10.63: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.63#Ex5 10.63#Ex5] || [[Item:Q3812|<math>f_{\nu-1}(x)+f_{\nu+1}(x) = -(\nu\sqrt{2}/x)\left(f_{\nu}(x)-g_{\nu}(x)\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>f_{\nu-1}(x)+f_{\nu+1}(x) = -(\nu\sqrt{2}/x)\left(f_{\nu}(x)-g_{\nu}(x)\right)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">f[nu - 1](x)+ f[nu + 1](x) = -(nu*sqrt(2)/x)*(f[nu](x)- g[nu](x))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[f, \[Nu]- 1][x]+ Subscript[f, \[Nu]+ 1][x] == -(\[Nu]*Sqrt[2]/x)*(Subscript[f, \[Nu]][x]- Subscript[g, \[Nu]][x])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/10.63#Ex5 10.63#Ex5] || <math qid="Q3812">f_{\nu-1}(x)+f_{\nu+1}(x) = -(\nu\sqrt{2}/x)\left(f_{\nu}(x)-g_{\nu}(x)\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>f_{\nu-1}(x)+f_{\nu+1}(x) = -(\nu\sqrt{2}/x)\left(f_{\nu}(x)-g_{\nu}(x)\right)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">f[nu - 1](x)+ f[nu + 1](x) = -(nu*sqrt(2)/x)*(f[nu](x)- g[nu](x))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[f, \[Nu]- 1][x]+ Subscript[f, \[Nu]+ 1][x] == -(\[Nu]*Sqrt[2]/x)*(Subscript[f, \[Nu]][x]- Subscript[g, \[Nu]][x])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.63#Ex9 10.63#Ex9] || [[Item:Q3816|<math>\sqrt{2}\Kelvinber{}'@@{x} = \Kelvinber{1}@@{x}+\Kelvinbei{1}@@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinber{}'@@{x} = \Kelvinber{1}@@{x}+\Kelvinbei{1}@@{x}</syntaxhighlight> || <math>\realpart@@{(1+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinBer(, x), x$(1) ) = KelvinBer(1, x)+ KelvinBei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinBer[, x], {x, 1}] == KelvinBer[1, x]+ KelvinBei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.297000428957679, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 1.5]], KelvinBei[Plus[1.0, Null], 1.5], Times[-1.0, KelvinBer[Plus[-1.0, Null], 1.5]], KelvinBer[Plus[1.0, Null], 1.5]]]]
| [https://dlmf.nist.gov/10.63#Ex9 10.63#Ex9] || <math qid="Q3816">\sqrt{2}\Kelvinber{}'@@{x} = \Kelvinber{1}@@{x}+\Kelvinbei{1}@@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinber{}'@@{x} = \Kelvinber{1}@@{x}+\Kelvinbei{1}@@{x}</syntaxhighlight> || <math>\realpart@@{(1+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinBer(, x), x$(1) ) = KelvinBer(1, x)+ KelvinBei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinBer[, x], {x, 1}] == KelvinBer[1, x]+ KelvinBei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.297000428957679, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 1.5]], KelvinBei[Plus[1.0, Null], 1.5], Times[-1.0, KelvinBer[Plus[-1.0, Null], 1.5]], KelvinBer[Plus[1.0, Null], 1.5]]]]
Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.011047944038096752, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 0.5]], KelvinBei[Plus[1.0, Null], 0.5], Times[-1.0, KelvinBer[Plus[-1.0, Null], 0.5]], KelvinBer[Plus[1.0, Null], 0.5]]]]
Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[0.011047944038096752, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 0.5]], KelvinBei[Plus[1.0, Null], 0.5], Times[-1.0, KelvinBer[Plus[-1.0, Null], 0.5]], KelvinBer[Plus[1.0, Null], 0.5]]]]
Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.63#Ex10 10.63#Ex10] || [[Item:Q3817|<math>\sqrt{2}\Kelvinbei{}'@@{x} = -\Kelvinber{1}x+\Kelvinbei{1}x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinbei{}'@@{x} = -\Kelvinber{1}x+\Kelvinbei{1}x</syntaxhighlight> || <math>\realpart@@{(1+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinBei(, x), x$(1) ) = - KelvinBer(1, x)+ KelvinBei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinBei[, x], {x, 1}] == - KelvinBer[1, x]+ KelvinBei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-1.0327304069618592, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 1.5]], KelvinBei[Plus[1.0, Null], 1.5], KelvinBer[Plus[-1.0, Null], 1.5], Times[-1.0, KelvinBer[Plus[1.0, Null], 1.5]]]]]
| [https://dlmf.nist.gov/10.63#Ex10 10.63#Ex10] || <math qid="Q3817">\sqrt{2}\Kelvinbei{}'@@{x} = -\Kelvinber{1}x+\Kelvinbei{1}x</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinbei{}'@@{x} = -\Kelvinber{1}x+\Kelvinbei{1}x</syntaxhighlight> || <math>\realpart@@{(1+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinBei(, x), x$(1) ) = - KelvinBer(1, x)+ KelvinBei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinBei[, x], {x, 1}] == - KelvinBer[1, x]+ KelvinBei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-1.0327304069618592, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 1.5]], KelvinBei[Plus[1.0, Null], 1.5], KelvinBer[Plus[-1.0, Null], 1.5], Times[-1.0, KelvinBer[Plus[1.0, Null], 1.5]]]]]
Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.35343830347212746, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 0.5]], KelvinBei[Plus[1.0, Null], 0.5], KelvinBer[Plus[-1.0, Null], 0.5], Times[-1.0, KelvinBer[Plus[1.0, Null], 0.5]]]]]
Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.35343830347212746, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 0.5]], KelvinBei[Plus[1.0, Null], 0.5], KelvinBer[Plus[-1.0, Null], 0.5], Times[-1.0, KelvinBer[Plus[1.0, Null], 0.5]]]]]
Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.63#Ex11 10.63#Ex11] || [[Item:Q3818|<math>\sqrt{2}\Kelvinker{}'@@{x} = \Kelvinker{1}x+\Kelvinkei{1}x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinker{}'@@{x} = \Kelvinker{1}x+\Kelvinkei{1}x</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinKer(, x), x$(1) ) = KelvinKer(1, x)+ KelvinKei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinKer[, x], {x, 1}] == KelvinKer[1, x]+ KelvinKei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.4160356041812476, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 1.5]], KelvinKei[Plus[1.0, Null], 1.5], Times[-1.0, KelvinKer[Plus[-1.0, Null], 1.5]], KelvinKer[Plus[1.0, Null], 1.5]]]]
| [https://dlmf.nist.gov/10.63#Ex11 10.63#Ex11] || <math qid="Q3818">\sqrt{2}\Kelvinker{}'@@{x} = \Kelvinker{1}x+\Kelvinkei{1}x</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinker{}'@@{x} = \Kelvinker{1}x+\Kelvinkei{1}x</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinKer(, x), x$(1) ) = KelvinKer(1, x)+ KelvinKei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinKer[, x], {x, 1}] == KelvinKer[1, x]+ KelvinKei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.4160356041812476, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 1.5]], KelvinKei[Plus[1.0, Null], 1.5], Times[-1.0, KelvinKer[Plus[-1.0, Null], 1.5]], KelvinKer[Plus[1.0, Null], 1.5]]]]
Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[2.5735854919446126, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 0.5]], KelvinKei[Plus[1.0, Null], 0.5], Times[-1.0, KelvinKer[Plus[-1.0, Null], 0.5]], KelvinKer[Plus[1.0, Null], 0.5]]]]
Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[2.5735854919446126, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 0.5]], KelvinKei[Plus[1.0, Null], 0.5], Times[-1.0, KelvinKer[Plus[-1.0, Null], 0.5]], KelvinKer[Plus[1.0, Null], 0.5]]]]
Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.63#Ex12 10.63#Ex12] || [[Item:Q3819|<math>\sqrt{2}\Kelvinkei{}'@@{x} = -\Kelvinker{1}x+\Kelvinkei{1}x</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinkei{}'@@{x} = -\Kelvinker{1}x+\Kelvinkei{1}x</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinKei(, x), x$(1) ) = - KelvinKer(1, x)+ KelvinKei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinKei[, x], {x, 1}] == - KelvinKer[1, x]+ KelvinKei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.418052966151267, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 1.5]], KelvinKei[Plus[1.0, Null], 1.5], KelvinKer[Plus[-1.0, Null], 1.5], Times[-1.0, KelvinKer[Plus[1.0, Null], 1.5]]]]]
| [https://dlmf.nist.gov/10.63#Ex12 10.63#Ex12] || <math qid="Q3819">\sqrt{2}\Kelvinkei{}'@@{x} = -\Kelvinker{1}x+\Kelvinkei{1}x</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sqrt{2}\Kelvinkei{}'@@{x} = -\Kelvinker{1}x+\Kelvinkei{1}x</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sqrt(2)*diff( KelvinKei(, x), x$(1) ) = - KelvinKer(1, x)+ KelvinKei(1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sqrt[2]*D[KelvinKei[, x], {x, 1}] == - KelvinKer[1, x]+ KelvinKei[1, x]</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.418052966151267, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 1.5]], KelvinKei[Plus[1.0, Null], 1.5], KelvinKer[Plus[-1.0, Null], 1.5], Times[-1.0, KelvinKer[Plus[1.0, Null], 1.5]]]]]
Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.47122132111956727, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 0.5]], KelvinKei[Plus[1.0, Null], 0.5], KelvinKer[Plus[-1.0, Null], 0.5], Times[-1.0, KelvinKer[Plus[1.0, Null], 0.5]]]]]
Test Values: {Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.47122132111956727, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 0.5]], KelvinKei[Plus[1.0, Null], 0.5], KelvinKer[Plus[-1.0, Null], 0.5], Times[-1.0, KelvinKer[Plus[1.0, Null], 0.5]]]]]
Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[x, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.63#Ex17 10.63#Ex17] || [[Item:Q3824|<math>p_{\nu+1} = p_{\nu-1}-(4\nu/x)r_{\nu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{\nu+1} = p_{\nu-1}-(4\nu/x)r_{\nu}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[nu + 1] = p[nu - 1]-(4*nu/x)*r[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, \[Nu]+ 1] == Subscript[p, \[Nu]- 1]-(4*\[Nu]/x)*Subscript[r, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/10.63#Ex17 10.63#Ex17] || <math qid="Q3824">p_{\nu+1} = p_{\nu-1}-(4\nu/x)r_{\nu}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{\nu+1} = p_{\nu-1}-(4\nu/x)r_{\nu}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[nu + 1] = p[nu - 1]-(4*nu/x)*r[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, \[Nu]+ 1] == Subscript[p, \[Nu]- 1]-(4*\[Nu]/x)*Subscript[r, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.63#Ex18 10.63#Ex18] || [[Item:Q3825|<math>q_{\nu+1} = -(\nu/x)p_{\nu}+r_{\nu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>q_{\nu+1} = -(\nu/x)p_{\nu}+r_{\nu}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">q[nu + 1] = -(nu/x)*p[nu]+ r[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[q, \[Nu]+ 1] == -(\[Nu]/x)*Subscript[p, \[Nu]]+ Subscript[r, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/10.63#Ex18 10.63#Ex18] || <math qid="Q3825">q_{\nu+1} = -(\nu/x)p_{\nu}+r_{\nu}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>q_{\nu+1} = -(\nu/x)p_{\nu}+r_{\nu}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">q[nu + 1] = -(nu/x)*p[nu]+ r[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[q, \[Nu]+ 1] == -(\[Nu]/x)*Subscript[p, \[Nu]]+ Subscript[r, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.63#Ex19 10.63#Ex19] || [[Item:Q3826|<math>r_{\nu+1} = -((\nu+1)/x)p_{\nu+1}+q_{\nu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>r_{\nu+1} = -((\nu+1)/x)p_{\nu+1}+q_{\nu}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">r[nu + 1] = -((nu + 1)/x)*p[nu + 1]+ q[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[r, \[Nu]+ 1] == -((\[Nu]+ 1)/x)*Subscript[p, \[Nu]+ 1]+ Subscript[q, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/10.63#Ex19 10.63#Ex19] || <math qid="Q3826">r_{\nu+1} = -((\nu+1)/x)p_{\nu+1}+q_{\nu}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>r_{\nu+1} = -((\nu+1)/x)p_{\nu+1}+q_{\nu}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">r[nu + 1] = -((nu + 1)/x)*p[nu + 1]+ q[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[r, \[Nu]+ 1] == -((\[Nu]+ 1)/x)*Subscript[p, \[Nu]+ 1]+ Subscript[q, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/10.63#Ex20 10.63#Ex20] || [[Item:Q3827|<math>s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-(\nu^{2}/x^{2})p_{\nu}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-(\nu^{2}/x^{2})p_{\nu}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2)) = (1)/(2)*p[nu + 1]+(1)/(2)*p[nu - 1]-((nu)^(2)/(x)^(2))*p[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2)) == Divide[1,2]*Subscript[p, \[Nu]+ 1]+Divide[1,2]*Subscript[p, \[Nu]- 1]-(\[Nu]^(2)/(x)^(2))*Subscript[p, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/10.63#Ex20 10.63#Ex20] || <math qid="Q3827">s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-(\nu^{2}/x^{2})p_{\nu}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-(\nu^{2}/x^{2})p_{\nu}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2)) = (1)/(2)*p[nu + 1]+(1)/(2)*p[nu - 1]-((nu)^(2)/(x)^(2))*p[nu]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2)) == Divide[1,2]*Subscript[p, \[Nu]+ 1]+Divide[1,2]*Subscript[p, \[Nu]- 1]-(\[Nu]^(2)/(x)^(2))*Subscript[p, \[Nu]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.63.E7 10.63.E7] || [[Item:Q3828|<math>p_{\nu}s_{\nu} = r_{\nu}^{2}+q_{\nu}^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{\nu}s_{\nu} = r_{\nu}^{2}+q_{\nu}^{2}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[nu]*((diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2)) = (r[nu])^(2)+ (q[nu])^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, \[Nu]]*((D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2)) == (Subscript[r, \[Nu]])^(2)+ (Subscript[q, \[Nu]])^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/10.63.E7 10.63.E7] || <math qid="Q3828">p_{\nu}s_{\nu} = r_{\nu}^{2}+q_{\nu}^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>p_{\nu}s_{\nu} = r_{\nu}^{2}+q_{\nu}^{2}</syntaxhighlight> || <math>\realpart@@{(\nu+k+1)} > 0</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">p[nu]*((diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2)) = (r[nu])^(2)+ (q[nu])^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[p, \[Nu]]*((D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2)) == (Subscript[r, \[Nu]])^(2)+ (Subscript[q, \[Nu]])^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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Latest revision as of 11:28, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.63#Ex5 f ν - 1 ⁢ ( x ) + f ν + 1 ⁢ ( x ) = - ( ν ⁢ 2 / x ) ⁢ ( f ν ⁢ ( x ) - g ν ⁢ ( x ) ) subscript 𝑓 𝜈 1 𝑥 subscript 𝑓 𝜈 1 𝑥 𝜈 2 𝑥 subscript 𝑓 𝜈 𝑥 subscript 𝑔 𝜈 𝑥 {\displaystyle{\displaystyle f_{\nu-1}(x)+f_{\nu+1}(x)=-(\nu\sqrt{2}/x)\left(f% _{\nu}(x)-g_{\nu}(x)\right)}}
f_{\nu-1}(x)+f_{\nu+1}(x) = -(\nu\sqrt{2}/x)\left(f_{\nu}(x)-g_{\nu}(x)\right)		 

f[nu - 1](x)+ f[nu + 1](x) = -(nu*sqrt(2)/x)*(f[nu](x)- g[nu](x))
 
Subscript[f, \[Nu]- 1][x]+ Subscript[f, \[Nu]+ 1][x] == -(\[Nu]*Sqrt[2]/x)*(Subscript[f, \[Nu]][x]- Subscript[g, \[Nu]][x])
 Skipped - no semantic math Skipped - no semantic math - -
10.63#Ex9 2 ⁢ ber ′ ⁡ x = ber 1 ⁡ x + bei 1 ⁡ x 2 diffop Kelvin-ber 1 𝑥 Kelvin-ber 1 𝑥 Kelvin-bei 1 𝑥 {\displaystyle{\displaystyle\sqrt{2}\operatorname{ber}'x=\operatorname{ber}_{1% }x+\operatorname{bei}_{1}x}}
\sqrt{2}\Kelvinber{}'@@{x} = \Kelvinber{1}@@{x}+\Kelvinbei{1}@@{x}		 ℜ ⁡ ( 1 + k + 1 ) > 0 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(1+k+1)>0}} 
sqrt(2)*diff( KelvinBer(, x), x$(1) ) = KelvinBer(1, x)+ KelvinBei(1, x)

Sqrt[2]*D[KelvinBer[, x], {x, 1}] == KelvinBer[1, x]+ KelvinBei[1, x]
Error Failure -
Failed [3 / 3]
Result: Plus[0.297000428957679, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 1.5]], KelvinBei[Plus[1.0, Null], 1.5], Times[-1.0, KelvinBer[Plus[-1.0, Null], 1.5]], KelvinBer[Plus[1.0, Null], 1.5]]]]
Test Values: {Rule[x, 1.5]}


Result: Plus[0.011047944038096752, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 0.5]], KelvinBei[Plus[1.0, Null], 0.5], Times[-1.0, KelvinBer[Plus[-1.0, Null], 0.5]], KelvinBer[Plus[1.0, Null], 0.5]]]]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
10.63#Ex10 2 ⁢ bei ′ ⁡ x = - ber 1 ⁢ x + bei 1 ⁢ x 2 diffop Kelvin-bei 1 𝑥 Kelvin-ber 1 𝑥 Kelvin-bei 1 𝑥 {\displaystyle{\displaystyle\sqrt{2}\operatorname{bei}'x=-\operatorname{ber}_{% 1}x+\operatorname{bei}_{1}x}}
\sqrt{2}\Kelvinbei{}'@@{x} = -\Kelvinber{1}x+\Kelvinbei{1}x		 ℜ ⁡ ( 1 + k + 1 ) > 0 1 𝑘 1 0 {\displaystyle{\displaystyle\Re(1+k+1)>0}} 
sqrt(2)*diff( KelvinBei(, x), x$(1) ) = - KelvinBer(1, x)+ KelvinBei(1, x)

Sqrt[2]*D[KelvinBei[, x], {x, 1}] == - KelvinBer[1, x]+ KelvinBei[1, x]
Error Failure -
Failed [3 / 3]
Result: Plus[-1.0327304069618592, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 1.5]], KelvinBei[Plus[1.0, Null], 1.5], KelvinBer[Plus[-1.0, Null], 1.5], Times[-1.0, KelvinBer[Plus[1.0, Null], 1.5]]]]]
Test Values: {Rule[x, 1.5]}


Result: Plus[-0.35343830347212746, Times[0.35355339059327373, Plus[Times[-1.0, KelvinBei[Plus[-1.0, Null], 0.5]], KelvinBei[Plus[1.0, Null], 0.5], KelvinBer[Plus[-1.0, Null], 0.5], Times[-1.0, KelvinBer[Plus[1.0, Null], 0.5]]]]]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
10.63#Ex11 2 ⁢ ker ′ ⁡ x = ker 1 ⁢ x + kei 1 ⁢ x 2 diffop Kelvin-ker 1 𝑥 Kelvin-ker 1 𝑥 Kelvin-kei 1 𝑥 {\displaystyle{\displaystyle\sqrt{2}\operatorname{ker}'x=\operatorname{ker}_{1% }x+\operatorname{kei}_{1}x}}
\sqrt{2}\Kelvinker{}'@@{x} = \Kelvinker{1}x+\Kelvinkei{1}x		 

sqrt(2)*diff( KelvinKer(, x), x$(1) ) = KelvinKer(1, x)+ KelvinKei(1, x)

Sqrt[2]*D[KelvinKer[, x], {x, 1}] == KelvinKer[1, x]+ KelvinKei[1, x]
Error Failure -
Failed [3 / 3]
Result: Plus[0.4160356041812476, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 1.5]], KelvinKei[Plus[1.0, Null], 1.5], Times[-1.0, KelvinKer[Plus[-1.0, Null], 1.5]], KelvinKer[Plus[1.0, Null], 1.5]]]]
Test Values: {Rule[x, 1.5]}


Result: Plus[2.5735854919446126, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 0.5]], KelvinKei[Plus[1.0, Null], 0.5], Times[-1.0, KelvinKer[Plus[-1.0, Null], 0.5]], KelvinKer[Plus[1.0, Null], 0.5]]]]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
10.63#Ex12 2 ⁢ kei ′ ⁡ x = - ker 1 ⁢ x + kei 1 ⁢ x 2 diffop Kelvin-kei 1 𝑥 Kelvin-ker 1 𝑥 Kelvin-kei 1 𝑥 {\displaystyle{\displaystyle\sqrt{2}\operatorname{kei}'x=-\operatorname{ker}_{% 1}x+\operatorname{kei}_{1}x}}
\sqrt{2}\Kelvinkei{}'@@{x} = -\Kelvinker{1}x+\Kelvinkei{1}x		 

sqrt(2)*diff( KelvinKei(, x), x$(1) ) = - KelvinKer(1, x)+ KelvinKei(1, x)

Sqrt[2]*D[KelvinKei[, x], {x, 1}] == - KelvinKer[1, x]+ KelvinKei[1, x]
Error Failure -
Failed [3 / 3]
Result: Plus[-0.418052966151267, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 1.5]], KelvinKei[Plus[1.0, Null], 1.5], KelvinKer[Plus[-1.0, Null], 1.5], Times[-1.0, KelvinKer[Plus[1.0, Null], 1.5]]]]]
Test Values: {Rule[x, 1.5]}


Result: Plus[-0.47122132111956727, Times[0.35355339059327373, Plus[Times[-1.0, KelvinKei[Plus[-1.0, Null], 0.5]], KelvinKei[Plus[1.0, Null], 0.5], KelvinKer[Plus[-1.0, Null], 0.5], Times[-1.0, KelvinKer[Plus[1.0, Null], 0.5]]]]]
Test Values: {Rule[x, 0.5]}

... skip entries to safe data
10.63#Ex17 p ν + 1 = p ν - 1 - ( 4 ⁢ ν / x ) ⁢ r ν subscript 𝑝 𝜈 1 subscript 𝑝 𝜈 1 4 𝜈 𝑥 subscript 𝑟 𝜈 {\displaystyle{\displaystyle p_{\nu+1}=p_{\nu-1}-(4\nu/x)r_{\nu}}}
p_{\nu+1} = p_{\nu-1}-(4\nu/x)r_{\nu}		 

p[nu + 1] = p[nu - 1]-(4*nu/x)*r[nu]
 
Subscript[p, \[Nu]+ 1] == Subscript[p, \[Nu]- 1]-(4*\[Nu]/x)*Subscript[r, \[Nu]]
 Skipped - no semantic math Skipped - no semantic math - -
10.63#Ex18 q ν + 1 = - ( ν / x ) ⁢ p ν + r ν subscript 𝑞 𝜈 1 𝜈 𝑥 subscript 𝑝 𝜈 subscript 𝑟 𝜈 {\displaystyle{\displaystyle q_{\nu+1}=-(\nu/x)p_{\nu}+r_{\nu}}}
q_{\nu+1} = -(\nu/x)p_{\nu}+r_{\nu}		 

q[nu + 1] = -(nu/x)*p[nu]+ r[nu]
 
Subscript[q, \[Nu]+ 1] == -(\[Nu]/x)*Subscript[p, \[Nu]]+ Subscript[r, \[Nu]]
 Skipped - no semantic math Skipped - no semantic math - -
10.63#Ex19 r ν + 1 = - ( ( ν + 1 ) / x ) ⁢ p ν + 1 + q ν subscript 𝑟 𝜈 1 𝜈 1 𝑥 subscript 𝑝 𝜈 1 subscript 𝑞 𝜈 {\displaystyle{\displaystyle r_{\nu+1}=-((\nu+1)/x)p_{\nu+1}+q_{\nu}}}
r_{\nu+1} = -((\nu+1)/x)p_{\nu+1}+q_{\nu}		 

r[nu + 1] = -((nu + 1)/x)*p[nu + 1]+ q[nu]
 
Subscript[r, \[Nu]+ 1] == -((\[Nu]+ 1)/x)*Subscript[p, \[Nu]+ 1]+ Subscript[q, \[Nu]]
 Skipped - no semantic math Skipped - no semantic math - -
10.63#Ex20 s ν = 1 2 ⁢ p ν + 1 + 1 2 ⁢ p ν - 1 - ( ν 2 / x 2 ) ⁢ p ν subscript 𝑠 𝜈 1 2 subscript 𝑝 𝜈 1 1 2 subscript 𝑝 𝜈 1 superscript 𝜈 2 superscript 𝑥 2 subscript 𝑝 𝜈 {\displaystyle{\displaystyle s_{\nu}=\tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-% 1}-(\nu^{2}/x^{2})p_{\nu}}}
s_{\nu} = \tfrac{1}{2}p_{\nu+1}+\tfrac{1}{2}p_{\nu-1}-(\nu^{2}/x^{2})p_{\nu}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
((diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2)) = (1)/(2)*p[nu + 1]+(1)/(2)*p[nu - 1]-((nu)^(2)/(x)^(2))*p[nu]
 
((D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2)) == Divide[1,2]*Subscript[p, \[Nu]+ 1]+Divide[1,2]*Subscript[p, \[Nu]- 1]-(\[Nu]^(2)/(x)^(2))*Subscript[p, \[Nu]]
 Skipped - no semantic math Skipped - no semantic math - -
10.63.E7 p ν ⁢ s ν = r ν 2 + q ν 2 subscript 𝑝 𝜈 subscript 𝑠 𝜈 superscript subscript 𝑟 𝜈 2 superscript subscript 𝑞 𝜈 2 {\displaystyle{\displaystyle p_{\nu}s_{\nu}=r_{\nu}^{2}+q_{\nu}^{2}}}
p_{\nu}s_{\nu} = r_{\nu}^{2}+q_{\nu}^{2}		 ℜ ⁡ ( ν + k + 1 ) > 0 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re(\nu+k+1)>0}} 
p[nu]*((diff( KelvinBer(nu, x), x$(1) ))^(2)+(diff( KelvinBei(nu, x), x$(1) ))^(2)) = (r[nu])^(2)+ (q[nu])^(2)
 
Subscript[p, \[Nu]]*((D[KelvinBer[\[Nu], x], {x, 1}])^(2)+(D[KelvinBei[\[Nu], x], {x, 1}])^(2)) == (Subscript[r, \[Nu]])^(2)+ (Subscript[q, \[Nu]])^(2)
 Skipped - no semantic math Skipped - no semantic math - -
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