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 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 {\displaystyle{\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)		 

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 ( 1 z ⁢ d d z ) m ⁡ ( z n + 1 ⁢ f n ⁢ ( z ) ) = z n - m + 1 ⁢ f n - m ⁢ ( z ) superscript 1 𝑧 derivative 𝑧 𝑚 superscript 𝑧 𝑛 1 subscript 𝑓 𝑛 𝑧 superscript 𝑧 𝑛 𝑚 1 subscript 𝑓 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}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)		 m = 0 𝑚 0 {\displaystyle{\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 ( 1 z ⁢ d d z ) m ⁡ ( z - n ⁢ f n ⁢ ( z ) ) = ( - 1 ) m ⁢ z - n - m ⁢ f n + m ⁢ ( z ) superscript 1 𝑧 derivative 𝑧 𝑚 superscript 𝑧 𝑛 subscript 𝑓 𝑛 𝑧 superscript 1 𝑚 superscript 𝑧 𝑛 𝑚 subscript 𝑓 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}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)		 

(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 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 {\displaystyle{\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)		 

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 ( 1 z ⁢ d d z ) m ⁡ ( z n + 1 ⁢ g n ⁢ ( z ) ) = z n - m + 1 ⁢ g n - m ⁢ ( z ) superscript 1 𝑧 derivative 𝑧 𝑚 superscript 𝑧 𝑛 1 subscript 𝑔 𝑛 𝑧 superscript 𝑧 𝑛 𝑚 1 subscript 𝑔 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}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)		 m = 0 𝑚 0 {\displaystyle{\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 ( 1 z ⁢ d d z ) m ⁡ ( z - n ⁢ g n ⁢ ( z ) ) = z - n - m ⁢ g n + m ⁢ ( z ) superscript 1 𝑧 derivative 𝑧 𝑚 superscript 𝑧 𝑛 subscript 𝑔 𝑛 𝑧 superscript 𝑧 𝑛 𝑚 subscript 𝑔 𝑛 𝑚 𝑧 {\displaystyle{\displaystyle\left(\frac{1}{z}\frac{\mathrm{d}}{\mathrm{d}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)		 

(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
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