modified Struve Function L of order v and argument z. (A&S 12.2.1)
</p>
<div class=categorybox>
·
<p>@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-Hypergeometric-Functions"></a>
</p><hr size="6">
<a name="Hypergeometric-Functions"></a>
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<h2 class="section"> 15.8 Hypergeometric Functions </h2>
<p>The Hypergeometric Functions are defined in Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, Chapters 13 and 15.
</p>
<p>Maxima has very limited knowledge of these functions. They
can be returned from function <code>hgfred</code>.
</p>
<p><a name="Item_003a-_0025m"></a>
</p><dl>
<dt><u>Function:</u> <b>%m</b><i> [<var>k</var>,<var>u</var>] (<var>z</var>) </i>
<a name="IDX769"></a>
</dt>
<dd><p>Whittaker M function
<code>M[k,u](z) = exp(-z/2)*z^(1/2+u)*M(1/2+u-k,1+2*u,z)</code>.
(A&S 13.1.32)
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-_0025w"></a>
</p><dl>
<dt><u>Function:</u> <b>%w</b><i> [<var>k</var>,<var>u</var>] (<var>z</var>) </i>
<a name="IDX770"></a>
</dt>
<dd><p>Whittaker W function. (A&S 13.1.33)
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-_0025f"></a>
</p><dl>
<dt><u>Function:</u> <b>%f</b><i> [<var>p</var>,<var>q</var>] (<var>[a],[b],z</var>) </i>
<a name="IDX771"></a>
</dt>
<dd><p>The pFq(a1,a2,..ap;b1,b2,..bq;z) hypergeometric function,
where <code>a</code> a list of length <code>p</code> and
<code>b</code> a list of length <code>q</code>.
<div class=categorybox>
·
@ref{Category: Bessel functions} ·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-hypergeometric"></a>
</p><dl>
<dt><u>Function:</u> <b>hypergeometric</b><i> ([<var>a1</var>, ..., <var>ap</var>],[<var>b1</var>, ... ,<var>bq</var>], x)</i>
<a name="IDX772"></a>
</dt>
<dd><p>The hypergeometric function. Unlike Maxima's <code>%f</code> hypergeometric
function, the function <code>hypergeometric</code> is a simplifying
function; also, <code>hypergeometric</code> supports complex double and
big floating point evaluation. For the Gauss hypergeometric function,
that is <em>p = 2</em> and <em>q = 1</em>, floating point evaluation
outside the unit circle is supported, but in general, it is not
supported.
</p>
<p>When the option variable <code>expand_hypergeometric</code> is true (default
is false) and one of the arguments <code>a1</code> through <code>ap</code> is a
negative integer (a polynomial case), <code>hypergeometric</code> returns an
expanded polynomial.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) hypergeometric([],[],x);
(%o1) %e^x
</pre>
<p>Polynomial cases automatically expand when <code>expand_hypergeometric</code> is true:
</p>
<pre class="example">(%i2) hypergeometric([-3],[7],x);
(%o2) hypergeometric([-3],[7],x)
(%i3) hypergeometric([-3],[7],x), expand_hypergeometric : true;
(%o3) -x^3/504+3*x^2/56-3*x/7+1
</pre>
<p>Both double float and big float evaluation is supported:
</p>
<pre class="example">(%i4) hypergeometric([5.1],[7.1 + %i],0.42);
(%o4) 1.346250786375334 - 0.0559061414208204 %i
(%i5) hypergeometric([5,6],[8], 5.7 - %i);
(%o5) .007375824009774946 - .001049813688578674 %i
(%i6) hypergeometric([5,6],[8], 5.7b0 - %i), fpprec : 30;
(%o6) 7.37582400977494674506442010824b-3
- 1.04981368857867315858055393376b-3 %i
</pre></dd></dl>
<p><a name="Item_003a-Parabolic-Cylinder-Functions"></a>
</p><hr size="6">
<a name="Parabolic-Cylinder-Functions"></a>
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<h2 class="section"> 15.9 Parabolic Cylinder Functions </h2>
<p>The Parabolic Cylinder Functions are defined in Abramowitz and Stegun,
<i>Handbook of Mathematical Functions</i>, Chapter 19.
</p>
<p>Maxima has very limited knowledge of these functions. They
can be returned from function <code>hgfred</code>.
</p>
<p><a name="Item_003a-parabolic_005fcylinder_005fd"></a>
</p><dl>
<dt><u>Function:</u> <b>parabolic_cylinder_d</b><i> (<var>v</var>, <var>z</var>) </i>
<a name="IDX773"></a>
</dt>
<dd><p>The parabolic cylinder function <code>parabolic_cylinder_d(v,z)</code>. (A&S 19.3.1)
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-Functions-and-Variables-for-Special-Functions"></a>
</p><hr size="6">
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<h2 class="section"> 15.10 Functions and Variables for Special Functions </h2>
<p><a name="Item_003a-specint"></a>
</p><dl>
<dt><u>Function:</u> <b>specint</b><i> (exp(- s*<var>t</var>) * <var>expr</var>, <var>t</var>)</i>
<a name="IDX774"></a>
</dt>
<dd><p>Compute the Laplace transform of <var>expr</var> with respect to the variable <var>t</var>.
The integrand <var>expr</var> may contain special functions.
</p>
<p>The following special functions are handled by <code>specint</code>: incomplete gamma
function, error functions (but not the error function <code>erfi</code>, it is easy to
transform <code>erfi</code> e.g. to the error function <code>erf</code>), exponential
integrals, bessel functions (including products of bessel functions), hankel
functions, hermite and the laguerre polynomials.
</p>
<p>Furthermore, <code>specint</code> can handle the hypergeometric function
<code>%f[p,q]([],[],z)</code>, the whittaker function of the first kind
<code>%m[u,k](z)</code> and of the second kind <code>%w[u,k](z)</code>.
</p>
<p>The result may be in terms of special functions and can include unsimplified
hypergeometric functions.
</p>
<p>When <code>laplace</code> fails to find a Laplace transform, <code>specint</code> is called.
Because <code>laplace</code> knows more general rules for Laplace transforms, it is
preferable to use <code>laplace</code> and not <code>specint</code>.
</p>
<p><code>demo(hypgeo)</code> displays several examples of Laplace transforms computed by
<code>specint</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) assume (p > 0, a > 0)$
(%i2) specint (t^(1/2) * exp(-a*t/4) * exp(-p*t), t);
sqrt(%pi)
(%o2) ------------
a 3/2
2 (p + -)
4
(%i3) specint (t^(1/2) * bessel_j(1, 2 * a^(1/2) * t^(1/2))
* exp(-p*t), t);
- a/p
sqrt(a) %e
(%o3) ---------------
2
p
</pre>
<p>Examples for exponential integrals:
</p>
<pre class="example">(%i4) assume(s>0,a>0,s-a>0)$
(%i5) ratsimp(specint(%e^(a*t)
*(log(a)+expintegral_e1(a*t))*%e^(-s*t),t));
log(s)
(%o5) ------
s - a
(%i6) logarc:true$
(%i7) gamma_expand:true$
radcan(specint((cos(t)*expintegral_si(t)
-sin(t)*expintegral_ci(t))*%e^(-s*t),t));
log(s)
(%o8) ------
2
s + 1
ratsimp(specint((2*t*log(a)+2/a*sin(a*t)
-2*t*expintegral_ci(a*t))*%e^(-s*t),t));
2 2
log(s + a )
(%o9) ------------
2
s
</pre>
<p>Results when using the expansion of <code>gamma_incomplete</code> and when changing
the representation to <code>expintegral_e1</code>:
</p>
<pre class="example">(%i10) assume(s>0)$
(%i11) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
1
gamma_incomplete(-, k s)
2
(%o11) ------------------------
sqrt(%pi) sqrt(s)
(%i12) gamma_expand:true$
(%i13) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
erfc(sqrt(k) sqrt(s))
(%o13) ---------------------
sqrt(s)
(%i14) expintrep:expintegral_e1$
(%i15) ratsimp(specint(1/(t+a)^2*%e^(-s*t),t));
a s
a s %e expintegral_e1(a s) - 1
(%o15) - ---------------------------------
a
</pre>
<div class=categorybox>
·
<p>@ref{Category: Laplace transform}
</div>
</p></dd></dl>
<p><a name="Item_003a-hgfred"></a>
</p><dl>
<dt><u>Function:</u> <b>hgfred</b><i> (<var>a</var>, <var>b</var>, <var>t</var>)</i>
<a name="IDX775"></a>
</dt>
<dd><p>Simplify the generalized hypergeometric function in terms of other,
simpler, forms. <var>a</var> is a list of numerator parameters and <var>b</var>
is a list of the denominator parameters.
</p>
<p>If <code>hgfred</code> cannot simplify the hypergeometric function, it returns
an expression of the form <code>%f[p,q]([a], [b], x)</code> where <var>p</var> is
the number of elements in <var>a</var>, and <var>q</var> is the number of elements
in <var>b</var>. This is the usual <code>pFq</code> generalized hypergeometric
function.
</p>
<pre class="example">(%i1) assume(not(equal(z,0)));
(%o1) [notequal(z, 0)]
(%i2) hgfred([v+1/2],[2*v+1],2*%i*z);
v/2 %i z
4 bessel_j(v, z) gamma(v + 1) %e
(%o2) ---------------------------------------
v
z
(%i3) hgfred([1,1],[2],z);
log(1 - z)
(%o3) - ----------
z
(%i4) hgfred([a,a+1/2],[3/2],z^2);
1 - 2 a 1 - 2 a
(z + 1) - (1 - z)
(%o4) -------------------------------
2 (1 - 2 a) z
</pre>
<p>It can be beneficial to load orthopoly too as the following example
shows. Note that <var>L</var> is the generalized Laguerre polynomial.
</p>
<pre class="example">(%i5) load(orthopoly)$
(%i6) hgfred([-2],[a],z);
(a - 1)
2 L (z)
2
(%o6) -------------
a (a + 1)
(%i7) ev(%);
2
z 2 z
(%o7) --------- - --- + 1
a (a + 1) a
</pre></dd></dl>
<p><a name="Item_003a-lambert_005fw"></a>
</p><dl>
<dt><u>Function:</u> <b>lambert_w</b><i> (<var>z</var>)</i>
<a name="IDX776"></a>
</dt>
<dd><p>The principal branch of Lambert's W function W(z), the solution of
<code>z = W(z) * exp(W(z))</code>. (DLMF 4.13)
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-generalized_005flambert_005fw"></a>
</p><dl>
<dt><u>Function:</u> <b>generalized_lambert_w</b><i> (<var>k</var>, <var>z</var>)</i>
<a name="IDX777"></a>
</dt>
<dd><p>The <var>k</var>-th branch of Lambert's W function W(z), the solution of
<code>z = W(z) * exp(W(z))</code>. (DLMF 4.13)
</p>
<p>The principal branch, denoted Wp(z) in DLMF, is <code>lambert_w(z) = generalized_lambert_w(0,z)</code>.
</p>
<p>The other branch with real values, denoted Wm(z) in DLMF, is <code>generalized_lambert_w(-1,z)</code>.
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-nzeta"></a>
</p><dl>
<dt><u>Function:</u> <b>nzeta</b><i> (<var>z</var>)</i>
<a name="IDX778"></a>
</dt>
<dd><p>The Plasma Dispersion Function
<code>nzeta(z) = %i*sqrt(%pi)*exp(-z^2)*(1-erf(-%i*z))</code>
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-nzetar"></a>
</p><dl>
<dt><u>Function:</u> <b>nzetar</b><i> (<var>z</var>)</i>
<a name="IDX779"></a>
</dt>
<dd><p>Returns <code>realpart(nzeta(z))</code>.
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-nzetai"></a>
</p><dl>
<dt><u>Function:</u> <b>nzetai</b><i> (<var>z</var>)</i>
<a name="IDX780"></a>
</dt>
<dd><p>Returns <code>imagpart(nzeta(z))</code>.
<div class=categorybox>
·
@ref{Category: Special functions}
</div>
</p></dd></dl>
<p><a name="Item_003a-Elliptic-Functions"></a>
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