[doc] bessel page: hardly readable MML equations => LaTeX 52/20952/5
Samuel GOUGEON [Thu, 18 Apr 2019 00:59:09 +0000 (02:59 +0200)]
  + alt=".." tags added for rendering as text
  + <subscript> & <superscript> used when applicable
  + (pt_BR): <scilab:image> added
  The non-localization of <scilab:image> is intentional.

Change-Id: Ia0bc8af1e0126bb4f6b8420ada0cda1485729d1c

14 files changed:
scilab/modules/helptools/etc/images_md5.txt
scilab/modules/helptools/images/_LaTeX_bessel.xml_1.png [new file with mode: 0644]
scilab/modules/helptools/images/_LaTeX_bessel.xml_2.png [new file with mode: 0644]
scilab/modules/helptools/images/_LaTeX_bessel.xml_3.png [new file with mode: 0644]
scilab/modules/helptools/images/bessel_equation1.mml.png [deleted file]
scilab/modules/helptools/images/bessel_equation2.mml.png [deleted file]
scilab/modules/helptools/images/bessel_equation3.mml.png [deleted file]
scilab/modules/special_functions/help/en_US/bessel.xml
scilab/modules/special_functions/help/ja_JP/bessel.xml
scilab/modules/special_functions/help/ja_JP/oldbessel.xml [deleted file]
scilab/modules/special_functions/help/mml/bessel_equation1.mml [deleted file]
scilab/modules/special_functions/help/mml/bessel_equation2.mml [deleted file]
scilab/modules/special_functions/help/mml/bessel_equation3.mml [deleted file]
scilab/modules/special_functions/help/pt_BR/bessel.xml

index 61ef1ce..ecd6f48 100644 (file)
@@ -242,6 +242,9 @@ _LaTeX_airy.xml_5.png=24e27ac7cb803a2c15a19e7fe81cd0fa
 _LaTeX_airy.xml_6.png=99b138cc89d2519f769a3fa07261a4e0
 _LaTeX_assert_computedigits.xml_1.png=082a70de2c216b6eb1951a75720fdcfc
 _LaTeX_assert_computedigits.xml_2.png=27dba5ee1401d31d2106736651e99140
+_LaTeX_bessel.xml_1.png=75b18324986d95f0ce739c723eff803f
+_LaTeX_bessel.xml_2.png=91604506c2c5f30b0fab1ef9da180a89
+_LaTeX_bessel.xml_3.png=5e4298e19e224d62ecbd7b6db2a80a31
 _LaTeX_blockdiag.xml_1.png=26bcad469ac051064707c071f102456c
 _LaTeX_bvode.xml_1.png=7d919bf8f33698749d88e30e99c33d7c
 _LaTeX_bvode.xml_2.png=417a5301693b60807fa658e5ef9f9535
diff --git a/scilab/modules/helptools/images/_LaTeX_bessel.xml_1.png b/scilab/modules/helptools/images/_LaTeX_bessel.xml_1.png
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diff --git a/scilab/modules/helptools/images/_LaTeX_bessel.xml_3.png b/scilab/modules/helptools/images/_LaTeX_bessel.xml_3.png
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diff --git a/scilab/modules/helptools/images/bessel_equation1.mml.png b/scilab/modules/helptools/images/bessel_equation1.mml.png
deleted file mode 100644 (file)
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diff --git a/scilab/modules/helptools/images/bessel_equation2.mml.png b/scilab/modules/helptools/images/bessel_equation2.mml.png
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diff --git a/scilab/modules/helptools/images/bessel_equation3.mml.png b/scilab/modules/helptools/images/bessel_equation3.mml.png
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index 02f7583..2ac8bcf 100644 (file)
@@ -3,8 +3,8 @@
  * Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
  * Copyright (C) 2008 - INRIA
  * Copyright (C) 2012 - Scilab Enterprises - Adeline CARNIS
- *
  * Copyright (C) 2012 - 2016 - Scilab Enterprises
+ * Copyright (C) 2019 - Samuel GOUGEON
  *
  * This file is hereby licensed under the terms of the GNU GPL v2.0,
  * pursuant to article 5.3.4 of the CeCILL v.2.1.
  * along with this program.
  *
  -->
-<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org"  xml:id="bessel" xml:lang="en">
+<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink"
+          xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml"
+          xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook"
+          xmlns:scilab="http://www.scilab.org"  xml:id="bessel" xml:lang="en">
     <refnamediv xml:id="besseli">
         <refname>besseli</refname>
         <refpurpose>Modified Bessel functions of the first kind (I sub
     </refnamediv>
     <refnamediv xml:id="besselj">
         <refname>besselj</refname>
-        <refpurpose>Bessel functions of the first kind (J sub alpha).</refpurpose>
+      <refpurpose>Bessel functions of the first kind (J<subscript>α</subscript>).</refpurpose>
     </refnamediv>
     <refnamediv xml:id="besselk">
         <refname>besselk</refname>
-        <refpurpose>Modified Bessel functions of the second kind (K sub
-            alpha).
+        <refpurpose>Modified Bessel functions of the second kind (K<subscript>α</subscript>).
         </refpurpose>
     </refnamediv>
     <refnamediv xml:id="bessely">
         <refname>bessely</refname>
-        <refpurpose>Bessel functions of the second kind (Y sub
-            alpha).
+        <refpurpose>Bessel functions of the second kind (Y<subscript>α</subscript>).
         </refpurpose>
     </refnamediv>
     <refnamediv xml:id="besselh">
         <refname>besselh</refname>
-        <refpurpose>Bessel functions of the third kind (aka Hankel
-            functions)
+        <refpurpose>Bessel functions of the third kind (aka Hankel functions)
         </refpurpose>
     </refnamediv>
     <refsynopsisdiv>
         <title>Syntax</title>
-        <synopsis>y = besseli(alpha,x [,ice])
-            y = besselj(alpha,x [,ice])
-            y = besselk(alpha,x [,ice])
-            y = bessely(alpha,x [,ice])
-            y = besselh(alpha,x)
-            y = besselh(alpha,K,x [,ice])
+        <synopsis>
+            y = besseli(alpha, x [,ice])
+            y = besselj(alpha, x [,ice])
+            y = besselk(alpha, x [,ice])
+            y = bessely(alpha, x [,ice])
+            y = besselh(alpha, x)
+            y = besselh(alpha, K, x [,ice])
         </synopsis>
     </refsynopsisdiv>
     <refsection>
@@ -77,8 +78,7 @@
             <varlistentry>
                 <term>K</term>
                 <listitem>
-                    <para>integer, with possible values 1 or 2, the Hankel function
-                        type.
+                    <para>integer, with possible values 1 or 2, the Hankel function type.
                     </para>
                 </listitem>
             </varlistentry>
             <listitem>
                 <para>
                     <literal>besseli(alpha,x)</literal> computes modified Bessel
-                    functions of the first kind (I sub alpha), for real order
-                    <literal>alpha</literal> and argument <literal>x</literal>.
+                    functions of the first kind (I<subscript>α</subscript>), for real order
+                    <varname>alpha</varname> and argument <varname>x</varname>.
                     <literal>besseli(alpha,x,1)</literal> computes
                     <literal>besseli(alpha,x).*exp(-abs(real(x)))</literal>.
                 </para>
             </listitem>
             <listitem>
                 <para>
-                    <literal>besselj(alpha,x)</literal> computes Bessel functions of
-                    the first kind (J sub alpha), for real order <literal>alpha</literal>
-                    and argument <literal>x</literal>.
+                    <literal>besselj(alpha,x)</literal> computes Bessel functions of the fisrt
+                    kind (J<subscript>α</subscript>), for real order <varname>alpha</varname>
+                    and argument <varname>x</varname>.
                     <literal>besselj(alpha,x,1)</literal> computes
                     <literal>besselj(alpha,x).*exp(-abs(imag(x)))</literal>.
                 </para>
             <listitem>
                 <para>
                     <literal>besselk(alpha,x)</literal> computes modified Bessel
-                    functions of the second kind (K sub alpha), for real order
-                    <literal>alpha</literal> and argument <literal>x</literal>.
+                    functions of the second kind (K<subscript>α</subscript>), for real order
+                    <varname>alpha</varname> and argument <varname>x</varname>.
                     <literal>besselk(alpha,x,1)</literal> computes
                     <literal>besselk(alpha,x).*exp(x)</literal>.
                 </para>
             </listitem>
             <listitem>
                 <para>
-                    <literal>bessely(alpha,x)</literal> computes Bessel functions of
-                    the second kind (Y sub alpha), for real order <literal>alpha</literal>
-                    and argument <literal>x</literal>.
+                    <literal>bessely(alpha,x)</literal> computes Bessel functions of the second
+                    kind (Y<subscript>alpha</subscript>), for real order <varname>alpha</varname>
+                    and argument <varname>x</varname>.
                     <literal>bessely(alpha,x,1)</literal> computes
                     <literal>bessely(alpha,x).*exp(-abs(imag(x)))</literal>.
                 </para>
                 <para>
                     <literal>besselh(alpha [,K] ,x)</literal> computes Bessel
                     functions of the third kind (Hankel function H1 or H2 depending on
-                    <literal>K</literal>), for real order <literal>alpha</literal> and
-                    argument <literal>x</literal>. If omitted <literal>K</literal> is
+                    <literal>K</literal>), for real order <varname>alpha</varname> and
+                    argument <varname>x</varname>. If omitted <literal>K</literal> is
                     supposed to be equal to 1. <literal>besselh(alpha,1,x,1)</literal>
                     computes <literal>besselh(alpha,1,x).*exp(-%i*x)</literal> and
                     <literal>besselh(alpha,2,x,1)</literal> computes
     <refsection>
         <title>Remarks</title>
         <para>
-            If <literal>alpha</literal> and <literal>x</literal> are arrays of
+            If <varname>alpha</varname> and <varname>x</varname> are arrays of
             the same size, the result <literal>y</literal> is also that size. If
             either input is a scalar, it is expanded to the other input's size. If one
             input is a row vector and the other is a column vector, the
-            result<literal>y</literal> is a two-dimensional table of function
+            result <literal>y</literal> is a two-dimensional table of function
             values.
         </para>
-        <para>Y_alpha and J_alpha Bessel functions are 2 independent solutions of
-            the Bessel 's differential equation :
+        <para>
+            Y<subscript>α</subscript> and J<subscript>α</subscript> Bessel functions are 2
+            independent solutions of the Bessel 's differential equation :
         </para>
-        <informalequation>
-            <mediaobject>
-                <imageobject>
-                    <imagedata align="center" fileref="../mml/bessel_equation1.mml"/>
-                </imageobject>
-            </mediaobject>
-        </informalequation>
-        <para>K_alpha and I_alpha modified Bessel functions are 2 independent
-            solutions of the modified Bessel 's differential equation :
+        <latex style="display" alt="x^2.(d^2y/d^2x) + x.dy/dx + (x^2 - alpha^2).y = 0,  alpha ≥ 0">
+            {x^2} \cdot {{d^2 y} \over {dx^2}} + x \cdot {{dy} \over {dx}} + (x^2 - \alpha^2) \cdot y = 0,
+            \quad\alpha\ge0
+        </latex>
+        <para>
+            K<subscript>α</subscript> and I<subscript>α</subscript> modified Bessel functions
+            are 2 independent solutions of the modified Bessel 's differential equation :
         </para>
-        <informalequation>
-            <mediaobject>
-                <imageobject>
-                    <imagedata align="center" fileref="../mml/bessel_equation2.mml"/>
-                </imageobject>
-            </mediaobject>
-        </informalequation>
-        <para>H^1_alpha and H^2_alpha, the Hankel functions of first and second
-            kind, are linear linear combinations of Bessel functions of the first and
+        <latex style="display" alt="x^2.(d^2y/d^2x) + x.dy/dx + (alpha^2 - x^2).y = 0,  alpha ≥ 0">
+            {x^2} \cdot {{d^2 y} \over {dx^2}} + x \cdot {{dy} \over {dx}} + (\alpha^2 - x^2) \cdot y = 0,
+            \quad\alpha\ge0
+        </latex>
+        <para>
+            H<subscript>α</subscript><superscript>1</superscript> and
+            H<subscript>α</subscript><superscript>2</superscript>, the Hankel functions of first
+            and second kind, are linear linear combinations of Bessel functions of the first and
             second kinds:
         </para>
-        <informalequation>
-            <mediaobject>
-                <imageobject>
-                    <imagedata align="center" fileref="../mml/bessel_equation3.mml"/>
-                </imageobject>
-            </mediaobject>
-        </informalequation>
+        <latex style="display" alt="H^1_α(z) = J_α(z) + i \cdot Y_α(z)  \n
+H^2_α(z) = J_α(z) - i \cdot Y_α(z)">
+            H^1_{\alpha}(z) = J_{\alpha}(z) + i \cdot Y_{\alpha}(z) \\
+            H^2_{\alpha}(z) = J_{\alpha}(z) - i \cdot Y_{\alpha}(z)
+        </latex>
     </refsection>
     <refsection>
         <title>Examples</title>
         <programlisting role="example"><![CDATA[
-//  besselI functions
-// ==================
-   x = linspace(0.01,10,5000)';
-   clf()
-   subplot(2,1,1)
-   plot2d(x,besseli(0:4,x), style=2:6)
-   legend('I'+string(0:4),2);
-   xtitle("Some modified Bessel functions of the first kind")
-   subplot(2,1,2)
-   plot2d(x,besseli(0:4,x,1), style=2:6)
-   legend('I'+string(0:4),1);
-   xtitle("Some modified scaled Bessel functions of the first kind")
+    // besselI functions
+    // -----------------
+    x = linspace(0.01,10,5000)';
+    clf
+    subplot(2,1,1)
+    plot2d(x,besseli(0:4,x), style=2:6)
+    legend('I'+string(0:4),2);
+    xtitle("Some modified Bessel functions of the first kind")
+    subplot(2,1,2)
+    plot2d(x,besseli(0:4,x,1), style=2:6)
+    legend('I'+string(0:4),1);
+    xtitle("Some modified scaled Bessel functions of the first kind")
  ]]></programlisting>
-
         <scilab:image>
             x = linspace(0.01,10,5000)';
             clf()
             plot2d(x,besseli(0:4,x,1), style=2:6)
             legend('I'+string(0:4),1);
             xtitle("Some modified scaled Bessel functions of the first kind")
-
         </scilab:image>
-
-
+        <para/>
         <programlisting role="example"><![CDATA[
-// besselJ functions
-// =================
-   clf()
-   x = linspace(0,40,5000)';
-   plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4")
-   legend('I'+string(0:4),1);
-   xtitle("Some Bessel functions of the first kind")
+    // besselJ functions
+    // -----------------
+    clf
+    x = linspace(0,40,5000)';
+    plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4")
+    legend('I'+string(0:4),1);
+    xtitle("Some Bessel functions of the first kind")
  ]]></programlisting>
 
         <scilab:image>
             legend('I'+string(0:4),1);
             xtitle("Some Bessel functions of the first kind")
         </scilab:image>
-
+        <para/>
         <programlisting role="example"><![CDATA[
 // use the fact that J_(1/2)(x) = sqrt(2/(x pi)) sin(x)
 // to compare the algorithm of besselj(0.5,x) with a more direct formula
    plot2d(x(ind), er(ind), style=2, logflag="nl")
    xtitle("relative error between 2 formulae for besselj(0.5,x)")
  ]]></programlisting>
-
         <scilab:image><![CDATA[
    x = linspace(0.1,40,5000)';
    y1 = besselj(0.5, x);
    xtitle("relative error between 2 formulae for besselj(0.5,x)")
  ]]></scilab:image>
 
+        <para/>
         <programlisting role="example"><![CDATA[
 // besselK functions
 // =================
    legend('K'+string(0:4),1);
    xtitle("Some modified scaled Bessel functions of the second kind")
  ]]></programlisting>
-
         <scilab:image>
             x = linspace(0.01,10,5000)';
             clf()
             xtitle("Some modified scaled Bessel functions of the second kind")
         </scilab:image>
 
+        <para/>
         <programlisting role="example"><![CDATA[
 // besselY functions
 // =================
    legend('Y'+string(0:4),4);
    xtitle("Some Bessel functions of the second kind")
  ]]></programlisting>
-
         <scilab:image>
             x = linspace(0.1,40,5000)'; // Y Bessel functions are unbounded  for x -> 0+
             clf()
             xtitle("Some Bessel functions of the second kind")
         </scilab:image>
 
+        <para/>
         <programlisting role="example"><![CDATA[
 // besselH functions
 // =================
    legends(string(0.2:0.2:3.2),1:16,"ur")
    xtitle("Level curves of |H1(0,z)|")
  ]]></programlisting>
-
         <scilab:image>
             x=-4:0.025:2; y=-1.5:0.025:1.5;
             [X,Y] = ndgrid(x,y);
             legends(string(0.2:0.2:3.2),1:16,"ur")
             xtitle("Level curves of |H1(0,z)|")
         </scilab:image>
-
     </refsection>
     <refsection>
         <title>Used Functions</title>
index 4f5ad89..5fe5f27 100644 (file)
@@ -2,8 +2,8 @@
 <!--
  * Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
  * Copyright (C) 2008 - INRIA
- *
  * Copyright (C) 2012 - 2016 - Scilab Enterprises
+ * Copyright (C) 2019 - Samuel GOUGEON
  *
  * This file is hereby licensed under the terms of the GNU GPL v2.0,
  * pursuant to article 5.3.4 of the CeCILL v.2.1.
  * along with this program.
  *
  -->
-<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org"  xml:id="bessel" xml:lang="ja">
+<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink"
+          xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml"
+          xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook"
+          xmlns:scilab="http://www.scilab.org"  xml:id="bessel" xml:lang="ja">
     <refnamediv xml:id="besseli">
         <refname>besseli</refname>
-        <refpurpose>第1種修正ベッセル関数 (I_alpha).</refpurpose>
+      <refpurpose>第1種修正ベッセル関数 (I<subscript>α</subscript>).</refpurpose>
     </refnamediv>
     <refnamediv xml:id="besselj">
         <refname>besselj</refname>
-        <refpurpose>第1種ベッセル関数 (J_alpha).</refpurpose>
+        <refpurpose>第1種ベッセル関数 (J<subscript>α</subscript>).</refpurpose>
     </refnamediv>
     <refnamediv xml:id="besselk">
         <refname>besselk</refname>
-        <refpurpose>第2種修正ベッセル関数 (K_alpha).</refpurpose>
+        <refpurpose>第2種修正ベッセル関数 (K<subscript>α</subscript>).</refpurpose>
     </refnamediv>
     <refnamediv xml:id="bessely">
         <refname>bessely</refname>
-        <refpurpose>第2種ベッセル関数 (Y_alpha).</refpurpose>
+        <refpurpose>第2種ベッセル関数 (Y<subscript>α</subscript>).</refpurpose>
     </refnamediv>
     <refnamediv xml:id="besselh">
         <refname>besselh</refname>
     </refnamediv>
     <refsynopsisdiv>
         <title>呼び出し手順</title>
-        <synopsis>y = besseli(alpha,x [,ice])
-            y = besselj(alpha,x [,ice])
-            y = besselk(alpha,x [,ice])
-            y = bessely(alpha,x [,ice])
-            y = besselh(alpha,x)
-            y = besselh(alpha,K,x [,ice])
+        <synopsis>
+            y = besseli(alpha, x [,ice])
+            y = besselj(alpha, x [,ice])
+            y = besselk(alpha, x [,ice])
+            y = bessely(alpha, x [,ice])
+            y = besselh(alpha, x)
+            y = besselh(alpha, K, x [,ice])
         </synopsis>
     </refsynopsisdiv>
     <refsection>
             <listitem>
                 <para>
                     <literal>besseli(alpha,x)</literal> は,
-                    実数の次数<literal>alpha</literal> および引数 <literal>x</literal>に関する
-                    第1種修正ベッセル関数(I_alpha)を計算します,
+                    実数の次数<varname>alpha</varname> および引数 <varname>x</varname>に関する
+                    第1種修正ベッセル関数(I<subscript>α</subscript>)を計算します,
                     <literal>besseli(alpha,x,1)</literal> は
                     <literal>besseli(alpha,x).*exp(-abs(real(x)))</literal>を計算します.
                 </para>
             </listitem>
             <listitem>
                 <para>
-                    <literal>besselj(alpha,x)</literal> は第1種のベッセル関数(J_alpha)を
-                    実数の次数<literal>alpha</literal> および引数 <literal>x</literal>に関して
+                    <literal>besselj(alpha,x)</literal> は第1種のベッセル関数(J<subscript>α</subscript>)を
+                    実数の次数<varname>alpha</varname> および引数 <varname>x</varname>に関して
                     計算します.
                     <literal>besselj(alpha,x,1)</literal> は
                     <literal>besselj(alpha,x).*exp(-abs(imag(x)))</literal>を計算します.
             <listitem>
                 <para>
                     <literal>besselk(alpha,x)</literal> は第2種修正ベッセル関数
-                    (K_alpha)を
-                    実数の次数<literal>alpha</literal> および引数 <literal>x</literal>に関して
+                    (K<subscript>α</subscript>)を
+                    実数の次数<varname>alpha</varname> および引数 <varname>x</varname>に関して
                     計算します.
                     <literal>besselk(alpha,x,1)</literal> は
                     <literal>besselk(alpha,x).*exp(x)</literal>を計算します.
             </listitem>
             <listitem>
                 <para>
-                    <literal>bessely(alpha,x)</literal>は第2種のベッセル関数(Y_alpha)を
-                    実数の次数<literal>alpha</literal> および引数 <literal>x</literal>に関して
+                    <literal>bessely(alpha,x)</literal>は第2種のベッセル関数(Y<subscript>α</subscript>)を
+                    実数の次数<varname>alpha</varname> および引数 <varname>x</varname>に関して
                     計算します.
                     <literal>bessely(alpha,x,1)</literal> は
                     <literal>bessely(alpha,x).*exp(-abs(imag(x)))</literal>を計算します.
                 <para>
                     <literal>besselh(alpha [,K] ,x)</literal> は第3種のベッセル関数
                     (<literal>K</literal>に依存してハンケル関数 H1 または H2)を
-                    実数の次数<literal>alpha</literal> および引数 <literal>x</literal>に関して
+                    実数の次数<varname>alpha</varname> および引数 <varname>x</varname>に関して
                     計算します.<literal>K</literal>が省略された場合,
                     1に等しいと仮定されます.
                     <literal>besselh(alpha,1,x,1)</literal>は
     <refsection>
         <title>注意</title>
         <para>
-            <literal>alpha</literal>および <literal>x</literal>が同じ大きさの
+            <varname>alpha</varname>および <varname>x</varname>が同じ大きさの
             配列の場合,結果<literal>y</literal>も同じ大きさとなります.
             入力のどちらかがスカラーの場合,
             もう片方の大きさにまで拡張されます.
             片方の入力が行ベクトルでもう片方が列ベクトルの場合,
             結果<literal>y</literal>は関数値の二次元テーブルとなります.
         </para>
-        <para>Y_alpha および J_alpha ベッセル関数はベッセルの微分方程式の
-            2つの独立解です:
+        <para>
+            Y<subscript>α</subscript> および J<subscript>α</subscript>
+            ベッセル関数はベッセルの微分方程式の 2つの独立解です:
         </para>
-        <informalequation>
-            <mediaobject>
-                <imageobject>
-                    <imagedata align="center" fileref="../mml/bessel_equation1.mml"/>
-                </imageobject>
-            </mediaobject>
-        </informalequation>
-        <para>修正ベッセル関数K_alpha および I_alphaは
+        <latex style="display" alt="x^2.(d^2y/d^2x) + x.dy/dx + (x^2 - alpha^2).y = 0,  alpha ≥ 0">
+            {x^2} \cdot {{d^2 y} \over {dx^2}} + x \cdot {{dy} \over {dx}} + (x^2 - \alpha^2) \cdot y = 0,
+            \quad\alpha\ge0
+        </latex>
+        <para>修正ベッセル関数K<subscript>α</subscript> および I<subscript>α</subscript>は
             修正ベッセル微分方程式の2つの独立解です:
         </para>
-        <informalequation>
-            <mediaobject>
-                <imageobject>
-                    <imagedata align="center" fileref="../mml/bessel_equation2.mml"/>
-                </imageobject>
-            </mediaobject>
-        </informalequation>
-        <para>H^1_alpha および H^2_alphaは第1種および第2種のハンケル関数
+        <latex style="display" alt="x^2.(d^2y/d^2x) + x.dy/dx + (alpha^2 - x^2).y = 0,  alpha ≥ 0">
+            {x^2} \cdot {{d^2 y} \over {dx^2}} + x \cdot {{dy} \over {dx}} + (\alpha^2 - x^2) \cdot y = 0,
+            \quad\alpha\ge0
+        </latex>
+        <para>
+            H<subscript>α</subscript><superscript>1</superscript> および
+            H<subscript>α</subscript><superscript>2</superscript>は第1種および第2種のハンケル関数
             で,第1種および第2種のベッセル関数の線形結合です:
         </para>
-        <informalequation>
-            <mediaobject>
-                <imageobject>
-                    <imagedata align="center" fileref="../mml/bessel_equation3.mml"/>
-                </imageobject>
-            </mediaobject>
-        </informalequation>
+        <latex style="display" alt="H^1_α(z) = J_α(z) + i \cdot Y_α(z)  \n
+H^2_α(z) = J_α(z) - i \cdot Y_α(z)">
+            H^1_{\alpha}(z) = J_{\alpha}(z) + i \cdot Y_{\alpha}(z) \\
+            H^2_{\alpha}(z) = J_{\alpha}(z) - i \cdot Y_{\alpha}(z)
+        </latex>
     </refsection>
+
     <refsection>
         <title>例</title>
         <programlisting role="example"><![CDATA[
             legend('I'+string(0:4),1);
             xtitle("Some modified scaled Bessel functions of the first kind")
         </scilab:image>
+
+        <para/>
         <programlisting role="example"><![CDATA[
 // besselJ 関数
 // =================
             legend('I'+string(0:4),1);
             xtitle("Some Bessel functions of the first kind")
         </scilab:image>
+
+        <para/>
         <programlisting role="example"><![CDATA[
 // J_(1/2)(x) = sqrt(2/(x pi)) sin(x) の関係を用いて
 //  besselj(0.5,x) のアルゴリズムをより直接的な式と比較します
    plot2d(x(ind), er(ind), style=2, logflag="nl")
    xtitle("relative error between 2 formulae for besselj(0.5,x)")
  ]]></scilab:image>
+
+        <para/>
         <programlisting role="example"><![CDATA[
 // besselK 関数
 // =================
             legend('K'+string(0:4),1);
             xtitle("Some modified scaled Bessel functions of the second kind")
         </scilab:image>
+
+        <para/>
         <programlisting role="example"><![CDATA[
 // besselY 関数
 // =================
             legend('Y'+string(0:4),4);
             xtitle("Some Bessel functions of the second kind")
         </scilab:image>
+
+        <para/>
         <programlisting role="example"><![CDATA[
 // besselH 関数
 // =================
diff --git a/scilab/modules/special_functions/help/ja_JP/oldbessel.xml b/scilab/modules/special_functions/help/ja_JP/oldbessel.xml
deleted file mode 100644 (file)
index a5bc427..0000000
+++ /dev/null
@@ -1,189 +0,0 @@
-<?xml version="1.0" encoding="UTF-8"?>
-<!--
- * Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
- * Copyright (C) 2008 - INRIA
- *
- * Copyright (C) 2012 - 2016 - Scilab Enterprises
- *
- * This file is hereby licensed under the terms of the GNU GPL v2.0,
- * pursuant to article 5.3.4 of the CeCILL v.2.1.
- * This file was originally licensed under the terms of the CeCILL v2.1,
- * and continues to be available under such terms.
- * For more information, see the COPYING file which you should have received
- * along with this program.
- *
- -->
-<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="oldbessel" xml:lang="ja">
-    <refnamediv xml:id="oldbesseli">
-        <refname>oldbesseli</refname>
-        <refpurpose>第1種の修正ベッセル関数 (I_alpha).</refpurpose>
-    </refnamediv>
-    <refnamediv xml:id="oldbesselj">
-        <refname>oldbesselj</refname>
-        <refpurpose>第1種のベッセル関数 (J_alpha).</refpurpose>
-    </refnamediv>
-    <refnamediv xml:id="oldbesselk">
-        <refname>oldbesselk</refname>
-        <refpurpose>第2種の修正ベッセル関数 (K_alpha).</refpurpose>
-    </refnamediv>
-    <refnamediv xml:id="oldbessely">
-        <refname>oldbessely</refname>
-        <refpurpose>第2種のベッセル関数 (Y_alpha).</refpurpose>
-    </refnamediv>
-    <refsynopsisdiv>
-        <title>呼び出し手順</title>
-        <synopsis>y = oldbesseli(alpha,x)
-            y = oldbesseli(alpha,x,ice)
-            y = oldbesselj(alpha,x)
-            y = oldbesselk(alpha,x)
-            y = oldbesselk(alpha,x,ice)
-            y = oldbessely(alpha,x)
-        </synopsis>
-    </refsynopsisdiv>
-    <refsection>
-        <title>パラメータ</title>
-        <variablelist>
-            <varlistentry>
-                <term>x</term>
-                <listitem>
-                    <para>real vector with non negative entries</para>
-                </listitem>
-            </varlistentry>
-            <varlistentry>
-                <term>alpha</term>
-                <listitem>
-                    <para>real vector with non negative entries regularly spaced with
-                        increment equal to one
-                        <literal>alpha=alpha0+(n1:n2)</literal>
-                    </para>
-                </listitem>
-            </varlistentry>
-            <varlistentry>
-                <term>ice</term>
-                <listitem>
-                    <para>integer flag, with default value 1</para>
-                </listitem>
-            </varlistentry>
-        </variablelist>
-    </refsection>
-    <refsection>
-        <title>説明</title>
-        <para>これらの関数は古い関数であり,
-            <link linkend="besseli">besseli</link>, <link linkend="besselj">besselj</link>,
-            <link linkend="besselk">besselk</link>, <link linkend="bessely">bessely</link> を代わりに使用してください.
-            しかし,これらの2組の関数の構文は異なっていることに注意してください.
-        </para>
-        <para>
-            <literal>oldbesseli(alpha,x)</literal> computes modified Bessel
-            functions of the first kind (I sub alpha), for real, non-negative order
-            <literal>alpha</literal> and real non negative argument
-            <literal>x</literal>. <literal>besseli(alpha,x,2)</literal> computes
-            <literal>besseli(alpha,x).*exp(-x)</literal>.
-        </para>
-        <para>
-            <literal>oldbesselj(alpha,x)</literal> computes Bessel functions of
-            the first kind (J sub alpha), for real, non-negative order
-            <literal>alpha</literal> and real non negative argument
-            <literal>x</literal>.
-        </para>
-        <para>
-            <literal>oldbesselk(alpha,x)</literal> computes modified Bessel
-            functions of the second kind (K sub alpha), for real, non-negative order
-            <literal>alpha</literal> and real non negative argument
-            <literal>x</literal>. <literal>besselk(alpha,x,2)</literal> computes
-            <literal>besselk(alpha,x).*exp(x)</literal>.
-        </para>
-        <para>
-            <literal>oldbessely(alpha,x)</literal> computes Bessel functions of
-            the second kind (Y sub alpha), for real, non-negative order
-            <literal>alpha</literal> and real non negative argument
-            <literal>x</literal>.
-        </para>
-        <para>
-            <literal>alpha</literal> and <literal>x</literal> may be vectors.
-            The output is <literal>m</literal>-by-<literal>n</literal> with <literal>m
-                = size(x,'*')
-            </literal>
-            ,<literal>n = size(alpha,'*')</literal> whose
-            <literal>(i,j)</literal> entry is
-            <literal>oldbessel?(alpha(j),x(i))</literal>.
-        </para>
-    </refsection>
-    <refsection>
-        <title>Remarks</title>
-        <para>Y_alpha and J_alpha Bessel functions are 2 independent solutions of
-            the Bessel 's differential equation :
-        </para>
-        <informalequation>
-            <mediaobject>
-                <imageobject>
-                    <imagedata align="center" fileref="../mml/bessel_equation1.mml"/>
-                </imageobject>
-            </mediaobject>
-        </informalequation>
-        <para>K_alpha and I_alpha modified Bessel functions are 2 independent
-            solutions of the modified Bessel 's differential equation :
-        </para>
-        <informalequation>
-            <mediaobject>
-                <imageobject>
-                    <imagedata align="center" fileref="../mml/bessel_equation2.mml"/>
-                </imageobject>
-            </mediaobject>
-        </informalequation>
-    </refsection>
-    <refsection>
-        <title>Examples</title>
-        <programlisting role="example"><![CDATA[
-// example #1: display some I Bessel functions
-x = linspace(0.01,10,5000)';
-y = oldbesseli(0:4,x);
-ys = oldbesseli(0:4,x,2);
-clf()
-subplot(2,1,1)
-plot2d(x,y, style=2:6, leg="I0@I1@I2@I3@I4", rect=[0,0,6,10])
-xtitle("Some modified Bessel functions of the first kind")
-subplot(2,1,2)
-plot2d(x,ys, style=2:6, leg="I0s@I1s@I2s@I3s@I4s", rect=[0,0,6,1])
-xtitle("Some modified scaled Bessel functions of the first kind")
-// example #2 : display some J Bessel functions
-x = linspace(0,40,5000)';
-y = besselj(0:4,x);
-clf()
-plot2d(x,y, style=2:6, leg="J0@J1@J2@J3@J4")
-xtitle("Some Bessel functions of the first kind")
-// example #3 : use the fact that J_(1/2)(x) = sqrt(2/(x pi)) sin(x)
-//              to compare the algorithm of besselj(0.5,x) with
-//              a more direct formula
-x = linspace(0.1,40,5000)';
-y1 = besselj(0.5, x);
-y2 = sqrt(2 ./(%pi*x)).*sin(x);
-er = abs((y1-y2)./y2);
-ind = find(er &gt; 0 &amp; y2 ~= 0);
-clf()
-subplot(2,1,1)
-plot2d(x,y1,style=2)
-xtitle("besselj(0.5,x)")
-subplot(2,1,2)
-plot2d(x(ind), er(ind), style=2, logflag="nl")
-xtitle("relative error between 2 formulae for besselj(0.5,x)")
-// example #4: display some K Bessel functions
-x = linspace(0.01,10,5000)';
-y = besselk(0:4,x);
-ys = besselk(0:4,x,1);
-clf()
-subplot(2,1,1)
-plot2d(x,y, style=0:4, leg="K0@K1@K2@K3@K4", rect=[0,0,6,10])
-xtitle("Some modified Bessel functions of the second kind")
-subplot(2,1,2)
-plot2d(x,ys, style=0:4, leg="K0s@K1s@K2s@K3s@K4s", rect=[0,0,6,10])
-xtitle("Some modified scaled Bessel functions of the second kind")
-// example #5: plot severals Y Bessel functions
-x = linspace(0.1,40,5000)'; // Y Bessel functions are unbounded  for x -> 0+
-y = bessely(0:4,x);
-clf()
-plot2d(x,y, style=0:4, leg="Y0@Y1@Y2@Y3@Y4", rect=[0,-1.5,40,0.6])
-xtitle("Some Bessel functions of the second kind")
- ]]></programlisting>
-    </refsection>
-</refentry>
diff --git a/scilab/modules/special_functions/help/mml/bessel_equation1.mml b/scilab/modules/special_functions/help/mml/bessel_equation1.mml
deleted file mode 100644 (file)
index 39878a0..0000000
+++ /dev/null
@@ -1,71 +0,0 @@
-<?xml version="1.0" encoding="UTF-8"?>
-<!DOCTYPE math:math PUBLIC "-//OpenOffice.org//DTD Modified W3C MathML 1.01//EN" "math.dtd">
-<math:math xmlns:math="http://www.w3.org/1998/Math/MathML">
- <math:semantics>
-  <math:mrow>
-   <math:mrow>
-    <math:mrow>
-     <math:mrow>
-      <math:mrow>
-       <math:msup>
-        <math:mi>x</math:mi>
-        <math:mn>2</math:mn>
-       </math:msup>
-       <math:mo math:stretchy="false">⋅</math:mo>
-       <math:mfrac>
-        <math:mrow>
-         <math:msup>
-          <math:mi>d</math:mi>
-          <math:mn>2</math:mn>
-         </math:msup>
-         <math:mi>y</math:mi>
-        </math:mrow>
-        <math:msup>
-         <math:mi math:fontstyle="italic">dx</math:mi>
-         <math:mn>2</math:mn>
-        </math:msup>
-       </math:mfrac>
-      </math:mrow>
-      <math:mo math:stretchy="false">+</math:mo>
-      <math:mrow>
-       <math:mi>x</math:mi>
-       <math:mo math:stretchy="false">⋅</math:mo>
-       <math:mfrac>
-        <math:mi math:fontstyle="italic">dy</math:mi>
-        <math:mi math:fontstyle="italic">dx</math:mi>
-       </math:mfrac>
-      </math:mrow>
-     </math:mrow>
-     <math:mo math:stretchy="false">+</math:mo>
-     <math:mrow>
-      <math:mrow>
-       <math:mo math:stretchy="false">(</math:mo>
-       <math:mrow>
-        <math:msup>
-         <math:mi>x</math:mi>
-         <math:mn>2</math:mn>
-        </math:msup>
-        <math:mo math:stretchy="false">−</math:mo>
-        <math:msup>
-         <math:mo math:stretchy="false">α</math:mo>
-         <math:mn>2</math:mn>
-        </math:msup>
-       </math:mrow>
-       <math:mo math:stretchy="false">)</math:mo>
-      </math:mrow>
-      <math:mo math:stretchy="false">⋅</math:mo>
-      <math:mi>y</math:mi>
-     </math:mrow>
-    </math:mrow>
-    <math:mo math:stretchy="false">=</math:mo>
-    <math:mn>0,</math:mn>
-   </math:mrow>
-   <math:mrow>
-    <math:mo math:stretchy="false">α</math:mo>
-    <math:mo math:stretchy="false">≥</math:mo>
-    <math:mn>0</math:mn>
-   </math:mrow>
-  </math:mrow>
-  <math:annotation math:encoding="StarMath 5.0">x^2 cdot {{d^2 y} over {dx^2}} + x cdot {{dy} over {dx}} + (x^2 - %alpha^2) cdot y = 0, %alpha&gt;=0</math:annotation>
- </math:semantics>
-</math:math>
\ No newline at end of file
diff --git a/scilab/modules/special_functions/help/mml/bessel_equation2.mml b/scilab/modules/special_functions/help/mml/bessel_equation2.mml
deleted file mode 100644 (file)
index f62d246..0000000
+++ /dev/null
@@ -1,71 +0,0 @@
-<?xml version="1.0" encoding="UTF-8"?>
-<!DOCTYPE math:math PUBLIC "-//OpenOffice.org//DTD Modified W3C MathML 1.01//EN" "math.dtd">
-<math:math xmlns:math="http://www.w3.org/1998/Math/MathML">
- <math:semantics>
-  <math:mrow>
-   <math:mrow>
-    <math:mrow>
-     <math:mrow>
-      <math:mrow>
-       <math:msup>
-        <math:mi>x</math:mi>
-        <math:mn>2</math:mn>
-       </math:msup>
-       <math:mo math:stretchy="false">⋅</math:mo>
-       <math:mfrac>
-        <math:mrow>
-         <math:msup>
-          <math:mi>d</math:mi>
-          <math:mn>2</math:mn>
-         </math:msup>
-         <math:mi>y</math:mi>
-        </math:mrow>
-        <math:msup>
-         <math:mi math:fontstyle="italic">dx</math:mi>
-         <math:mn>2</math:mn>
-        </math:msup>
-       </math:mfrac>
-      </math:mrow>
-      <math:mo math:stretchy="false">+</math:mo>
-      <math:mrow>
-       <math:mi>x</math:mi>
-       <math:mo math:stretchy="false">⋅</math:mo>
-       <math:mfrac>
-        <math:mi math:fontstyle="italic">dy</math:mi>
-        <math:mi math:fontstyle="italic">dx</math:mi>
-       </math:mfrac>
-      </math:mrow>
-     </math:mrow>
-     <math:mo math:stretchy="false">−</math:mo>
-     <math:mrow>
-      <math:mrow>
-       <math:mo math:stretchy="false">(</math:mo>
-       <math:mrow>
-        <math:msup>
-         <math:mi>x</math:mi>
-         <math:mn>2</math:mn>
-        </math:msup>
-        <math:mo math:stretchy="false">+</math:mo>
-        <math:msup>
-         <math:mo math:stretchy="false">α</math:mo>
-         <math:mn>2</math:mn>
-        </math:msup>
-       </math:mrow>
-       <math:mo math:stretchy="false">)</math:mo>
-      </math:mrow>
-      <math:mo math:stretchy="false">⋅</math:mo>
-      <math:mi>y</math:mi>
-     </math:mrow>
-    </math:mrow>
-    <math:mo math:stretchy="false">=</math:mo>
-    <math:mn>0,</math:mn>
-   </math:mrow>
-   <math:mrow>
-    <math:mo math:stretchy="false">α</math:mo>
-    <math:mo math:stretchy="false">≥</math:mo>
-    <math:mn>0</math:mn>
-   </math:mrow>
-  </math:mrow>
-  <math:annotation math:encoding="StarMath 5.0">x^2 cdot {{d^2 y} over {dx^2}} + x cdot {{dy} over {dx}} - (x^2 + %alpha^2) cdot y = 0, %alpha &gt;= 0</math:annotation>
- </math:semantics>
-</math:math>
\ No newline at end of file
diff --git a/scilab/modules/special_functions/help/mml/bessel_equation3.mml b/scilab/modules/special_functions/help/mml/bessel_equation3.mml
deleted file mode 100644 (file)
index de5200b..0000000
+++ /dev/null
@@ -1,98 +0,0 @@
-<?xml version="1.0" encoding="UTF-8"?>
-<!DOCTYPE math:math PUBLIC "-//OpenOffice.org//DTD Modified W3C MathML 1.01//EN" "math.dtd">
-<math:math xmlns:math="http://www.w3.org/1998/Math/MathML">
- <math:semantics>
-  <math:mrow>
-   <math:mtable>
-    <math:mtr>
-     <math:mrow>
-      <math:msubsup>
-       <math:mi>H</math:mi>
-       <math:mo math:stretchy="false">α</math:mo>
-       <math:mn>1</math:mn>
-      </math:msubsup>
-      <math:mrow>
-       <math:mrow>
-        <math:mo math:stretchy="false">(</math:mo>
-        <math:mi>z</math:mi>
-        <math:mo math:stretchy="false">)</math:mo>
-       </math:mrow>
-       <math:mo math:stretchy="false">=</math:mo>
-       <math:msub>
-        <math:mi>J</math:mi>
-        <math:mo math:stretchy="false">α</math:mo>
-       </math:msub>
-      </math:mrow>
-      <math:mrow>
-       <math:mrow>
-        <math:mo math:stretchy="false">(</math:mo>
-        <math:mi>z</math:mi>
-        <math:mo math:stretchy="false">)</math:mo>
-       </math:mrow>
-       <math:mo math:stretchy="false">+</math:mo>
-       <math:mrow>
-        <math:mi>i</math:mi>
-        <math:mo math:stretchy="false">⋅</math:mo>
-        <math:msub>
-         <math:mi>Y</math:mi>
-         <math:mo math:stretchy="false">α</math:mo>
-        </math:msub>
-       </math:mrow>
-      </math:mrow>
-      <math:mrow>
-       <math:mo math:stretchy="false">(</math:mo>
-       <math:mi>z</math:mi>
-       <math:mo math:stretchy="false">)</math:mo>
-      </math:mrow>
-     </math:mrow>
-    </math:mtr>
-    <math:mtr>
-     <math:mrow>
-      <math:msubsup>
-       <math:mi>H</math:mi>
-       <math:mo math:stretchy="false">α</math:mo>
-       <math:mn>2</math:mn>
-      </math:msubsup>
-      <math:mrow>
-       <math:mrow>
-        <math:mo math:stretchy="false">(</math:mo>
-        <math:mi>z</math:mi>
-        <math:mo math:stretchy="false">)</math:mo>
-       </math:mrow>
-       <math:mo math:stretchy="false">=</math:mo>
-       <math:msub>
-        <math:mi>J</math:mi>
-        <math:mo math:stretchy="false">α</math:mo>
-       </math:msub>
-      </math:mrow>
-      <math:mrow>
-       <math:mrow>
-        <math:mo math:stretchy="false">(</math:mo>
-        <math:mi>z</math:mi>
-        <math:mo math:stretchy="false">)</math:mo>
-       </math:mrow>
-       <math:mo math:stretchy="false">−</math:mo>
-       <math:mrow>
-        <math:mi>i</math:mi>
-        <math:mo math:stretchy="false">⋅</math:mo>
-        <math:msub>
-         <math:mi>Y</math:mi>
-         <math:mo math:stretchy="false">α</math:mo>
-        </math:msub>
-       </math:mrow>
-      </math:mrow>
-      <math:mrow>
-       <math:mo math:stretchy="false">(</math:mo>
-       <math:mi>z</math:mi>
-       <math:mo math:stretchy="false">)</math:mo>
-      </math:mrow>
-     </math:mrow>
-    </math:mtr>
-   </math:mtable>
-  </math:mrow>
-  <math:annotation math:encoding="StarMath 5.0">alignl stack {
-H^1_%alpha(z) = J_%alpha(z) + i cdot Y_%alpha(z) #
-H^2_%alpha(z) = J_%alpha(z) - i cdot Y_%alpha(z)
-}</math:annotation>
- </math:semantics>
-</math:math>
\ No newline at end of file
index 3c93730..90cf0b2 100644 (file)
@@ -2,8 +2,8 @@
 <!--
  * Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
  * Copyright (C) 2008 - INRIA
- *
  * Copyright (C) 2012 - 2016 - Scilab Enterprises
+ * Copyright (C) 2019 - Samuel GOUGEON
  *
  * This file is hereby licensed under the terms of the GNU GPL v2.0,
  * pursuant to article 5.3.4 of the CeCILL v.2.1.
  * along with this program.
  *
  -->
-<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:id="bessel" xml:lang="pt">
+<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink"
+          xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml"
+          xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook"
+          xmlns:scilab="http://www.scilab.org" xml:id="bessel" xml:lang="pt">
     <refnamediv xml:id="besseli">
         <refname>besseli</refname>
-        <refpurpose>funções modificadas de Bessel do primeiro tipo (I sub
-            alfa).
+        <refpurpose>
+            funções modificadas de Bessel do primeiro tipo (I<subscript>α</subscript>).
         </refpurpose>
     </refnamediv>
     <refnamediv xml:id="besselj">
         <refname>besselj</refname>
-        <refpurpose>funções de Bessel do primeiro tipo (J sub alpha).</refpurpose>
+        <refpurpose>
+            funções de Bessel do primeiro tipo (J<subscript>α</subscript>).
+        </refpurpose>
     </refnamediv>
     <refnamediv xml:id="besselk">
         <refname>besselk</refname>
-        <refpurpose>funções modificadas de Bessel do segundo tipo (K sub
-            alpha).
+        <refpurpose>
+            funções modificadas de Bessel do segundo tipo (K<subscript>α</subscript>).
         </refpurpose>
     </refnamediv>
     <refnamediv xml:id="bessely">
         <refname>bessely</refname>
-        <refpurpose>funções de Bessel do segundo tipo (Y sub alpha).</refpurpose>
+        <refpurpose>
+            funções de Bessel do segundo tipo (Y<subscript>α</subscript>).
+        </refpurpose>
     </refnamediv>
     <refnamediv xml:id="besselh">
         <refname>besselh</refname>
-        <refpurpose>funções de Bessel do terceiro tipo (também conhecidas como
-            funções de Hankel)
+        <refpurpose>funções de Bessel do terceiro tipo (também conhecidas como funções de Hankel)
         </refpurpose>
     </refnamediv>
     <refsynopsisdiv>
         <title>Seqüência de Chamamento</title>
-        <synopsis>y = besseli(alpha,x [,ice])
-            y = besselj(alpha,x [,ice])
-            y = besselk(alpha,x [,ice])
-            y = bessely(alpha,x [,ice])
-            y = besselh(alpha,x)
-            y = besselh(alpha,K,x [,ice])
+        <synopsis>
+            y = besseli(alpha, x [,ice])
+            y = besselj(alpha, x [,ice])
+            y = besselk(alpha, x [,ice])
+            y = bessely(alpha, x [,ice])
+            y = besselh(alpha, x)
+            y = besselh(alpha, K, x [,ice])
         </synopsis>
     </refsynopsisdiv>
     <refsection>
@@ -87,8 +94,8 @@
             <listitem>
                 <para>
                     <literal>besseli(alpha,x)</literal> computa as funções de Bessel
-                    modificadas do primeiro tipo (I sub alfa), para ordem real
-                    <literal>alpha</literal> e argumento <literal>x</literal>.
+                    modificadas do primeiro tipo (I<subscript>α</subscript>), para ordem real
+                    <varname>alpha</varname> e argumento <varname>x</varname>.
                     <literal>besseli(alpha,x,1)</literal> computa
                     <literal>besseli(alpha,x).*exp(-abs(real(x)))</literal>.
                 </para>
             <listitem>
                 <para>
                     <literal>besselj(alpha,x)</literal> computa as funções de Bessel
-                    do primeiro tipo (J sub alfa), para ordem real
-                    <literal>alpha</literal> e argumento <literal>x</literal>.
+                    do primeiro tipo (J<subscript>α</subscript>), para ordem real
+                    <varname>alpha</varname> e argumento <varname>x</varname>.
                     <literal>besselj(alpha,x,1)</literal> computa
                     <literal>besselj(alpha,x).*exp(-abs(imag(x)))</literal>.
                 </para>
             <listitem>
                 <para>
                     <literal>besselk(alpha,x)</literal> computa as funções de Bessel
-                    modificadas do segundo tipo (K sub alfa), para ordem real
-                    <literal>alpha</literal> e argumento <literal>x</literal>.
+                    modificadas do segundo tipo (K<subscript>α</subscript>), para ordem real
+                    <varname>alpha</varname> e argumento <varname>x</varname>.
                     <literal>besselk(alpha,x,1)</literal> computa
                     <literal>besselk(alpha,x).*exp(x)</literal>.
                 </para>
             </listitem>
             <listitem>
                 <para>
-                    <literal>bessely(alpha,x)</literal> computa as funções de Bessel
-                    do segundo tipo (Y sub alfa), para ordem real <literal>alpha</literal>
-                    e argumento <literal>x</literal>.
+                    <literal>bessely(alpha,x)</literal> computa as funções de Bessel do segundo
+                    tipo (Y<subscript>α</subscript>), para ordem real <varname>alpha</varname>
+                    e argumento <varname>x</varname>.
                     <literal>bessely(alpha,x,1)</literal> computa
                     <literal>bessely(alpha,x).*exp(-abs(imag(x)))</literal>.
                 </para>
                 <para>
                     <literal>besselh(alpha [,K] ,x)</literal> computa as funções de
                     Bessel do terceiro tipo (função de Hankel H1 ou H2, dependendo do
-                    <literal>K</literal>), para ordem real <literal>alpha</literal> e
-                    argumentot <literal>x</literal>. Se omitido, <literal>K</literal> é
+                    <literal>K</literal>), para ordem real <varname>alpha</varname> e
+                    argumentot <varname>x</varname>. Se omitido, <literal>K</literal> é
                     suposto como sendo 1. <literal>besselh(alpha,1,x,1)</literal> computa
                     <literal>besselh(alpha,1,x).*exp(-%i*x)</literal> e
                     <literal>besselh(alpha,2,x,1)</literal> computa
     <refsection>
         <title>Observações </title>
         <para>
-            Se <literal>alpha</literal> e <literal>x</literal> são arrays de
+            Se <varname>alpha</varname> e <varname>x</varname> são arrays de
             mesmo tamanho, o resultado <literal>y</literal> também terá este tamanho.
             Se uma entrada é um escalar, ela é expandida para o tamanho da outra
             entrada. Se uma entrada é um vetor linha e a outra é um vetor coluna, o
             resultado <literal>y</literal> é um table 2-dimensional ("tabela") de
             valores de funções.
         </para>
-        <para>As funções de Bessel Y_alfa e J_alfa são duas soluções independentes
-            da equação diferencial de Bessel:
+        <para>
+            As funções de Bessel Y<subscript>α</subscript> e J<subscript>α</subscript>
+            são duas soluções independentes da equação diferencial de Bessel:
         </para>
-        <programlisting role=""><![CDATA[
-2 2 2 x y" + x y' + (x - alfa ) y = 0 , alfa >= 0
- ]]></programlisting>
-        <para>As funções modificadas de Bessel K_alfa e I_alfa são duas soluções
-            independentes para a equação diferencial de Bessel :
+        <latex style="display" alt="x^2.(d^2y/d^2x) + x.dy/dx + (x^2 - alpha^2).y = 0,  alpha ≥ 0">
+            {x^2} \cdot {{d^2 y} \over {dx^2}} + x \cdot {{dy} \over {dx}} + (x^2 - \alpha^2) \cdot y = 0,
+            \quad\alpha\ge0
+        </latex>
+        <para>
+            As funções modificadas de Bessel K<subscript>α</subscript> e I<subscript>α</subscript>
+            são duas soluções independentes para a equação diferencial de Bessel :
         </para>
-        <programlisting role=""><![CDATA[
-2 2 2 x y" + x y' - (x + alfa ) y = 0 , alfa >= 0
- ]]></programlisting>
-        <para>As funções de Hankel de primeiro e segundo tipos H^1_alfa e
-            H^2_alfa, são combinações lineares das funções de Bessel de primeiro e
-            segundo tipos:
+        <latex style="display" alt="x^2.(d^2y/d^2x) + x.dy/dx + (alpha^2 - x^2).y = 0,  alpha ≥ 0">
+            {x^2} \cdot {{d^2 y} \over {dx^2}} + x \cdot {{dy} \over {dx}} + (\alpha^2 - x^2) \cdot y = 0,
+            \quad\alpha\ge0
+        </latex>
+        <para>
+            As funções de Hankel de primeiro e segundo tipos
+            H<superscript>2</superscript><subscript>α</subscript> e
+            H<superscript>2</superscript><subscript>α</subscript>, são combinações
+            lineares das funções de Bessel de primeiro e segundo tipos:
         </para>
-        <programlisting role=""><![CDATA[
-H^1_alfa(z) = J_alfa(z) + i Y_alfa(z) H^2_alfa(z) = J_alfa(z) - i Y_alfa(z)
- ]]></programlisting>
+        <latex style="display" alt="H^1_α(z) = J_α(z) + i \cdot Y_α(z)  \n
+H^2_α(z) = J_α(z) - i \cdot Y_α(z)">
+            H^1_{\alpha}(z) = J_{\alpha}(z) + i \cdot Y_{\alpha}(z) \\
+            H^2_{\alpha}(z) = J_{\alpha}(z) - i \cdot Y_{\alpha}(z)
+        </latex>
     </refsection>
+
     <refsection>
         <title>Exemplos</title>
         <programlisting role="example"><![CDATA[
@@ -179,6 +195,21 @@ subplot(2,1,2)
 plot2d(x,besseli(0:4,x,1), style=2:6)
 legend('I'+string(0:4),1);
 xtitle("Algumas funções modificadas de Bessel do primeiro tipo escaladas")
+ ]]></programlisting>
+        <scilab:image>
+            x = linspace(0.01,10,5000)';
+            clf()
+            subplot(2,1,1)
+            plot2d(x,besseli(0:4,x), style=2:6)
+            legend('I'+string(0:4),2);
+            xtitle("Some modified Bessel functions of the first kind")
+            subplot(2,1,2)
+            plot2d(x,besseli(0:4,x,1), style=2:6)
+            legend('I'+string(0:4),1);
+            xtitle("Some modified scaled Bessel functions of the first kind")
+        </scilab:image>
+        <para/>
+        <programlisting role="example"><![CDATA[
 
 // Funções J de Bessel
 // =================
@@ -187,7 +218,16 @@ clf()
 plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4")
 legend('I'+string(0:4),1);
 xtitle("Algumas funções de Bessel do primeiro tipo")
+ ]]></programlisting>
+        <scilab:image>
+            x = linspace(0,40,5000)';
+            plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4")
+            legend('I'+string(0:4),1);
+            xtitle("Some Bessel functions of the first kind")
+        </scilab:image>
 
+        <para/>
+        <programlisting role="example"><![CDATA[
 // Usando o fato de que J_(1/2)(x) = sqrt(2/(x pi)) sin(x)
 // Para comparar o algoritmo de besselj(0.5,x) com uma fórmula mais direta
 x = linspace(0.1,40,5000)';
@@ -202,7 +242,24 @@ xtitle("besselj(0.5,x)")
 subplot(2,1,2)
 plot2d(x(ind), er(ind), style=2, logflag="nl")
 xtitle("Erro relativo entre as duas fórmulas para besselj(0.5,x)")
+ ]]></programlisting>
+        <scilab:image><![CDATA[
+   x = linspace(0.1,40,5000)';
+   y1 = besselj(0.5, x);
+   y2 = sqrt(2 ./(%pi*x)).*sin(x);
+   er = abs((y1-y2)./y2);
+   ind = find(er > 0 & y2 ~= 0);
+   clf()
+   subplot(2,1,1)
+   plot2d(x,y1,style=2)
+   xtitle("besselj(0.5,x)")
+   subplot(2,1,2)
+   plot2d(x(ind), er(ind), style=2, logflag="nl")
+   xtitle("relative error between 2 formulae for besselj(0.5,x)")
+ ]]></scilab:image>
 
+        <para/>
+        <programlisting role="example"><![CDATA[
 // Funções K de Bessel
 // =================
 x = linspace(0.01,10,5000)';
@@ -215,7 +272,22 @@ subplot(2,1,2)
 plot2d(x,besselk(0:4,x,1), style=0:4, rect=[0,0,6,10])
 legend('K'+string(0:4),1);
 xtitle("Algumas funções modificadas de Bessel do segundo tipo escaladas")
+ ]]></programlisting>
+        <scilab:image>
+            x = linspace(0.01,10,5000)';
+            clf()
+            subplot(2,1,1)
+            plot2d(x,besselk(0:4,x), style=0:4, rect=[0,0,6,10])
+            legend('K'+string(0:4),1);
+            xtitle("Some modified Bessel functions of the second kind")
+            subplot(2,1,2)
+            plot2d(x,besselk(0:4,x,1), style=0:4, rect=[0,0,6,10])
+            legend('K'+string(0:4),1);
+            xtitle("Some modified scaled Bessel functions of the second kind")
+        </scilab:image>
 
+        <para/>
+        <programlisting role="example"><![CDATA[
 // Funções Y de Bessel
 // =================
 x = linspace(0.1,40,5000)'; // funções Y de Bessel não possuem limite para x -> 0+
@@ -223,7 +295,17 @@ clf()
 plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6])
 legend('Y'+string(0:4),4);
 xtitle("Algumas funções de Bessel do segundo tipo")
+ ]]></programlisting>
+        <scilab:image>
+            x = linspace(0.1,40,5000)'; // Y Bessel functions are unbounded  for x -> 0+
+            clf()
+            plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6])
+            legend('Y'+string(0:4),4);
+            xtitle("Some Bessel functions of the second kind")
+        </scilab:image>
 
+        <para/>
+        <programlisting role="example"><![CDATA[
 // Funções H de Bessel
 // =================
 x=-4:0.025:2; y=-1.5:0.025:1.5;
@@ -236,6 +318,17 @@ contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5])
 legends(string(0.2:0.2:3.2),1:16,"ur")
 xtitle("Curvas de nível de |H1(0,z)|")
  ]]></programlisting>
+        <scilab:image>
+            x=-4:0.025:2; y=-1.5:0.025:1.5;
+            [X,Y] = ndgrid(x,y);
+            H = besselh(0,1,X+%i*Y);
+            clf();f=gcf();
+            xset("fpf"," ")
+            f.color_map=jetcolormap(16);
+            contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5])
+            legends(string(0.2:0.2:3.2),1:16,"ur")
+            xtitle("Level curves of |H1(0,z)|")
+        </scilab:image>
     </refsection>
     <refsection>
         <title>Autores</title>