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Hypergeometric: Start adding docs.
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jzmaddock
58
doc/equations/hyper_0f1.mml
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doc/equations/hyper_0f1_bessel_i.mml
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doc/equations/hyper_0f1_cf.mml
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<mo>;</mo>
|
||||
<mo>;</mo>
|
||||
<mi>z</mi>
|
||||
</mrow>
|
||||
</mfenced>
|
||||
<mo>=</mo>
|
||||
<munderover>
|
||||
<mo>∑</mo>
|
||||
<mrow>
|
||||
<mi>k</mi>
|
||||
<mo>=</mo>
|
||||
<mn>1</mn>
|
||||
</mrow>
|
||||
<mi>∞</mi>
|
||||
</munderover>
|
||||
<mfrac>
|
||||
<mrow>
|
||||
<msub>
|
||||
<mfenced>
|
||||
<mrow>
|
||||
<msub>
|
||||
<mi>a</mi>
|
||||
<mn>1</mn>
|
||||
</msub>
|
||||
</mrow>
|
||||
</mfenced>
|
||||
<mi>k</mi>
|
||||
</msub>
|
||||
<msub>
|
||||
<mfenced>
|
||||
<mrow>
|
||||
<msub>
|
||||
<mi>a</mi>
|
||||
<mn>2</mn>
|
||||
</msub>
|
||||
</mrow>
|
||||
</mfenced>
|
||||
<mi>k</mi>
|
||||
</msub>
|
||||
<msup>
|
||||
<mi>z</mi>
|
||||
<mi>k</mi>
|
||||
</msup>
|
||||
</mrow>
|
||||
<mrow>
|
||||
<mi>k</mi>
|
||||
<mo>!</mo>
|
||||
</mrow>
|
||||
</mfrac>
|
||||
</mrow>
|
||||
</math></body>
|
||||
</html>
|
||||
2
doc/equations/hyper_2f0.svg
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doc/equations/hyper_2f0.svg
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|
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126
doc/equations/hyper_2f0_cf.mml
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doc/equations/hyper_2f0_cf.mml
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@@ -0,0 +1,126 @@
|
||||
<?xml version="1.0" encoding="UTF-8"?>
|
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|
||||
<html xmlns="http://www.w3.org/1999/xhtml">
|
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<head><title>hyper_2f0_cf</title>
|
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|
||||
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|
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|
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|
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<mo>;</mo>
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<mfenced>
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<mrow>
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<mi>k</mi>
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<mo>+</mo>
|
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<mi>b</mi>
|
||||
</mrow>
|
||||
</mfenced>
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||||
</mrow>
|
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</mfrac>
|
||||
<mo>+</mo>
|
||||
<mn>1</mn>
|
||||
</mrow>
|
||||
</mfenced>
|
||||
<mrow>
|
||||
<mn>1</mn>
|
||||
</mrow>
|
||||
<mi>∞</mi>
|
||||
</msubsup>
|
||||
</mrow>
|
||||
</mfenced>
|
||||
</mrow>
|
||||
</mfrac>
|
||||
</mrow>
|
||||
</math></body>
|
||||
</html>
|
||||
2
doc/equations/hyper_2f0_cf.svg
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|
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84
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<?xml version="1.0" encoding="UTF-8"?>
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<html xmlns="http://www.w3.org/1999/xhtml">
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<head><title>hyper_2f0_hermite</title>
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<mi/>
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<mn>2</mn>
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<msub>
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<mi>F</mi>
|
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<mn>0</mn>
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</msub>
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<mfenced>
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<mrow>
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<mrow>
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<mo>−</mo>
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<mfrac>
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<mi>n</mi>
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<mn>2</mn>
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</mfrac>
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</mrow>
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<mo>,</mo>
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<mfrac>
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<mrow>
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<mn>1</mn>
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<mo>=</mo>
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||||
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<mfenced>
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<mrow>
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<mfrac>
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<mn>2</mn>
|
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<mrow>
|
||||
<msqrt>
|
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<mrow>
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<mo>−</mo>
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<mi>z</mi>
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</mrow>
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</msqrt>
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</mfrac>
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<mrow>
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<mo>−</mo>
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<mi>n</mi>
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</mrow>
|
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</msup>
|
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<msub>
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<mi>H</mi>
|
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<mi>n</mi>
|
||||
</msub>
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<mn>1</mn>
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<mrow>
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<mo>−</mo>
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<mi>z</mi>
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|
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</mfrac>
|
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|
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</mfenced>
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</mrow>
|
||||
</math></body>
|
||||
</html>
|
||||
2
doc/equations/hyper_2f0_hermite.svg
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|
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61
doc/equations/hyper_2f0_laguerre.mml
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|
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<?xml version="1.0" encoding="UTF-8"?>
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<html xmlns="http://www.w3.org/1999/xhtml">
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<head><title>hyper_2f0_laguerre</title>
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|
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|
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|
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|
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<mo>,</mo>
|
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|
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|
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|
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|
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|
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|
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|
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<mi>L</mi>
|
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|
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|
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|
||||
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|
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|
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|
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|
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|
||||
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|
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|
||||
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|
||||
<mi>z</mi>
|
||||
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|
||||
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|
||||
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|
||||
</mrow>
|
||||
</math></body>
|
||||
</html>
|
||||
2
doc/equations/hyper_2f0_laguerre.svg
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doc/equations/hyper_2f0_laguerre.svg
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|
After Width: | Height: | Size: 6.6 KiB |
@@ -600,6 +600,8 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22.
|
||||
|
||||
[include sf/expint.qbk]
|
||||
|
||||
[include sf/hypergeometric.qbk]
|
||||
|
||||
[include sf/powers.qbk]
|
||||
[include sf/sinc.qbk]
|
||||
[include sf/inv_hyper.qbk]
|
||||
|
||||
122
doc/sf/hypergeometric.qbk
Normal file
122
doc/sf/hypergeometric.qbk
Normal file
@@ -0,0 +1,122 @@
|
||||
|
||||
[section:hypergeometric Hypergeometric Functions]
|
||||
|
||||
[section:hyper_geometric_1f0 Hypergeometric 1F0]
|
||||
|
||||
template <class T1, class T2>
|
||||
``__sf_result`` hypergeometric_1f0(T1 a, T2 z);
|
||||
|
||||
template <class T1, class T2, class ``__Policy``>
|
||||
``__sf_result`` hypergeometric_1f0(T1 a, T2 z, const ``__Policy``&);
|
||||
|
||||
[h4 Description]
|
||||
|
||||
The function hypergeometric_1f0 returns the result of:
|
||||
|
||||
[equation hyper_1f0]
|
||||
|
||||
The return type of these functions is computed using the __arg_promotion_rules
|
||||
when T1 and T2 are different types.
|
||||
|
||||
[optional_policy]
|
||||
|
||||
The functions return the result of __domain_error whenever the result is
|
||||
undefined or complex. This occurs for `z == 1` or `1 - z < 0` and `a` not an integer.
|
||||
|
||||
[h4 Implementation]
|
||||
|
||||
Implementation is trivially via:
|
||||
|
||||
[sub 1]F[sub 0](a, z) = (1-z)[super -a]
|
||||
|
||||
|
||||
[endsect]
|
||||
|
||||
[section:hypergeometric_0f1 Hypergeometric 0F1]
|
||||
|
||||
template <class T1, class T2>
|
||||
``__sf_result`` hypergeometric_0f1(T1 b, T2 z);
|
||||
|
||||
template <class T1, class T2, class ``__Policy``>
|
||||
``__sf_result`` hypergeometric_0f1(T1 b, T2 z, const ``__Policy``&);
|
||||
|
||||
[h4 Description]
|
||||
|
||||
The function hypergeometric_0f1 returns the result of:
|
||||
|
||||
[equation hyper_0f1]
|
||||
|
||||
The return type of these functions is computed using the __arg_promotion_rules
|
||||
when T1 and T2 are different types.
|
||||
|
||||
[optional_policy]
|
||||
|
||||
The functions return the result of __domain_error whenever the result is
|
||||
undefined or complex. This occurs only when `b` is an integer <= 0.
|
||||
|
||||
[h4 Implementation]
|
||||
|
||||
The function is implemented via it's defining series whenever that series is non-divergent.
|
||||
|
||||
For a divergent series we use the continued fraction as long as the result is not too small:
|
||||
|
||||
[equation hyper_0f1_cf]
|
||||
|
||||
Otherwise one of the Bessel function relations:
|
||||
|
||||
[equation hyper_0f1_bessel_j]
|
||||
|
||||
[equation hyper_0f1_bessel_i]
|
||||
|
||||
[endsect]
|
||||
|
||||
[section:hyper_geometric_1f0 Hypergeometric 2F0]
|
||||
|
||||
template <class T1, class T2, class T3>
|
||||
``__sf_result`` hypergeometric_2f0(T1 a1, T2 a2, T3 z);
|
||||
|
||||
template <class T1, class T2, class T3, class ``__Policy``>
|
||||
``__sf_result`` hypergeometric_2f0(T1 a1, T2 a2, T3 z, const ``__Policy``&);
|
||||
|
||||
[h4 Description]
|
||||
|
||||
The function hypergeometric_2f0 returns the result of:
|
||||
|
||||
[equation hyper_2f0]
|
||||
|
||||
The return type of these functions is computed using the __arg_promotion_rules
|
||||
when T1 and T2 are different types.
|
||||
|
||||
[optional_policy]
|
||||
|
||||
The functions return the result of __domain_error whenever the result is
|
||||
undefined or complex. The valid domain for this function occurs only when one of `a1` or
|
||||
`a2` is a negative integer: ie the polynomial case.
|
||||
|
||||
[h4 Implementation]
|
||||
|
||||
When a1 == a2 - 0.5 then the function is implemented in terms of the Hermite polynomial:
|
||||
|
||||
[equation hyper_2f0_hermite]
|
||||
|
||||
When both a1 and a2 are integers then the function is implemented in terms of the associated-Laguerre polynomial:
|
||||
|
||||
[equation hyper_2f0_laguerre]
|
||||
|
||||
Otherwise if the defining series would be divergent, then we use the continued fraction form:
|
||||
|
||||
[equation hyper_2f0_cf]
|
||||
|
||||
Otherwise we use the defining series.
|
||||
|
||||
[endsect]
|
||||
|
||||
|
||||
[endsect]
|
||||
|
||||
[/
|
||||
Copyright 2017 John Maddock.
|
||||
Distributed under the Boost Software License, Version 1.0.
|
||||
(See accompanying file LICENSE_1_0.txt or copy at
|
||||
http://www.boost.org/LICENSE_1_0.txt).
|
||||
]
|
||||
Reference in New Issue
Block a user