Bessel Functions - 10.51 Recurrence Relations and Derivatives
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 10.51#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle f_{n-1}(z)+f_{n+1}(z) = ((2n+1)/z)f_{n}(z)}
f_{n-1}(z)+f_{n+1}(z) = ((2n+1)/z)f_{n}(z) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | f[n - 1](z)+ f[n + 1](z) = ((2*n + 1)/z)*f[n](z) |
Subscript[f, n - 1][z]+ Subscript[f, n + 1][z] == ((2*n + 1)/z)*Subscript[f, n][z] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.51#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}f_{n}(z)) = z^{n-m+1}f_{n-m}(z)}
\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}f_{n}(z)) = z^{n-m+1}f_{n-m}(z) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m = 0} | (diff((1)/(z), z))^(m)*((z)^(n + 1)* f[n](z)) = (z)^(n - m + 1)* f[n - m](z)
|
(D[Divide[1,z], z])^(m)*((z)^(n + 1)* Subscript[f, n][z]) == (z)^(n - m + 1)* Subscript[f, n - m][z]
|
Failure | Failure | Error | Failed [288 / 300]
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]]]}
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]]]}
... skip entries to safe data |
| 10.51#Ex6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}f_{n}(z)) = (-1)^{m}z^{-n-m}f_{n+m}(z)}
\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}f_{n}(z)) = (-1)^{m}z^{-n-m}f_{n+m}(z) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (diff((1)/(z), z))^(m)*((z)^(- n)* f[n](z)) = (- 1)^(m)* (z)^(- n - m)* f[n + m](z)
|
(D[Divide[1,z], z])^(m)*((z)^(- n)* Subscript[f, n][z]) == (- 1)^(m)* (z)^(- n - m)* Subscript[f, n + m][z]
|
Failure | Failure | Failed [288 / 300] 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}
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}
... skip entries to safe data |
Failed [288 / 300]
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]]]}
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]]]}
... skip entries to safe data |
| 10.51#Ex7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle g_{n-1}(z)-g_{n+1}(z) = ((2n+1)/z)g_{n}(z)}
g_{n-1}(z)-g_{n+1}(z) = ((2n+1)/z)g_{n}(z) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | g[n - 1](z)- g[n + 1](z) = ((2*n + 1)/z)*g[n](z) |
Subscript[g, n - 1][z]- Subscript[g, n + 1][z] == ((2*n + 1)/z)*Subscript[g, n][z] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.51#Ex11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}g_{n}(z)) = z^{n-m+1}g_{n-m}(z)}
\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{n+1}g_{n}(z)) = z^{n-m+1}g_{n-m}(z) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m = 0} | (diff((1)/(z), z))^(m)*((z)^(n + 1)* g[n](z)) = (z)^(n - m + 1)* g[n - m](z)
|
(D[Divide[1,z], z])^(m)*((z)^(n + 1)* Subscript[g, n][z]) == (z)^(n - m + 1)* Subscript[g, n - m][z]
|
Failure | Failure | Error | Failed [288 / 300]
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]]]}
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]]]}
... skip entries to safe data |
| 10.51#Ex12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}g_{n}(z)) = z^{-n-m}g_{n+m}(z)}
\left(\frac{1}{z}\deriv{}{z}\right)^{m}(z^{-n}g_{n}(z)) = z^{-n-m}g_{n+m}(z) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (diff((1)/(z), z))^(m)*((z)^(- n)* g[n](z)) = (z)^(- n - m)* g[n + m](z)
|
(D[Divide[1,z], z])^(m)*((z)^(- n)* Subscript[g, n][z]) == (z)^(- n - m)* Subscript[g, n + m][z]
|
Failure | Failure | Failed [288 / 300] 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}
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}
... skip entries to safe data |
Failed [288 / 300]
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]]]}
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]]]}
... skip entries to safe data |