View source for 10.51

{{DISPLAYTITLE:Bessel Functions - 10.51 Recurrence Relations and Derivatives}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/10.51#Ex1 10.51#Ex1] || <math qid="Q3735">f_{n-1}(z)+f_{n+1}(z) = ((2n+1)/z)f_{n}(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>f_{n-1}(z)+f_{n+1}(z) = ((2n+1)/z)f_{n}(z)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">f[n - 1](z)+ f[n + 1](z) = ((2*n + 1)/z)*f[n](z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[f, n - 1][z]+ Subscript[f, n + 1][z] == ((2*n + 1)/z)*Subscript[f, n][z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.51#Ex5 10.51#Ex5] || <math qid="Q3739">\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}f_{n}(z)) = z^{n-m+1}f_{n-m}(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}f_{n}(z)) = z^{n-m+1}f_{n-m}(z)</syntaxhighlight> || <math>m = 0</math> || <syntaxhighlight lang=mathematica>(diff((1)/(z), z))^(m)*((z)^(n + 1)* f[n](z)) = (z)^(n - m + 1)* f[n - m](z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Divide[1,z], z])^(m)*((z)^(n + 1)* Subscript[f, n][z]) == (z)^(n - m + 1)* Subscript[f, n - m][z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.49999999999999994, -1.8660254037844388]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.49999999999999994, -1.8660254037844388]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.51#Ex6 10.51#Ex6] || <math qid="Q3740">\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}f_{n}(z)) = (-1)^{m}z^{-n-m}f_{n+m}(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}f_{n}(z)) = (-1)^{m}z^{-n-m}f_{n+m}(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(diff((1)/(z), z))^(m)*((z)^(- n)* f[n](z)) = (- 1)^(m)* (z)^(- n - m)* f[n + m](z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Divide[1,z], z])^(m)*((z)^(- n)* Subscript[f, n][z]) == (- 1)^(m)* (z)^(- n - m)* Subscript[f, n + m][z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.366025403-.3660254033*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, f[n+m] = 1/2*3^(1/2)+1/2*I, n = 1, m = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9999999993-.9999999984*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, f[n+m] = 1/2*3^(1/2)+1/2*I, n = 2, m = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.1339745962155613, 0.49999999999999994]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.3660254037844386, 0.36602540378443865]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.51#Ex7 10.51#Ex7] || <math qid="Q3741">g_{n-1}(z)-g_{n+1}(z) = ((2n+1)/z)g_{n}(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>g_{n-1}(z)-g_{n+1}(z) = ((2n+1)/z)g_{n}(z)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">g[n - 1](z)- g[n + 1](z) = ((2*n + 1)/z)*g[n](z)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[g, n - 1][z]- Subscript[g, n + 1][z] == ((2*n + 1)/z)*Subscript[g, n][z]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/10.51#Ex11 10.51#Ex11] || <math qid="Q3745">\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}g_{n}(z)) = z^{n-m+1}g_{n-m}(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}g_{n}(z)) = z^{n-m+1}g_{n-m}(z)</syntaxhighlight> || <math>m = 0</math> || <syntaxhighlight lang=mathematica>(diff((1)/(z), z))^(m)*((z)^(n + 1)* g[n](z)) = (z)^(n - m + 1)* g[n - m](z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Divide[1,z], z])^(m)*((z)^(n + 1)* Subscript[g, n][z]) == (z)^(n - m + 1)* Subscript[g, n - m][z]</syntaxhighlight> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.49999999999999994, -1.8660254037844388]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.49999999999999994, -1.8660254037844388]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[Times[-1, m], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.51#Ex12 10.51#Ex12] || <math qid="Q3746">\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}g_{n}(z)) = z^{-n-m}g_{n+m}(z)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}g_{n}(z)) = z^{-n-m}g_{n+m}(z)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(diff((1)/(z), z))^(m)*((z)^(- n)* g[n](z)) = (z)^(- n - m)* g[n + m](z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Divide[1,z], z])^(m)*((z)^(- n)* Subscript[g, n][z]) == (z)^(- n - m)* Subscript[g, n + m][z]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .3660254028+1.366025403*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, g[n] = 1/2*3^(1/2)+1/2*I, g[n+m] = 1/2*3^(1/2)+1/2*I, n = 1, m = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .9999999987+.9999999996*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, g[n] = 1/2*3^(1/2)+1/2*I, g[n+m] = 1/2*3^(1/2)+1/2*I, n = 2, m = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [288 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-1.8660254037844388, 0.49999999999999994]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[-1.3660254037844388, 1.3660254037844386]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[g, Plus[m, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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