scilab/modules/special_functions/help/ja_JP/bessel.xml

   1 <?xml version="1.0" encoding="UTF-8"?>
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   3  * Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
   4  * Copyright (C) 2008 - INRIA
   5  * Copyright (C) 2012 - 2016 - Scilab Enterprises
   6  * Copyright (C) 2019 - Samuel GOUGEON
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   8  * This file is hereby licensed under the terms of the GNU GPL v2.0,
   9  * pursuant to article 5.3.4 of the CeCILL v.2.1.
  10  * This file was originally licensed under the terms of the CeCILL v2.1,
  11  * and continues to be available under such terms.
  12  * For more information, see the COPYING file which you should have received
  13  * along with this program.
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  16 <refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink"
  17           xmlns:svg="http://www.w3.org/2000/svg" xmlns:ns4="http://www.w3.org/1999/xhtml"
  18           xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook"
  19           xmlns:scilab="http://www.scilab.org"  xml:id="bessel" xml:lang="ja">
  20     <refnamediv xml:id="besseli">
  21         <refname>besseli</refname>
  22       <refpurpose>第1種修正ベッセル関数 (I<subscript>α</subscript>).</refpurpose>
  23     </refnamediv>
  24     <refnamediv xml:id="besselj">
  25         <refname>besselj</refname>
  26         <refpurpose>第1種ベッセル関数 (J<subscript>α</subscript>).</refpurpose>
  27     </refnamediv>
  28     <refnamediv xml:id="besselk">
  29         <refname>besselk</refname>
  30         <refpurpose>第2種修正ベッセル関数 (K<subscript>α</subscript>).</refpurpose>
  31     </refnamediv>
  32     <refnamediv xml:id="bessely">
  33         <refname>bessely</refname>
  34         <refpurpose>第2種ベッセル関数 (Y<subscript>α</subscript>).</refpurpose>
  35     </refnamediv>
  36     <refnamediv xml:id="besselh">
  37         <refname>besselh</refname>
  38         <refpurpose>第3種ベッセル関数 (ハンケル関数と同じ)</refpurpose>
  39     </refnamediv>
  40     <refsynopsisdiv>
  41         <title>呼び出し手順</title>
  42         <synopsis>
  43             y = besseli(alpha, x [,ice])
  44             y = besselj(alpha, x [,ice])
  45             y = besselk(alpha, x [,ice])
  46             y = bessely(alpha, x [,ice])
  47             y = besselh(alpha, x)
  48             y = besselh(alpha, K, x [,ice])
  49         </synopsis>
  50     </refsynopsisdiv>
  51     <refsection>
  52         <title>引数</title>
  53         <variablelist>
  54             <varlistentry>
  55                 <term>x</term>
  56                 <listitem>
  57                     <para>実数または複素数のベクトル.</para>
  58                 </listitem>
  59             </varlistentry>
  60             <varlistentry>
  61                 <term>alpha</term>
  62                 <listitem>
  63                     <para>r実数ベクトル</para>
  64                 </listitem>
  65             </varlistentry>
  66             <varlistentry>
  67                 <term>ice</term>
  68                 <listitem>
  69                     <para>整数フラグ, デフォルト値: 0</para>
  70                 </listitem>
  71             </varlistentry>
  72             <varlistentry>
  73                 <term>K</term>
  74                 <listitem>
  75                     <para>整数, 指定可能な値は 1 または 2, ハンケル関数の型.</para>
  76                 </listitem>
  77             </varlistentry>
  78         </variablelist>
  79     </refsection>
  80     <refsection>
  81         <title>説明</title>
  82         <itemizedlist>
  83             <listitem>
  84                 <para>
  85                     <literal>besseli(alpha,x)</literal> は,
  86                     実数の次数<varname>alpha</varname> および引数 <varname>x</varname>に関する
  87                     第1種修正ベッセル関数(I<subscript>α</subscript>)を計算します,
  88                     <literal>besseli(alpha,x,1)</literal> は
  89                     <literal>besseli(alpha,x).*exp(-abs(real(x)))</literal>を計算します.
  90                 </para>
  91             </listitem>
  92             <listitem>
  93                 <para>
  94                     <literal>besselj(alpha,x)</literal> は第1種のベッセル関数(J<subscript>α</subscript>)を
  95                     実数の次数<varname>alpha</varname> および引数 <varname>x</varname>に関して
  96                     計算します.
  97                     <literal>besselj(alpha,x,1)</literal> は
  98                     <literal>besselj(alpha,x).*exp(-abs(imag(x)))</literal>を計算します.
  99                 </para>
 100             </listitem>
 101             <listitem>
 102                 <para>
 103                     <literal>besselk(alpha,x)</literal> は第2種修正ベッセル関数
 104                     (K<subscript>α</subscript>)を
 105                     実数の次数<varname>alpha</varname> および引数 <varname>x</varname>に関して
 106                     計算します.
 107                     <literal>besselk(alpha,x,1)</literal> は
 108                     <literal>besselk(alpha,x).*exp(x)</literal>を計算します.
 109                 </para>
 110             </listitem>
 111             <listitem>
 112                 <para>
 113                     <literal>bessely(alpha,x)</literal>は第2種のベッセル関数(Y<subscript>α</subscript>)を
 114                     実数の次数<varname>alpha</varname> および引数 <varname>x</varname>に関して
 115                     計算します.
 116                     <literal>bessely(alpha,x,1)</literal> は
 117                     <literal>bessely(alpha,x).*exp(-abs(imag(x)))</literal>を計算します.
 118                 </para>
 119             </listitem>
 120             <listitem>
 121                 <para>
 122                     <literal>besselh(alpha [,K] ,x)</literal> は第3種のベッセル関数
 123                     (<literal>K</literal>に依存してハンケル関数 H1 または H2)を
 124                     実数の次数<varname>alpha</varname> および引数 <varname>x</varname>に関して
 125                     計算します.<literal>K</literal>が省略された場合,
 126                     1に等しいと仮定されます.
 127                     <literal>besselh(alpha,1,x,1)</literal>は
 128                     <literal>besselh(alpha,1,x).*exp(-%i*x)</literal>を計算し,
 129                     <literal>besselh(alpha,2,x,1)</literal> は
 130                     <literal>besselh(alpha,2,x).*exp(%i*x)</literal>を計算します.
 131                 </para>
 132             </listitem>
 133         </itemizedlist>
 134     </refsection>
 135     <refsection>
 136         <title>注意</title>
 137         <para>
 138             <varname>alpha</varname>および <varname>x</varname>が同じ大きさの
 139             配列の場合,結果<literal>y</literal>も同じ大きさとなります.
 140             入力のどちらかがスカラーの場合,
 141             もう片方の大きさにまで拡張されます.
 142             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,
 143             結果<literal>y</literal>は関数値の二次元テーブルとなります.
 144         </para>
 145         <para>
 146             Y<subscript>α</subscript> および J<subscript>α</subscript>
 147             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:
 148         </para>
 149         <latex style="display" alt="x^2.(d^2y/d^2x) + x.dy/dx + (x^2 - alpha^2).y = 0,  alpha ≥ 0">
 150             {x^2} \cdot {{d^2 y} \over {dx^2}} + x \cdot {{dy} \over {dx}} + (x^2 - \alpha^2) \cdot y = 0,
 151             \quad\alpha\ge0
 152         </latex>
 153         <para>修正ベッセル関数K<subscript>α</subscript> および I<subscript>α</subscript>は
 154             修正ベッセル微分方程式の2つの独立解です:
 155         </para>
 156         <latex style="display" alt="x^2.(d^2y/d^2x) + x.dy/dx + (alpha^2 - x^2).y = 0,  alpha ≥ 0">
 157             {x^2} \cdot {{d^2 y} \over {dx^2}} + x \cdot {{dy} \over {dx}} + (\alpha^2 - x^2) \cdot y = 0,
 158             \quad\alpha\ge0
 159         </latex>
 160         <para>
 161             H<subscript>α</subscript><superscript>1</superscript> および
 162             H<subscript>α</subscript><superscript>2</superscript>は第1種および第2種のハンケル関数
 163             で,第1種および第2種のベッセル関数の線形結合です:
 164         </para>
 165         <latex style="display" alt="H^1_α(z) = J_α(z) + i \cdot Y_α(z)  \n
 166 H^2_α(z) = J_α(z) - i \cdot Y_α(z)">
 167             H^1_{\alpha}(z) = J_{\alpha}(z) + i \cdot Y_{\alpha}(z) \\
 168             H^2_{\alpha}(z) = J_{\alpha}(z) - i \cdot Y_{\alpha}(z)
 169         </latex>
 170     </refsection>
 171
 172     <refsection>
 173         <title>例</title>
 174         <programlisting role="example"><![CDATA[
 175 //  besselI 関数
 176 // ==================
 177    x = linspace(0.01,10,5000)';
 178    clf()
 179    subplot(2,1,1)
 180    plot2d(x,besseli(0:4,x), style=2:6)
 181    legend('I'+string(0:4),2);
 182    xtitle("Some modified Bessel functions of the first kind")
 183    subplot(2,1,2)
 184    plot2d(x,besseli(0:4,x,1), style=2:6)
 185    legend('I'+string(0:4),1);
 186    xtitle("Some modified scaled Bessel functions of the first kind")
 187  ]]></programlisting>
 188         <scilab:image>
 189             x = linspace(0.01,10,5000)';
 190             clf()
 191             subplot(2,1,1)
 192             plot2d(x,besseli(0:4,x), style=2:6)
 193             legend('I'+string(0:4),2);
 194             xtitle("Some modified Bessel functions of the first kind")
 195             subplot(2,1,2)
 196             plot2d(x,besseli(0:4,x,1), style=2:6)
 197             legend('I'+string(0:4),1);
 198             xtitle("Some modified scaled Bessel functions of the first kind")
 199         </scilab:image>
 200
 201         <para/>
 202         <programlisting role="example"><![CDATA[
 203 // besselJ 関数
 204 // =================
 205    x = linspace(0,40,5000)';
 206    clf()
 207    plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4")
 208    legend('I'+string(0:4),1);
 209    xtitle("Some Bessel functions of the first kind")
 210  ]]></programlisting>
 211         <scilab:image>
 212             x = linspace(0,40,5000)';
 213             plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4")
 214             legend('I'+string(0:4),1);
 215             xtitle("Some Bessel functions of the first kind")
 216         </scilab:image>
 217
 218         <para/>
 219         <programlisting role="example"><![CDATA[
 220 // J_(1/2)(x) = sqrt(2/(x pi)) sin(x)　の関係を用いて
 221 //  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
 222    x = linspace(0.1,40,5000)';
 223    y1 = besselj(0.5, x);
 224    y2 = sqrt(2 ./(%pi*x)).*sin(x);
 225    er = abs((y1-y2)./y2);
 226    ind = find(er &gt; 0 &amp; y2 ~= 0);
 227    clf()
 228    subplot(2,1,1)
 229    plot2d(x,y1,style=2)
 230    xtitle("besselj(0.5,x)")
 231    subplot(2,1,2)
 232    plot2d(x(ind), er(ind), style=2, logflag="nl")
 233    xtitle("relative error between 2 formulae for besselj(0.5,x)")
 234  ]]></programlisting>
 235         <scilab:image><![CDATA[
 236    x = linspace(0.1,40,5000)';
 237    y1 = besselj(0.5, x);
 238    y2 = sqrt(2 ./(%pi*x)).*sin(x);
 239    er = abs((y1-y2)./y2);
 240    ind = find(er > 0 & y2 ~= 0);
 241    clf()
 242    subplot(2,1,1)
 243    plot2d(x,y1,style=2)
 244    xtitle("besselj(0.5,x)")
 245    subplot(2,1,2)
 246    plot2d(x(ind), er(ind), style=2, logflag="nl")
 247    xtitle("relative error between 2 formulae for besselj(0.5,x)")
 248  ]]></scilab:image>
 249
 250         <para/>
 251         <programlisting role="example"><![CDATA[
 252 // besselK 関数
 253 // =================
 254    x = linspace(0.01,10,5000)';
 255    clf()
 256    subplot(2,1,1)
 257    plot2d(x,besselk(0:4,x), style=0:4, rect=[0,0,6,10])
 258    legend('K'+string(0:4),1);
 259    xtitle("Some modified Bessel functions of the second kind")
 260    subplot(2,1,2)
 261    plot2d(x,besselk(0:4,x,1), style=0:4, rect=[0,0,6,10])
 262    legend('K'+string(0:4),1);
 263    xtitle("Some modified scaled Bessel functions of the second kind")
 264  ]]></programlisting>
 265         <scilab:image>
 266             x = linspace(0.01,10,5000)';
 267             clf()
 268             subplot(2,1,1)
 269             plot2d(x,besselk(0:4,x), style=0:4, rect=[0,0,6,10])
 270             legend('K'+string(0:4),1);
 271             xtitle("Some modified Bessel functions of the second kind")
 272             subplot(2,1,2)
 273             plot2d(x,besselk(0:4,x,1), style=0:4, rect=[0,0,6,10])
 274             legend('K'+string(0:4),1);
 275             xtitle("Some modified scaled Bessel functions of the second kind")
 276         </scilab:image>
 277
 278         <para/>
 279         <programlisting role="example"><![CDATA[
 280 // besselY 関数
 281 // =================
 282    x = linspace(0.1,40,5000)'; // Y ベッセル関数は x -&gt; 0+ については制限されません
 283    clf()
 284    plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6])
 285    legend('Y'+string(0:4),4);
 286    xtitle("Some Bessel functions of the second kind")
 287  ]]></programlisting>
 288         <scilab:image>
 289             x = linspace(0.1,40,5000)'; // Y Bessel functions are unbounded  for x -> 0+
 290             clf()
 291             plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6])
 292             legend('Y'+string(0:4),4);
 293             xtitle("Some Bessel functions of the second kind")
 294         </scilab:image>
 295
 296         <para/>
 297         <programlisting role="example"><![CDATA[
 298 // besselH 関数
 299 // =================
 300    x=-4:0.025:2; y=-1.5:0.025:1.5;
 301    [X,Y] = ndgrid(x,y);
 302    H = besselh(0,1,X+%i*Y);
 303    clf();f=gcf();
 304    xset("fpf"," ")
 305    f.color_map=jetcolormap(16);
 306    contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5])
 307    legends(string(0.2:0.2:3.2),1:16,"ur")
 308    xtitle("Level curves of |H1(0,z)|")
 309  ]]></programlisting>
 310         <scilab:image>
 311             x=-4:0.025:2; y=-1.5:0.025:1.5;
 312             [X,Y] = ndgrid(x,y);
 313             H = besselh(0,1,X+%i*Y);
 314             clf();f=gcf();
 315             xset("fpf"," ")
 316             f.color_map=jetcolormap(16);
 317             contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5])
 318             legends(string(0.2:0.2:3.2),1:16,"ur")
 319             xtitle("Level curves of |H1(0,z)|")
 320         </scilab:image>
 321     </refsection>
 322     <refsection>
 323         <title>使用される関数</title>
 324         <para>ソースコードは SCI/modules/special_functions/src/fortran/slatec および
 325             SCI/modules/special_functions/src/fortran にあります
 326         </para>
 327         <para>Slatec : dbesi.f, zbesi.f, dbesj.f, zbesj.f, dbesk.f, zbesk.f,
 328             dbesy.f, zbesy.f, zbesh.f
 329         </para>
 330         <para>拡張定義領域(Serge Steer INRIA)のドライバ (Serge Steer INRIA): dbesig.f,
 331             zbesig.f, dbesjg.f, zbesjg.f, dbeskg.f, zbeskg.f, dbesyg.f, zbesyg.f,
 332             zbeshg.f
 333         </para>
 334     </refsection>
 335 </refentry>