<title>Modified Bessel Functions of the First and Second Kinds</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
-<link rel="home" href="../../index.html" title="Math Toolkit 2.10.0">
+<link rel="home" href="../../index.html" title="Math Toolkit 2.11.0">
<link rel="up" href="../bessel.html" title="Bessel Functions">
<link rel="prev" href="bessel_root.html" title="Finding Zeros of Bessel Functions of the First and Second Kinds">
<link rel="next" href="sph_bessel.html" title="Spherical Bessel Functions of the First and Second Kinds">
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_i</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
-<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
-<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_i</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
+<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_i</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">></span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_k</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
-<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
-<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_k</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
+<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
+<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">cyl_bessel_k</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">v</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
</pre>
<h5>
<a name="math_toolkit.bessel.mbessel.h1"></a>
the result of the modified Bessel functions of the first and second kind
respectively:
</p>
-<p>
- cyl_bessel_i(v, x) = I<sub>v</sub>(x)
- </p>
-<p>
- cyl_bessel_k(v, x) = K<sub>v</sub>(x)
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ cyl_bessel_i(v, x) = I<sub>v</sub>(x)
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ cyl_bessel_k(v, x) = K<sub>v</sub>(x)
+ </p></blockquote></div>
<p>
where:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel2.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel3.svg"></span>
+
+ </p></blockquote></div>
<p>
The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a> when T1 and T2 are different types.
an integer.
</p>
<p>
- The final <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
+ The final <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
- what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">policy
+ what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
<p>
The following graph illustrates the exponential behaviour of I<sub>v</sub>.
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_i.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_i.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
The following graph illustrates the exponential decay of K<sub>v</sub>.
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_k.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/cyl_bessel_k.svg" align="middle"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.bessel.mbessel.h2"></a>
<span class="phrase"><a name="math_toolkit.bessel.mbessel.testing"></a></span><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.testing">Testing</a>
of the worst problem cases in other libraries:
</p>
<div class="table">
-<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i_integer_orders_"></a><p class="title"><b>Table 7.44. Error rates for cyl_bessel_i (integer orders)</b></p>
+<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i_integer_orders_"></a><p class="title"><b>Table 8.44. Error rates for cyl_bessel_i (integer orders)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i (integer orders)">
<colgroup>
<col>
</table></div>
</div>
<br class="table-break"><div class="table">
-<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i"></a><p class="title"><b>Table 7.45. Error rates for cyl_bessel_i</b></p>
+<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_i"></a><p class="title"><b>Table 8.45. Error rates for cyl_bessel_i</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_i">
<colgroup>
<col>
</table></div>
</div>
<br class="table-break"><div class="table">
-<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k_integer_orders_"></a><p class="title"><b>Table 7.46. Error rates for cyl_bessel_k (integer orders)</b></p>
+<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k_integer_orders_"></a><p class="title"><b>Table 8.46. Error rates for cyl_bessel_k (integer orders)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k (integer orders)">
<colgroup>
<col>
</table></div>
</div>
<br class="table-break"><div class="table">
-<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k"></a><p class="title"><b>Table 7.47. Error rates for cyl_bessel_k</b></p>
+<a name="math_toolkit.bessel.mbessel.table_cyl_bessel_k"></a><p class="title"><b>Table 8.47. Error rates for cyl_bessel_k</b></p>
<div class="table-contents"><table class="table" summary="Error rates for cyl_bessel_k">
<colgroup>
<col>
precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/i0__double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/i0__80_bit_long_double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/i0____float128.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/i1__double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/i1__80_bit_long_double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/i1____float128.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/k0__double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/k0__80_bit_long_double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/k0____float128.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/k1__double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/k1__80_bit_long_double.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/k1____float128.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/i0__double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/i0__80_bit_long_double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/i0____float128.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/i1__double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/i1__80_bit_long_double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/i1____float128.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/k0__double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/k0__80_bit_long_double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/k0____float128.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/k1__double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/k1__80_bit_long_double.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/k1____float128.svg" align="middle"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.bessel.mbessel.h4"></a>
<span class="phrase"><a name="math_toolkit.bessel.mbessel.implementation"></a></span><a class="link" href="mbessel.html#math_toolkit.bessel.mbessel.implementation">Implementation</a>
The following are handled as special cases first:
</p>
<p>
- When computing I<sub>v</sub>   for <span class="emphasis"><em>x < 0</em></span>, then ν   must be an integer
- or a domain error occurs. If ν   is an integer, then the function is odd if ν   is
- odd and even if ν   is even, and we can reflect to <span class="emphasis"><em>x > 0</em></span>.
+ When computing I<sub>v</sub> for <span class="emphasis"><em>x < 0</em></span>, then ν must be an integer
+ or a domain error occurs. If ν is an integer, then the function is odd if ν is
+ odd and even if ν is even, and we can reflect to <span class="emphasis"><em>x > 0</em></span>.
</p>
<p>
- For I<sub>v</sub>   with v equal to 0, 1 or 0.5 are handled as special cases.
+ For I<sub>v</sub> with v equal to 0, 1 or 0.5 are handled as special cases.
</p>
<p>
The 0 and 1 cases use polynomial approximations on finite and infinite intervals.
extended by us to deal with up to 128-bit precision (with different approximations
for each target precision).
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel21.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel20.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel17.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel18.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel21.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel20.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel17.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel18.svg"></span>
+
+ </p></blockquote></div>
<p>
Similarly we have:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel22.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel23.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel24.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/bessel25.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel22.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel23.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel24.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/bessel25.svg"></span>
+
+ </p></blockquote></div>
<p>
The 0.5 case is a simple trigonometric function:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ I<sub>0.5</sub>(x) = sqrt(2 / πx) * sinh(x)
+ </p></blockquote></div>
<p>
- I<sub>0.5</sub>(x) = sqrt(2 / πx) * sinh(x)
- </p>
-<p>
- For K<sub>v</sub>   with <span class="emphasis"><em>v</em></span> an integer, the result is calculated using
+ For K<sub>v</sub> with <span class="emphasis"><em>v</em></span> an integer, the result is calculated using
the recurrence relation:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel5.svg"></span>
- </p>
-<p>
- starting from K<sub>0</sub>   and K<sub>1</sub>   which are calculated using rational the approximations
+ starting from K<sub>0</sub> and K<sub>1</sub> which are calculated using rational the approximations
above. These rational approximations are accurate to around 19 digits, and
are therefore only used when T has no more than 64 binary digits of precision.
</p>
<p>
When <span class="emphasis"><em>x</em></span> is small compared to <span class="emphasis"><em>v</em></span>,
- I<sub>v</sub>x   is best computed directly from the series:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel17.svg"></span>
+ I<sub>v</sub>x is best computed directly from the series:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel17.svg"></span>
+
+ </p></blockquote></div>
<p>
- In the general case, we first normalize ν   to [<code class="literal">0, [inf]</code>)
+ In the general case, we first normalize ν to [<code class="literal">0, [inf]</code>)
with the help of the reflection formulae:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel9.svg"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel10.svg"></span>
- </p>
-<p>
- Let μ   = ν - floor(ν + 1/2), then μ   is the fractional part of ν   such that |μ| <= 1/2
+ Let μ = ν - floor(ν + 1/2), then μ is the fractional part of ν such that |μ| <= 1/2
(we need this for convergence later). The idea is to calculate K<sub>μ</sub>(x) and K<sub>μ+1</sub>(x),
and use them to obtain I<sub>ν</sub>(x) and K<sub>ν</sub>(x).
</p>
<p>
- The algorithm is proposed by Temme in N.M. Temme, <span class="emphasis"><em>On the numerical
- evaluation of the modified bessel function of the third kind</em></span>,
- Journal of Computational Physics, vol 19, 324 (1975), which needs two continued
- fractions as well as the Wronskian:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel11.svg"></span>
+ The algorithm is proposed by Temme in
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ N.M. Temme, <span class="emphasis"><em>On the numerical evaluation of the modified bessel
+ function of the third kind</em></span>, Journal of Computational Physics,
+ vol 19, 324 (1975),
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel12.svg"></span>
+ which needs two continued fractions as well as the Wronskian:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel11.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel12.svg"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel8.svg"></span>
+ The continued fractions are computed using the modified Lentz's method
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ (W.J. Lentz, <span class="emphasis"><em>Generating Bessel functions in Mie scattering calculations
+ using continued fractions</em></span>, Applied Optics, vol 15, 668 (1976)).
+ </p></blockquote></div>
<p>
- The continued fractions are computed using the modified Lentz's method (W.J.
- Lentz, <span class="emphasis"><em>Generating Bessel functions in Mie scattering calculations
- using continued fractions</em></span>, Applied Optics, vol 15, 668 (1976)).
Their convergence rates depend on <span class="emphasis"><em>x</em></span>, therefore we need
different strategies for large <span class="emphasis"><em>x</em></span> and small <span class="emphasis"><em>x</em></span>.
</p>
<p>
When <span class="emphasis"><em>x</em></span> is large (<span class="emphasis"><em>x</em></span> > 2), both
continued fractions converge (CF1 may be slow for really large <span class="emphasis"><em>x</em></span>).
- K<sub>μ</sub>   and K<sub>μ+1</sub>  
+ K<sub>μ</sub> and K<sub>μ+1</sub>
can be calculated by
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel13.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel13.svg"></span>
+
+ </p></blockquote></div>
<p>
where
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel14.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel14.svg"></span>
+
+ </p></blockquote></div>
<p>
<span class="emphasis"><em>S</em></span> is also a series that is summed along with CF2, see
- I.J. Thompson and A.R. Barnett, <span class="emphasis"><em>Modified Bessel functions I_v and
- K_v of real order and complex argument to selected accuracy</em></span>, Computer
- Physics Communications, vol 47, 245 (1987).
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ I.J. Thompson and A.R. Barnett, <span class="emphasis"><em>Modified Bessel functions I_v
+ and K_v of real order and complex argument to selected accuracy</em></span>,
+ Computer Physics Communications, vol 47, 245 (1987).
+ </p></blockquote></div>
<p>
When <span class="emphasis"><em>x</em></span> is small (<span class="emphasis"><em>x</em></span> <= 2), CF2
convergence may fail (but CF1 works very well). The solution here is Temme's
series:
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel15.svg"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel15.svg"></span>
+
+ </p></blockquote></div>
<p>
where
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/mbessel16.svg"></span>
+
+ </p></blockquote></div>
<p>
- <span class="inlinemediaobject"><img src="../../../equations/mbessel16.svg"></span>
+ f<sub>k</sub> and h<sub>k</sub>
+are also computed by recursions (involving gamma functions), but
+ the formulas are a little complicated, readers are referred to
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ N.M. Temme, <span class="emphasis"><em>On the numerical evaluation of the modified Bessel
+ function of the third kind</em></span>, Journal of Computational Physics,
+ vol 19, 324 (1975).
+ </p></blockquote></div>
<p>
- f<sub>k</sub>   and h<sub>k</sub>  
-are also computed by recursions (involving gamma functions), but
- the formulas are a little complicated, readers are referred to N.M. Temme,
- <span class="emphasis"><em>On the numerical evaluation of the modified Bessel function of
- the third kind</em></span>, Journal of Computational Physics, vol 19, 324
- (1975). Note: Temme's series converge only for |μ| <= 1/2.
+ Note: Temme's series converge only for |μ| <= 1/2.
</p>
<p>
K<sub>ν</sub>(x) is then calculated from the forward recurrence, as is K<sub>ν+1</sub>(x). With these
projects / platform / upstream / boost.git / blobdiff