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

   1 <?xml version="1.0" encoding="UTF-8"?>
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  16 <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">
  17     <refnamediv xml:id="besseli">
  18         <refname>besseli</refname>
  19         <refpurpose>第1種修正ベッセル関数 (I_alpha).</refpurpose>
  20     </refnamediv>
  21     <refnamediv xml:id="besselj">
  22         <refname>besselj</refname>
  23         <refpurpose>第1種ベッセル関数 (J_alpha).</refpurpose>
  24     </refnamediv>
  25     <refnamediv xml:id="besselk">
  26         <refname>besselk</refname>
  27         <refpurpose>第2種修正ベッセル関数 (K_alpha).</refpurpose>
  28     </refnamediv>
  29     <refnamediv xml:id="bessely">
  30         <refname>bessely</refname>
  31         <refpurpose>第2種ベッセル関数 (Y_alpha).</refpurpose>
  32     </refnamediv>
  33     <refnamediv xml:id="besselh">
  34         <refname>besselh</refname>
  35         <refpurpose>第3種ベッセル関数 (ハンケル関数と同じ)</refpurpose>
  36     </refnamediv>
  37     <refsynopsisdiv>
  38         <title>呼び出し手順</title>
  39         <synopsis>y = besseli(alpha,x [,ice])
  40             y = besselj(alpha,x [,ice])
  41             y = besselk(alpha,x [,ice])
  42             y = bessely(alpha,x [,ice])
  43             y = besselh(alpha,x)
  44             y = besselh(alpha,K,x [,ice])
  45         </synopsis>
  46     </refsynopsisdiv>
  47     <refsection>
  48         <title>引数</title>
  49         <variablelist>
  50             <varlistentry>
  51                 <term>x</term>
  52                 <listitem>
  53                     <para>実数または複素数のベクトル.</para>
  54                 </listitem>
  55             </varlistentry>
  56             <varlistentry>
  57                 <term>alpha</term>
  58                 <listitem>
  59                     <para>r実数ベクトル</para>
  60                 </listitem>
  61             </varlistentry>
  62             <varlistentry>
  63                 <term>ice</term>
  64                 <listitem>
  65                     <para>整数フラグ, デフォルト値: 0</para>
  66                 </listitem>
  67             </varlistentry>
  68             <varlistentry>
  69                 <term>K</term>
  70                 <listitem>
  71                     <para>整数, 指定可能な値は 1 または 2, ハンケル関数の型.</para>
  72                 </listitem>
  73             </varlistentry>
  74         </variablelist>
  75     </refsection>
  76     <refsection>
  77         <title>説明</title>
  78         <itemizedlist>
  79             <listitem>
  80                 <para>
  81                     <literal>besseli(alpha,x)</literal> は,
  82                     実数の次数<literal>alpha</literal> および引数 <literal>x</literal>に関する
  83                     第1種修正ベッセル関数(I_alpha)を計算します,
  84                     <literal>besseli(alpha,x,1)</literal> は
  85                     <literal>besseli(alpha,x).*exp(-abs(real(x)))</literal>を計算します.
  86                 </para>
  87             </listitem>
  88             <listitem>
  89                 <para>
  90                     <literal>besselj(alpha,x)</literal> は第1種のベッセル関数(J_alpha)を
  91                     実数の次数<literal>alpha</literal> および引数 <literal>x</literal>に関して
  92                     計算します.
  93                     <literal>besselj(alpha,x,1)</literal> は
  94                     <literal>besselj(alpha,x).*exp(-abs(imag(x)))</literal>を計算します.
  95                 </para>
  96             </listitem>
  97             <listitem>
  98                 <para>
  99                     <literal>besselk(alpha,x)</literal> は第2種修正ベッセル関数
 100                     (K_alpha)を
 101                     実数の次数<literal>alpha</literal> および引数 <literal>x</literal>に関して
 102                     計算します.
 103                     <literal>besselk(alpha,x,1)</literal> は
 104                     <literal>besselk(alpha,x).*exp(x)</literal>を計算します.
 105                 </para>
 106             </listitem>
 107             <listitem>
 108                 <para>
 109                     <literal>bessely(alpha,x)</literal>は第2種のベッセル関数(Y_alpha)を
 110                     実数の次数<literal>alpha</literal> および引数 <literal>x</literal>に関して
 111                     計算します.
 112                     <literal>bessely(alpha,x,1)</literal> は
 113                     <literal>bessely(alpha,x).*exp(-abs(imag(x)))</literal>を計算します.
 114                 </para>
 115             </listitem>
 116             <listitem>
 117                 <para>
 118                     <literal>besselh(alpha [,K] ,x)</literal> は第3種のベッセル関数
 119                     (<literal>K</literal>に依存してハンケル関数 H1 または H2)を
 120                     実数の次数<literal>alpha</literal> および引数 <literal>x</literal>に関して
 121                     計算します.<literal>K</literal>が省略された場合,
 122                     1に等しいと仮定されます.
 123                     <literal>besselh(alpha,1,x,1)</literal>は
 124                     <literal>besselh(alpha,1,x).*exp(-%i*x)</literal>を計算し,
 125                     <literal>besselh(alpha,2,x,1)</literal> は
 126                     <literal>besselh(alpha,2,x).*exp(%i*x)</literal>を計算します.
 127                 </para>
 128             </listitem>
 129         </itemizedlist>
 130     </refsection>
 131     <refsection>
 132         <title>注意</title>
 133         <para>
 134             <literal>alpha</literal>および <literal>x</literal>が同じ大きさの
 135             配列の場合,結果<literal>y</literal>も同じ大きさとなります.
 136             入力のどちらかがスカラーの場合,
 137             もう片方の大きさにまで拡張されます.
 138             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,
 139             結果<literal>y</literal>は関数値の二次元テーブルとなります.
 140         </para>
 141         <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
 142             2つの独立解です:
 143         </para>
 144         <informalequation>
 145             <mediaobject>
 146                 <imageobject>
 147                     <imagedata align="center" fileref="../mml/bessel_equation1.mml"/>
 148                 </imageobject>
 149             </mediaobject>
 150         </informalequation>
 151         <para>修正ベッセル関数K_alpha および I_alphaは
 152             修正ベッセル微分方程式の2つの独立解です:
 153         </para>
 154         <informalequation>
 155             <mediaobject>
 156                 <imageobject>
 157                     <imagedata align="center" fileref="../mml/bessel_equation2.mml"/>
 158                 </imageobject>
 159             </mediaobject>
 160         </informalequation>
 161         <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
 162             で,第1種および第2種のベッセル関数の線形結合です:
 163         </para>
 164         <informalequation>
 165             <mediaobject>
 166                 <imageobject>
 167                     <imagedata align="center" fileref="../mml/bessel_equation3.mml"/>
 168                 </imageobject>
 169             </mediaobject>
 170         </informalequation>
 171     </refsection>
 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         <programlisting role="example"><![CDATA[
 201 // besselJ 関数
 202 // =================
 203    x = linspace(0,40,5000)';
 204    clf()
 205    plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4")
 206    legend('I'+string(0:4),1);
 207    xtitle("Some Bessel functions of the first kind")
 208  ]]></programlisting>
 209         <scilab:image>
 210             x = linspace(0,40,5000)';
 211             plot2d(x,besselj(0:4,x), style=2:6, leg="J0@J1@J2@J3@J4")
 212             legend('I'+string(0:4),1);
 213             xtitle("Some Bessel functions of the first kind")
 214         </scilab:image>
 215         <programlisting role="example"><![CDATA[
 216 // J_(1/2)(x) = sqrt(2/(x pi)) sin(x)　の関係を用いて
 217 //  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
 218    x = linspace(0.1,40,5000)';
 219    y1 = besselj(0.5, x);
 220    y2 = sqrt(2 ./(%pi*x)).*sin(x);
 221    er = abs((y1-y2)./y2);
 222    ind = find(er &gt; 0 &amp; y2 ~= 0);
 223    clf()
 224    subplot(2,1,1)
 225    plot2d(x,y1,style=2)
 226    xtitle("besselj(0.5,x)")
 227    subplot(2,1,2)
 228    plot2d(x(ind), er(ind), style=2, logflag="nl")
 229    xtitle("relative error between 2 formulae for besselj(0.5,x)")
 230  ]]></programlisting>
 231         <scilab:image><![CDATA[
 232    x = linspace(0.1,40,5000)';
 233    y1 = besselj(0.5, x);
 234    y2 = sqrt(2 ./(%pi*x)).*sin(x);
 235    er = abs((y1-y2)./y2);
 236    ind = find(er > 0 & y2 ~= 0);
 237    clf()
 238    subplot(2,1,1)
 239    plot2d(x,y1,style=2)
 240    xtitle("besselj(0.5,x)")
 241    subplot(2,1,2)
 242    plot2d(x(ind), er(ind), style=2, logflag="nl")
 243    xtitle("relative error between 2 formulae for besselj(0.5,x)")
 244  ]]></scilab:image>
 245         <programlisting role="example"><![CDATA[
 246 // besselK 関数
 247 // =================
 248    x = linspace(0.01,10,5000)';
 249    clf()
 250    subplot(2,1,1)
 251    plot2d(x,besselk(0:4,x), style=0:4, rect=[0,0,6,10])
 252    legend('K'+string(0:4),1);
 253    xtitle("Some modified Bessel functions of the second kind")
 254    subplot(2,1,2)
 255    plot2d(x,besselk(0:4,x,1), style=0:4, rect=[0,0,6,10])
 256    legend('K'+string(0:4),1);
 257    xtitle("Some modified scaled Bessel functions of the second kind")
 258  ]]></programlisting>
 259         <scilab:image>
 260             x = linspace(0.01,10,5000)';
 261             clf()
 262             subplot(2,1,1)
 263             plot2d(x,besselk(0:4,x), style=0:4, rect=[0,0,6,10])
 264             legend('K'+string(0:4),1);
 265             xtitle("Some modified Bessel functions of the second kind")
 266             subplot(2,1,2)
 267             plot2d(x,besselk(0:4,x,1), style=0:4, rect=[0,0,6,10])
 268             legend('K'+string(0:4),1);
 269             xtitle("Some modified scaled Bessel functions of the second kind")
 270         </scilab:image>
 271         <programlisting role="example"><![CDATA[
 272 // besselY 関数
 273 // =================
 274    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
 275    clf()
 276    plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6])
 277    legend('Y'+string(0:4),4);
 278    xtitle("Some Bessel functions of the second kind")
 279  ]]></programlisting>
 280         <scilab:image>
 281             x = linspace(0.1,40,5000)'; // Y Bessel functions are unbounded  for x -> 0+
 282             clf()
 283             plot2d(x,bessely(0:4,x), style=0:4, rect=[0,-1.5,40,0.6])
 284             legend('Y'+string(0:4),4);
 285             xtitle("Some Bessel functions of the second kind")
 286         </scilab:image>
 287         <programlisting role="example"><![CDATA[
 288 // besselH 関数
 289 // =================
 290    x=-4:0.025:2; y=-1.5:0.025:1.5;
 291    [X,Y] = ndgrid(x,y);
 292    H = besselh(0,1,X+%i*Y);
 293    clf();f=gcf();
 294    xset("fpf"," ")
 295    f.color_map=jetcolormap(16);
 296    contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5])
 297    legends(string(0.2:0.2:3.2),1:16,"ur")
 298    xtitle("Level curves of |H1(0,z)|")
 299  ]]></programlisting>
 300         <scilab:image>
 301             x=-4:0.025:2; y=-1.5:0.025:1.5;
 302             [X,Y] = ndgrid(x,y);
 303             H = besselh(0,1,X+%i*Y);
 304             clf();f=gcf();
 305             xset("fpf"," ")
 306             f.color_map=jetcolormap(16);
 307             contour2d(x,y,abs(H),0.2:0.2:3.2,strf="034",rect=[-4,-1.5,3,1.5])
 308             legends(string(0.2:0.2:3.2),1:16,"ur")
 309             xtitle("Level curves of |H1(0,z)|")
 310         </scilab:image>
 311     </refsection>
 312     <refsection>
 313         <title>使用される関数</title>
 314         <para>ソースコードは SCI/modules/special_functions/src/fortran/slatec および
 315             SCI/modules/special_functions/src/fortran にあります
 316         </para>
 317         <para>Slatec : dbesi.f, zbesi.f, dbesj.f, zbesj.f, dbesk.f, zbesk.f,
 318             dbesy.f, zbesy.f, zbesh.f
 319         </para>
 320         <para>拡張定義領域(Serge Steer INRIA)のドライバ (Serge Steer INRIA): dbesig.f,
 321             zbesig.f, dbesjg.f, zbesjg.f, dbeskg.f, zbeskg.f, dbesyg.f, zbesyg.f,
 322             zbeshg.f
 323         </para>
 324     </refsection>
 325 </refentry>