<title>Finding Zeros of Bessel Functions of the First and Second Kinds</title>
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<span class="identifier">OutputIterator</span> <span class="identifier">out_it</span><span class="special">);</span> <span class="comment">// Destination for zeros.</span>
</pre>
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for error handling and precision.
</p>
<pre class="programlisting"> <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
number of zeros on the positive real axis. The real zeros on the positive
real axis can be found by solving for the roots of
</p>
-<p>
-   <span class="emphasis"><em>J<sub>ν</sub>(j<sub>ν, m</sub>) = 0</em></span>
- </p>
-<p>
-   <span class="emphasis"><em>Y<sub>ν</sub>(y<sub>ν, m</sub>) = 0</em></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="emphasis"><em>J<sub>ν</sub>(j<sub>ν, m</sub>) = 0</em></span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="emphasis"><em>Y<sub>ν</sub>(y<sub>ν, m</sub>) = 0</em></span>
+ </p></blockquote></div>
<p>
Here, <span class="emphasis"><em>j<sub>ν, m</sub></em></span> represents the <span class="emphasis"><em>m<sup>th</sup></em></span> root
of the cylindrical Bessel function of order <span class="emphasis"><em>ν</em></span>, and <span class="emphasis"><em>y<sub>ν,
In each case the index or rank of the zero returned is 1-based, which is
to say:
</p>
-<pre class="programlisting"><span class="identifier">cyl_bessel_j_zero</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
-</pre>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ cyl_bessel_j_zero(v, 1);
+ </p></blockquote></div>
<p>
returns the first zero of Bessel J.
</p>
the <code class="computeroutput"><span class="identifier">cyl_bessel_j</span></code> and <code class="computeroutput"><span class="identifier">cyl_neumann</span></code> functions, but not infinite
nor NaN.
</p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/bessel_j_zeros.svg" align="middle"></span>
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../graphs/neumann_y_zeros.svg" align="middle"></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/bessel_j_zeros.svg" align="middle"></span>
+
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../graphs/neumann_y_zeros.svg" align="middle"></span>
+
+ </p></blockquote></div>
<h5>
<a name="math_toolkit.bessel.bessel_root.h2"></a>
<span class="phrase"><a name="math_toolkit.bessel.bessel_root.examples_of_finding_bessel_and_n"></a></span><a class="link" href="bessel_root.html#math_toolkit.bessel.bessel_root.examples_of_finding_bessel_and_n">Examples
by M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, NBS
(1964). In particular,
</p>
-<p>
-   <span class="emphasis"><em>d/<sub>dx</sub> <span class="emphasis"><em>J<sub>ν</sub>(x)</em></span> = <span class="emphasis"><em>J<sub>ν-1</sub>(x)</em></span> - ν J<sub>ν</sub>(x)</em></span>
- / x
- </p>
-<p>
-   <span class="emphasis"><em>d/<sub>dx</sub> <span class="emphasis"><em>Y<sub>ν</sub>(x)</em></span> = <span class="emphasis"><em>Y<sub>ν-1</sub>(x)</em></span> - ν Y<sub>ν</sub>(x)</em></span>
- / x
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">d/<sub>dx</sub> <span class="emphasis"><em>J<sub>ν</sub>(x)</em></span> = <span class="emphasis"><em>J<sub>ν-1</sub>(x)</em></span>
+ - ν J<sub>ν</sub>(x) / x</span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">d/<sub>dx</sub> <span class="emphasis"><em>Y<sub>ν</sub>(x)</em></span> = <span class="emphasis"><em>Y<sub>ν-1</sub>(x)</em></span>
+ - ν Y<sub>ν</sub>(x) / x</span>
+ </p></blockquote></div>
<p>
Enumeration of the rank of a root (in other words the index of a root) begins
with one and counts up, in other words <span class="emphasis"><em>m,=1,2,3,…</em></span> The
<p>
In particular,
</p>
-<p>
-   <span class="emphasis"><em>j<sub>ν,1</sub> ≅ ν + 1.85575 ν<sup>⅓</sup> + 1.033150 ν<sup>-⅓</sup> - 0.00397 ν<sup>-1</sup> - 0.0908 ν<sup>-5/3</sup> + 0.043 ν<sup>-7/3</sup> +
- …</em></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">j<sub>ν,1</sub> ≅ ν + 1.85575 ν<sup>⅓</sup> + 1.033150 ν<sup>-⅓</sup> - 0.00397 ν<sup>-1</sup> - 0.0908
+ ν<sup>-5/3</sup> + 0.043 ν<sup>-7/3</sup> + …</span>
+ </p></blockquote></div>
<p>
and
</p>
-<p>
-   <span class="emphasis"><em>y<sub>ν,1</sub> ≅ ν + 0.93157 ν<sup>⅓</sup> + 0.26035 ν<sup>-⅓</sup> + 0.01198 ν<sup>-1</sup> - 0.0060 ν<sup>-5/3</sup> - 0.001 ν<sup>-7/3</sup> +
- …</em></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">y<sub>ν,1</sub> ≅ ν + 0.93157 ν<sup>⅓</sup> + 0.26035 ν<sup>-⅓</sup> + 0.01198 ν<sup>-1</sup> - 0.0060
+ ν<sup>-5/3</sup> - 0.001 ν<sup>-7/3</sup> + …</span>
+ </p></blockquote></div>
<p>
Calculations of the estimates of <span class="emphasis"><em>j<sub>ν, m</sub></em></span> and <span class="emphasis"><em>y<sub>ν,
m</sub></em></span> with rank <span class="emphasis"><em>m > 2</em></span> and <span class="emphasis"><em>0 ≤ ν <
2.2</em></span> use McMahon's approximation, as described in M. Abramowitz
and I. A. Stegan, Section 9.5 and 9.5.12. In particular,
</p>
-<p>
-   <span class="emphasis"><em>j<sub>ν,m</sub>, y<sub>ν,m</sub> ≅ β - (μ-1) / 8β</em></span>
- </p>
-<p>
-       <span class="emphasis"><em>- 4(μ-1)(7μ - 31) / 3(8β)<sup>3</sup></em></span>
- </p>
-<p>
-       <span class="emphasis"><em>-32(μ-1)(83μ² - 982μ + 3779) / 15(8β)<sup>5</sup></em></span>
- </p>
-<p>
-       <span class="emphasis"><em>-64(μ-1)(6949μ<sup>3</sup> - 153855μ² + 1585743μ- 6277237) / 105(8a)<sup>7</sup></em></span>
- </p>
-<p>
-       <span class="emphasis"><em>- …</em></span>     (5)
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="emphasis"><em>j<sub>ν,m</sub>, y<sub>ν,m</sub> ≅</em></span>
+ </p></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
+ β - (μ-1) / 8β
+ </p></blockquote></div></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="emphasis"><em>- 4(μ-1)(7μ - 31) / 3(8β)<sup>3</sup></em></span>
+ </p></blockquote></div></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="emphasis"><em>-32(μ-1)(83μ² - 982μ + 3779) / 15(8β)<sup>5</sup></em></span>
+ </p></blockquote></div></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="emphasis"><em>-64(μ-1)(6949μ<sup>3</sup> - 153855μ² + 1585743μ- 6277237) / 105(8a)<sup>7</sup></em></span>
+ </p></blockquote></div></blockquote></div>
+<div class="blockquote"><blockquote class="blockquote"><div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="emphasis"><em>- …</em></span>   (5)
+ </p></blockquote></div></blockquote></div>
<p>
where <span class="emphasis"><em>μ = 4ν<sup>2</sup></em></span> and <span class="emphasis"><em>β = (m + ½ν - ¼)π</em></span> for
<span class="emphasis"><em>j<sub>ν,m</sub></em></span> and <span class="emphasis"><em>β = (m + ½ν -¾)π for <span class="emphasis"><em>y<sub>ν,m</sub></em></span></em></span>.
<p>
In summary,
</p>
-<p>
-   <span class="emphasis"><em>j<sub>ν, m</sub> ∼ νx(-ζ) + f<sub>1</sub>(-ζ/ν)</em></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">j<sub>ν, m</sub> ∼ νx(-ζ) + f<sub>1</sub>(-ζ/ν)</span>
+ </p></blockquote></div>
<p>
where <span class="emphasis"><em>-ζ = ν<sup>-2/3</sup>a<sub>m</sub></em></span> and <span class="emphasis"><em>a<sub>m</sub></em></span> is the absolute
value of the <span class="emphasis"><em>m<sup>th</sup></em></span> root of <span class="emphasis"><em>Ai(x)</em></span>
<p>
Here <span class="emphasis"><em>x = x(-ζ)</em></span> is the inverse of the function
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">⅔(-ζ)<sup>3/2</sup> = √(x² - 1) - cos⁻¹(1/x)</span>
+ </p></blockquote></div>
<p>
-   <span class="emphasis"><em>⅔(-ζ)<sup>3/2</sup> = √(x² - 1) - cos⁻¹(1/x)</em></span>     (7)
+ (7)
</p>
<p>
Furthermore,
</p>
-<p>
-   <span class="emphasis"><em>f<sub>1</sub>(-ζ) = ½x(-ζ) {h(-ζ)}² ⋅ b<sub>0</sub>(-ζ)</em></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">f<sub>1</sub>(-ζ) = ½x(-ζ) {h(-ζ)}² ⋅ b<sub>0</sub>(-ζ)</span>
+ </p></blockquote></div>
<p>
where
</p>
-<p>
-   <span class="emphasis"><em>h(-ζ) = {4(-ζ) / (x² - 1)}<sup>4</sup></em></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">h(-ζ) = {4(-ζ) / (x² - 1)}<sup>4</sup></span>
+ </p></blockquote></div>
<p>
and
</p>
-<p>
-   <span class="emphasis"><em>b<sub>0</sub>(-ζ) = -5/(48ζ²) + 1/(-ζ)<sup>½</sup> ⋅ { 5/(24(x<sup>2</sup>-1)<sup>3/2</sup>) + 1/(8(x<sup>2</sup>-1)<sup>½)</sup>}</em></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">b<sub>0</sub>(-ζ) = -5/(48ζ²) + 1/(-ζ)<sup>½</sup> ⋅ { 5/(24(x<sup>2</sup>-1)<sup>3/2</sup>) +
+ 1/(8(x<sup>2</sup>-1)<sup>½)</sup>}</span>
+ </p></blockquote></div>
<p>
When solving for <span class="emphasis"><em>x(-ζ)</em></span> in Eq. 7 above, the right-hand-side
is expanded to order 2 in a Taylor series for large <span class="emphasis"><em>x</em></span>.
This results in
</p>
-<p>
-   <span class="emphasis"><em>⅔(-ζ)<sup>3/2</sup> ≈ x + 1/2x - π/2</em></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">⅔(-ζ)<sup>3/2</sup> ≈ x + 1/2x - π/2</span>
+ </p></blockquote></div>
<p>
The positive root of the resulting quadratic equation is used to find an
initial estimate <span class="emphasis"><em>x(-ζ)</em></span>. This initial estimate is subsequently
root of the Bessel function of corresponding positive integer order. In other
words,
</p>
-<p>
-   <span class="emphasis"><em>j<sub>nν,m</sub></em></span> < <span class="emphasis"><em>j<sub>-ν,m</sub></em></span> < <span class="emphasis"><em>j<sub>nν,m+1</sub></em></span>
- </p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="serif_italic">j<sub>nν, m</sub> < j<sub>-ν, m</sub> < j<sub>nν, m+1</sub></span>
+ </p></blockquote></div>
<p>
where <span class="emphasis"><em>m > 1</em></span> and <span class="emphasis"><em>n<sub>ν</sub></em></span> represents
the integral floor of the absolute value of <span class="emphasis"><em>|-ν|</em></span>.
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