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<a name="Special-Functions"></a>
<p>
Next: <a rel="next" accesskey="n" href="Coordinate-Transformations.html#Coordinate-Transformations">Coordinate Transformations</a>,
Previous: <a rel="previous" accesskey="p" href="Utility-Functions.html#Utility-Functions">Utility Functions</a>,
Up: <a rel="up" accesskey="u" href="Arithmetic.html#Arithmetic">Arithmetic</a>
<hr>
</div>
<h3 class="section">17.6 Special Functions</h3>
<!-- ./DLD-FUNCTIONS/besselj.cc -->
<p><a name="doc_002dairy"></a>
<div class="defun">
— Loadable Function: [<var>a</var>, <var>ierr</var>] = <b>airy</b> (<var>k, z, opt</var>)<var><a name="index-airy-1482"></a></var><br>
<blockquote><p>Compute Airy functions of the first and second kind, and their
derivatives.
<pre class="example"> K Function Scale factor (if 'opt' is supplied)
--- -------- ---------------------------------------
0 Ai (Z) exp ((2/3) * Z * sqrt (Z))
1 dAi(Z)/dZ exp ((2/3) * Z * sqrt (Z))
2 Bi (Z) exp (-abs (real ((2/3) * Z *sqrt (Z))))
3 dBi(Z)/dZ exp (-abs (real ((2/3) * Z *sqrt (Z))))
</pre>
<p>The function call <code>airy (</code><var>z</var><code>)</code> is equivalent to
<code>airy (0, </code><var>z</var><code>)</code>.
<p>The result is the same size as <var>z</var>.
<p>If requested, <var>ierr</var> contains the following status information and
is the same size as the result.
<ol type=1 start=0>
<li>Normal return.
<li>Input error, return <code>NaN</code>.
<li>Overflow, return <code>Inf</code>.
<li>Loss of significance by argument reduction results in less than half
of machine accuracy.
<li>Complete loss of significance by argument reduction, return <code>NaN</code>.
<li>Error—no computation, algorithm termination condition not met,
return <code>NaN</code>.
</ol>
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/besselj.cc -->
<p><a name="doc_002dbesselj"></a>
<div class="defun">
— Loadable Function: [<var>j</var>, <var>ierr</var>] = <b>besselj</b> (<var>alpha, x, opt</var>)<var><a name="index-besselj-1483"></a></var><br>
— Loadable Function: [<var>y</var>, <var>ierr</var>] = <b>bessely</b> (<var>alpha, x, opt</var>)<var><a name="index-bessely-1484"></a></var><br>
— Loadable Function: [<var>i</var>, <var>ierr</var>] = <b>besseli</b> (<var>alpha, x, opt</var>)<var><a name="index-besseli-1485"></a></var><br>
— Loadable Function: [<var>k</var>, <var>ierr</var>] = <b>besselk</b> (<var>alpha, x, opt</var>)<var><a name="index-besselk-1486"></a></var><br>
— Loadable Function: [<var>h</var>, <var>ierr</var>] = <b>besselh</b> (<var>alpha, k, x, opt</var>)<var><a name="index-besselh-1487"></a></var><br>
<blockquote><p>Compute Bessel or Hankel functions of various kinds:
<dl>
<dt><code>besselj</code><dd>Bessel functions of the first kind. If the argument <var>opt</var> is supplied,
the result is multiplied by <code>exp(-abs(imag(x)))</code>.
<br><dt><code>bessely</code><dd>Bessel functions of the second kind. If the argument <var>opt</var> is supplied,
the result is multiplied by <code>exp(-abs(imag(x)))</code>.
<br><dt><code>besseli</code><dd>Modified Bessel functions of the first kind. If the argument <var>opt</var> is supplied,
the result is multiplied by <code>exp(-abs(real(x)))</code>.
<br><dt><code>besselk</code><dd>Modified Bessel functions of the second kind. If the argument <var>opt</var> is supplied,
the result is multiplied by <code>exp(x)</code>.
<br><dt><code>besselh</code><dd>Compute Hankel functions of the first (<var>k</var> = 1) or second (<var>k</var>
= 2) kind. If the argument <var>opt</var> is supplied, the result is multiplied by
<code>exp (-I*</code><var>x</var><code>)</code> for <var>k</var> = 1 or <code>exp (I*</code><var>x</var><code>)</code> for
<var>k</var> = 2.
</dl>
<p>If <var>alpha</var> is a scalar, the result is the same size as <var>x</var>.
If <var>x</var> is a scalar, the result is the same size as <var>alpha</var>.
If <var>alpha</var> is a row vector and <var>x</var> is a column vector, the
result is a matrix with <code>length (</code><var>x</var><code>)</code> rows and
<code>length (</code><var>alpha</var><code>)</code> columns. Otherwise, <var>alpha</var> and
<var>x</var> must conform and the result will be the same size.
<p>The value of <var>alpha</var> must be real. The value of <var>x</var> may be
complex.
<p>If requested, <var>ierr</var> contains the following status information
and is the same size as the result.
<ol type=1 start=0>
<li>Normal return.
<li>Input error, return <code>NaN</code>.
<li>Overflow, return <code>Inf</code>.
<li>Loss of significance by argument reduction results in less than
half of machine accuracy.
<li>Complete loss of significance by argument reduction, return <code>NaN</code>.
<li>Error—no computation, algorithm termination condition not met,
return <code>NaN</code>.
</ol>
</p></blockquote></div>
<!-- ./specfun/beta.m -->
<p><a name="doc_002dbeta"></a>
<div class="defun">
— Mapping Function: <b>beta</b> (<var>a, b</var>)<var><a name="index-beta-1488"></a></var><br>
<blockquote><p>For real inputs, return the Beta function,
<pre class="example"> beta (a, b) = gamma (a) * gamma (b) / gamma (a + b).
</pre>
</blockquote></div>
<!-- ./DLD-FUNCTIONS/betainc.cc -->
<p><a name="doc_002dbetainc"></a>
<div class="defun">
— Mapping Function: <b>betainc</b> (<var>x, a, b</var>)<var><a name="index-betainc-1489"></a></var><br>
<blockquote><p>Return the incomplete Beta function,
<!-- Set example in small font to prevent overfull line -->
<pre class="smallexample"> x
/
betainc (x, a, b) = beta (a, b)^(-1) | t^(a-1) (1-t)^(b-1) dt.
/
t=0
</pre>
<p>If x has more than one component, both <var>a</var> and <var>b</var> must be
scalars. If <var>x</var> is a scalar, <var>a</var> and <var>b</var> must be of
compatible dimensions.
</p></blockquote></div>
<!-- ./specfun/betaln.m -->
<p><a name="doc_002dbetaln"></a>
<div class="defun">
— Mapping Function: <b>betaln</b> (<var>a, b</var>)<var><a name="index-betaln-1490"></a></var><br>
<blockquote><p>Return the log of the Beta function,
<pre class="example"> betaln (a, b) = gammaln (a) + gammaln (b) - gammaln (a + b)
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dbeta.html#doc_002dbeta">beta</a>, <a href="doc_002dbetainc.html#doc_002dbetainc">betainc</a>, <a href="doc_002dgammaln.html#doc_002dgammaln">gammaln</a>.
</p></blockquote></div>
<!-- ./miscellaneous/bincoeff.m -->
<p><a name="doc_002dbincoeff"></a>
<div class="defun">
— Mapping Function: <b>bincoeff</b> (<var>n, k</var>)<var><a name="index-bincoeff-1491"></a></var><br>
<blockquote><p>Return the binomial coefficient of <var>n</var> and <var>k</var>, defined as
<pre class="example"> / \
| n | n (n-1) (n-2) ... (n-k+1)
| | = -------------------------
| k | k!
\ /
</pre>
<p>For example,
<pre class="example"> bincoeff (5, 2)
⇒ 10
</pre>
<p>In most cases, the <code>nchoosek</code> function is faster for small
scalar integer arguments. It also warns about loss of precision for
big arguments.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dnchoosek.html#doc_002dnchoosek">nchoosek</a>.
</p></blockquote></div>
<!-- ./linear-algebra/commutation_matrix.m -->
<p><a name="doc_002dcommutation_005fmatrix"></a>
<div class="defun">
— Function File: <b>commutation_matrix</b> (<var>m, n</var>)<var><a name="index-commutation_005fmatrix-1492"></a></var><br>
<blockquote><p>Return the commutation matrix
K(m,n)
which is the unique
<var>m</var>*<var>n</var> by <var>m</var>*<var>n</var>
matrix such that
K(m,n) * vec(A) = vec(A')
for all
m by n
matrices
A.
<p>If only one argument <var>m</var> is given,
K(m,m)
is returned.
<p>See Magnus and Neudecker (1988), Matrix differential calculus with
applications in statistics and econometrics.
</p></blockquote></div>
<!-- ./linear-algebra/duplication_matrix.m -->
<p><a name="doc_002dduplication_005fmatrix"></a>
<div class="defun">
— Function File: <b>duplication_matrix</b> (<var>n</var>)<var><a name="index-duplication_005fmatrix-1493"></a></var><br>
<blockquote><p>Return the duplication matrix
Dn
which is the unique
n^2 by n*(n+1)/2
matrix such that
Dn vech (A) = vec (A)
for all symmetric
n by n
matrices
A.
<p>See Magnus and Neudecker (1988), Matrix differential calculus with
applications in statistics and econometrics.
</p></blockquote></div>
<!-- mappers.cc -->
<p><a name="doc_002derf"></a>
<div class="defun">
— Mapping Function: <b>erf</b> (<var>z</var>)<var><a name="index-erf-1494"></a></var><br>
<blockquote><p>Computes the error function,
<pre class="example"> z
/
erf (z) = (2/sqrt (pi)) | e^(-t^2) dt
/
t=0
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002derfc.html#doc_002derfc">erfc</a>, <a href="doc_002derfinv.html#doc_002derfinv">erfinv</a>.
</p></blockquote></div>
<!-- mappers.cc -->
<p><a name="doc_002derfc"></a>
<div class="defun">
— Mapping Function: <b>erfc</b> (<var>z</var>)<var><a name="index-erfc-1495"></a></var><br>
<blockquote><p>Computes the complementary error function,
<code>1 - erf (</code><var>z</var><code>)</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002derf.html#doc_002derf">erf</a>, <a href="doc_002derfinv.html#doc_002derfinv">erfinv</a>.
</p></blockquote></div>
<!-- ./specfun/erfinv.m -->
<p><a name="doc_002derfinv"></a>
<div class="defun">
— Mapping Function: <b>erfinv</b> (<var>z</var>)<var><a name="index-erfinv-1496"></a></var><br>
<blockquote><p>Computes the inverse of the error function.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002derf.html#doc_002derf">erf</a>, <a href="doc_002derfc.html#doc_002derfc">erfc</a>.
</p></blockquote></div>
<!-- mappers.cc -->
<p><a name="doc_002dgamma"></a>
<div class="defun">
— Mapping Function: <b>gamma</b> (<var>z</var>)<var><a name="index-gamma-1497"></a></var><br>
<blockquote><p>Computes the Gamma function,
<pre class="example"> infinity
/
gamma (z) = | t^(z-1) exp (-t) dt.
/
t=0
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dgammainc.html#doc_002dgammainc">gammainc</a>, <a href="doc_002dlgamma.html#doc_002dlgamma">lgamma</a>.
</p></blockquote></div>
<!-- ./DLD-FUNCTIONS/gammainc.cc -->
<p><a name="doc_002dgammainc"></a>
<div class="defun">
— Mapping Function: <b>gammainc</b> (<var>x, a</var>)<var><a name="index-gammainc-1498"></a></var><br>
<blockquote><p>Compute the normalized incomplete gamma function,
<pre class="smallexample"> x
1 /
gammainc (x, a) = --------- | exp (-t) t^(a-1) dt
gamma (a) /
t=0
</pre>
<p>with the limiting value of 1 as <var>x</var> approaches infinity.
The standard notation is P(a,x), e.g., Abramowitz and Stegun (6.5.1).
<p>If <var>a</var> is scalar, then <code>gammainc (</code><var>x</var><code>, </code><var>a</var><code>)</code> is returned
for each element of <var>x</var> and vice versa.
<p>If neither <var>x</var> nor <var>a</var> is scalar, the sizes of <var>x</var> and
<var>a</var> must agree, and <var>gammainc</var> is applied element-by-element.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dgamma.html#doc_002dgamma">gamma</a>, <a href="doc_002dlgamma.html#doc_002dlgamma">lgamma</a>.
</p></blockquote></div>
<!-- ./specfun/legendre.m -->
<p><a name="doc_002dlegendre"></a>
<div class="defun">
— Function File: <var>l</var> = <b>legendre</b> (<var>n, x</var>)<var><a name="index-legendre-1499"></a></var><br>
— Function File: <var>l</var> = <b>legendre</b> (<var>n, x, normalization</var>)<var><a name="index-legendre-1500"></a></var><br>
<blockquote><p>Compute the Legendre function of degree <var>n</var> and order
<var>m</var> = 0 <small class="dots">...</small> N. The optional argument, <var>normalization</var>,
may be one of <code>"unnorm"</code>, <code>"sch"</code>, or <code>"norm"</code>.
The default is <code>"unnorm"</code>. The value of <var>n</var> must be a
non-negative scalar integer.
<p>If the optional argument <var>normalization</var> is missing or is
<code>"unnorm"</code>, compute the Legendre function of degree <var>n</var> and
order <var>m</var> and return all values for <var>m</var> = 0 <small class="dots">...</small> <var>n</var>.
The return value has one dimension more than <var>x</var>.
<p>The Legendre Function of degree <var>n</var> and order <var>m</var>:
<pre class="example"> m m 2 m/2 d^m
P(x) = (-1) * (1-x ) * ---- P (x)
n dx^m n
</pre>
<p class="noindent">with Legendre polynomial of degree <var>n</var>:
<pre class="example"> 1 d^n 2 n
P (x) = ------ [----(x - 1) ]
n 2^n n! dx^n
</pre>
<p class="noindent"><code>legendre (3, [-1.0, -0.9, -0.8])</code> returns the matrix:
<pre class="example"> x | -1.0 | -0.9 | -0.8
------------------------------------
m=0 | -1.00000 | -0.47250 | -0.08000
m=1 | 0.00000 | -1.99420 | -1.98000
m=2 | 0.00000 | -2.56500 | -4.32000
m=3 | 0.00000 | -1.24229 | -3.24000
</pre>
<p>If the optional argument <code>normalization</code> is <code>"sch"</code>,
compute the Schmidt semi-normalized associated Legendre function.
The Schmidt semi-normalized associated Legendre function is related
to the unnormalized Legendre functions by the following:
<p>For Legendre functions of degree n and order 0:
<pre class="example"> 0 0
SP (x) = P (x)
n n
</pre>
<p>For Legendre functions of degree n and order m:
<pre class="example"> m m m 2(n-m)! 0.5
SP (x) = P (x) * (-1) * [-------]
n n (n+m)!
</pre>
<p>If the optional argument <var>normalization</var> is <code>"norm"</code>,
compute the fully normalized associated Legendre function.
The fully normalized associated Legendre function is related
to the unnormalized Legendre functions by the following:
<p>For Legendre functions of degree <var>n</var> and order <var>m</var>
<pre class="example"> m m m (n+0.5)(n-m)! 0.5
NP (x) = P (x) * (-1) * [-------------]
n n (n+m)!
</pre>
</blockquote></div>
<p><a name="doc_002dgammaln"></a><!-- mappers.cc -->
<a name="doc_002dlgamma"></a>
<div class="defun">
— Mapping Function: <b>lgamma</b> (<var>x</var>)<var><a name="index-lgamma-1501"></a></var><br>
— Mapping Function: <b>gammaln</b> (<var>x</var>)<var><a name="index-gammaln-1502"></a></var><br>
<blockquote><p>Return the natural logarithm of the gamma function of <var>x</var>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dgamma.html#doc_002dgamma">gamma</a>, <a href="doc_002dgammainc.html#doc_002dgammainc">gammainc</a>.
</p></blockquote></div>
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