Bessel Functions - 10.56 Generating Functions

From testwiki
Revision as of 17:00, 25 May 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
10.56.E1 cos ⁡ z 2 - 2 ⁢ z ⁢ t z = cos ⁡ z z + ∑ n = 1 ∞ t n n ! ⁢ 𝗃 n - 1 ⁡ ( z ) superscript 𝑧 2 2 𝑧 𝑡 𝑧 𝑧 𝑧 superscript subscript 𝑛 1 superscript 𝑡 𝑛 𝑛 spherical-Bessel-J 𝑛 1 𝑧 {\displaystyle{\displaystyle\frac{\cos\sqrt{z^{2}-2zt}}{z}=\frac{\cos z}{z}+% \sum_{n=1}^{\infty}\frac{t^{n}}{n!}\mathsf{j}_{n-1}\left(z\right)}}
\frac{\cos@@{\sqrt{z^{2}-2zt}}}{z} = \frac{\cos@@{z}}{z}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}\sphBesselJ{n-1}@{z}		 ℜ ⁡ ( ( ( n - 1 ) + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( n - 1 ) - 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( - ( n - 1 ) - 1 2 ) ) + k + 1 ) > 0 formulae-sequence 𝑛 1 1 2 𝑘 1 0 formulae-sequence 𝑛 1 1 2 𝑘 1 0 𝑛 1 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re(((n-1)+\frac{1}{2})+k+1)>0,\Re((-(n-1)-\frac{1% }{2})+k+1)>0,\Re((-(-(n-1)-\frac{1}{2}))+k+1)>0}} 
Error

Divide[Cos[Sqrt[(z)^(2)- 2*z*t]],z] == Divide[Cos[z],z]+ Sum[Divide[(t)^(n),(n)!]*SphericalBesselJ[n - 1, z], {n, 1, Infinity}, GenerateConditions->None]
Missing Macro Error Failure -
Failed [42 / 42]
Result: Plus[Complex[-1.0653161526495918, 0.32810386977400907], Times[-1.0, NSum[Times[Power[-1.5, n], Power[Factorial[n], -1], SphericalBesselJ[Plus[-1, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-1.8246723112251149, 0.13108435615091096], Times[-1.0, NSum[Times[Power[-1.5, n], Power[Factorial[n], -1], SphericalBesselJ[Plus[-1, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.56.E2 sin ⁡ z 2 - 2 ⁢ z ⁢ t z = sin ⁡ z z + ∑ n = 1 ∞ t n n ! ⁢ 𝗒 n - 1 ⁡ ( z ) superscript 𝑧 2 2 𝑧 𝑡 𝑧 𝑧 𝑧 superscript subscript 𝑛 1 superscript 𝑡 𝑛 𝑛 spherical-Bessel-Y 𝑛 1 𝑧 {\displaystyle{\displaystyle\frac{\sin\sqrt{z^{2}-2zt}}{z}=\frac{\sin z}{z}+% \sum_{n=1}^{\infty}\frac{t^{n}}{n!}\mathsf{y}_{n-1}\left(z\right)}}
\frac{\sin@@{\sqrt{z^{2}-2zt}}}{z} = \frac{\sin@@{z}}{z}+\sum_{n=1}^{\infty}\frac{t^{n}}{n!}\sphBesselY{n-1}@{z}		 ℜ ⁡ ( ( ( n - 1 ) + 1 2 ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( ( n - 1 ) + 1 2 ) ) + k + 1 ) > 0 , ℜ ⁡ ( ( - ( n - 1 ) - 1 2 ) + k + 1 ) > 0 formulae-sequence 𝑛 1 1 2 𝑘 1 0 formulae-sequence 𝑛 1 1 2 𝑘 1 0 𝑛 1 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re(((n-1)+\frac{1}{2})+k+1)>0,\Re((-((n-1)+\frac{% 1}{2}))+k+1)>0,\Re((-(n-1)-\frac{1}{2})+k+1)>0}} 
Error

Divide[Sin[Sqrt[(z)^(2)- 2*z*t]],z] == Divide[Sin[z],z]+ Sum[Divide[(t)^(n),(n)!]*SphericalBesselY[n - 1, z], {n, 1, Infinity}, GenerateConditions->None]
Missing Macro Error Aborted - Skipped - Because timed out
10.56.E3 cosh ⁡ z 2 + 2 ⁢ i ⁢ z ⁢ t z = cosh ⁡ z z + ∑ n = 1 ∞ ( i ⁢ t ) n n ! ⁢ 𝗂 n - 1 ( 1 ) ⁡ ( z ) superscript 𝑧 2 2 𝑖 𝑧 𝑡 𝑧 𝑧 𝑧 superscript subscript 𝑛 1 superscript 𝑖 𝑡 𝑛 𝑛 spherical-Bessel-I-1 𝑛 1 𝑧 {\displaystyle{\displaystyle\frac{\cosh\sqrt{z^{2}+2izt}}{z}=\frac{\cosh z}{z}% +\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}{\mathsf{i}^{(1)}_{n-1}}\left(z\right)}}
\frac{\cosh@@{\sqrt{z^{2}+2izt}}}{z} = \frac{\cosh@@{z}}{z}+\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}\modsphBesseli{1}{n-1}@{z}		 ℜ ⁡ ( ( ( n - 1 ) + 1 2 ) + k + 1 ) > 0 𝑛 1 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re(((n-1)+\frac{1}{2})+k+1)>0}} 
Error

Divide[Cosh[Sqrt[(z)^(2)+ 2*I*z*t]],z] == Divide[Cosh[z],z]+ Sum[Divide[(I*t)^(n),(n)!]*Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(1-1)*(n - 1 + 1/2), n - 1], {n, 1, Infinity}, GenerateConditions->None]
Missing Macro Error Failure -
Failed [42 / 42]
Result: Plus[Complex[-0.13108435615091052, -1.8246723112251153], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[-1, 2], n], Plus[-1, n]], Power[Factorial[n], -1]]
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-0.022834987510423566, -1.7127448295681993], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-2, 3]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[-1, 2], n], Plus[-1, n]], Power[Factorial[n], -1]]
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.56.E4 sinh ⁡ z 2 + 2 ⁢ i ⁢ z ⁢ t z = sinh ⁡ z z + ∑ n = 1 ∞ ( i ⁢ t ) n n ! ⁢ 𝗂 n - 1 ( 2 ) ⁡ ( z ) superscript 𝑧 2 2 𝑖 𝑧 𝑡 𝑧 𝑧 𝑧 superscript subscript 𝑛 1 superscript 𝑖 𝑡 𝑛 𝑛 spherical-Bessel-I-2 𝑛 1 𝑧 {\displaystyle{\displaystyle\frac{\sinh\sqrt{z^{2}+2izt}}{z}=\frac{\sinh z}{z}% +\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}{\mathsf{i}^{(2)}_{n-1}}\left(z\right)}}
\frac{\sinh@@{\sqrt{z^{2}+2izt}}}{z} = \frac{\sinh@@{z}}{z}+\sum_{n=1}^{\infty}\frac{(it)^{n}}{n!}\modsphBesseli{2}{n-1}@{z}		 ℜ ⁡ ( ( - ( n - 1 ) - 1 2 ) + k + 1 ) > 0 𝑛 1 1 2 𝑘 1 0 {\displaystyle{\displaystyle\Re((-(n-1)-\frac{1}{2})+k+1)>0}} 
Error

Divide[Sinh[Sqrt[(z)^(2)+ 2*I*z*t]],z] == Divide[Sinh[z],z]+ Sum[Divide[(I*t)^(n),(n)!]*Sqrt[Divide[Pi, z]/2] BesselI[(-1)^(2-1)*(n - 1 + 1/2), n - 1], {n, 1, Infinity}, GenerateConditions->None]
Missing Macro Error Failure -
Failed [42 / 42]
Result: Plus[Complex[-0.12983798012989667, -2.1935922908985273], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-1, 6]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], Times[-1, n]], Plus[-1, n]], Power[Factorial[n], -1]]
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Complex[-1.4886830119296848, -1.839102010336905], Times[-1.0, NSum[Times[Power[Complex[0.0, -1.5], n], Power[Power[E, Times[Complex[0, Rational[-2, 3]], Pi]], Rational[1, 2]], Power[Times[Rational[1, 2], Pi], Rational[1, 2]], BesselI[Plus[Rational[1, 2], Times[-1, n]], Plus[-1, n]], Power[Factorial[n], -1]]
Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[t, -1.5], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.56.E5 exp ⁡ ( - z 2 + 2 ⁢ i ⁢ z ⁢ t ) z = e - z z + 2 π ⁢ ∑ n = 1 ∞ ( - i ⁢ t ) n n ! ⁢ 𝗄 n - 1 ⁡ ( z ) superscript 𝑧 2 2 𝑖 𝑧 𝑡 𝑧 superscript 𝑒 𝑧 𝑧 2 𝜋 superscript subscript 𝑛 1 superscript 𝑖 𝑡 𝑛 𝑛 spherical-Bessel-K 𝑛 1 𝑧 {\displaystyle{\displaystyle\frac{\exp\left(-\sqrt{z^{2}+2izt}\right)}{z}=% \frac{e^{-z}}{z}+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-it)^{n}}{n!}\mathsf{k% }_{n-1}\left(z\right)}}
\frac{\exp@{-\sqrt{z^{2}+2izt}}}{z} = \frac{e^{-z}}{z}+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-it)^{n}}{n!}\modsphBesselK{n-1}@{z}		 

Error

Divide[Exp[-Sqrt[(z)^(2)+ 2*I*z*t]],z] == Divide[Exp[- z],z]+Divide[2,Pi]*Sum[Divide[(- I*t)^(n),(n)!]*Sqrt[1/2 Pi /z] BesselK[n - 1 + 1/2, z], {n, 1, Infinity}, GenerateConditions->None]
Missing Macro Error Aborted - Skipped - Because timed out
Retrieved from "https://lct.wmflabs.org/w/index.php?title=10.56&oldid=586"