10.23: Difference between revisions

Latest revision as of 11:24, 28 June 2021

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
10.23.E3	${\displaystyle{\displaystyle{J_{0}^{2}}\left(z\right)+2\sum_{k=1}^{\infty}{J_{% k}^{2}}\left(z\right)=1}}$ `\BesselJ{0}^{2}@{z}+2\sum_{k=1}^{\infty}\BesselJ{k}^{2}@{z} = 1`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re(k+k+1)>0}}$	(BesselJ(0, z))^(2)+ 2*sum((BesselJ(k, z))^(2), k = 1..infinity) = 1	(BesselJ[0, z])^(2)+ 2*Sum[(BesselJ[k, z])^(2), {k, 1, Infinity}, GenerateConditions->None] == 1	Aborted	Successful	Successful [Tested: 7]	Successful [Tested: 7]
10.23.E4	${\displaystyle{\displaystyle\sum_{k=0}^{2n}(-1)^{k}J_{k}\left(z\right)J_{2n-k}% \left(z\right)\\ +2\sum_{k=1}^{\infty}J_{k}\left(z\right)J_{2n+k}\left(z\right)=0}}$ `\sum_{k=0}^{2n}(-1)^{k}\BesselJ{k}@{z}\BesselJ{2n-k}@{z}\\ +2\sum_{k=1}^{\infty}\BesselJ{k}@{z}\BesselJ{2n+k}@{z} = 0`	${\displaystyle{\displaystyle n\geq 1,\Re(k+k+1)>0,\Re((2n-k)+k+1)>0,\Re((2n+k)% +k+1)>0}}$	sum((- 1)^(k)* BesselJ(k, z)BesselJ(2n - k, z); , k = 0..2n)+ 2sum(BesselJ(k, z)BesselJ(2*n + k, z), k = 1..infinity) = 0	Sum[(- 1)^(k)* BesselJ[k, z]BesselJ[2n - k, z], {k, 0, 2n}, GenerateConditions->None]+ 2Sum[BesselJ[k, z]BesselJ[2*n + k, z], {k, 1, Infinity}, GenerateConditions->None] == 0	Error	Failure	-	Failed [21 / 21] Result: Plus[Complex[0.00727987412712798, -0.017853077134921347], Times[2.0, NSum[Times[BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[2, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[2.4034761502300195^-4, -3.087748713313073^-5], Times[2.0, NSum[Times[BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[4, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.23.E5	${\displaystyle{\displaystyle\sum_{k=0}^{n}J_{k}\left(z\right)J_{n-k}\left(z% \right)+2\sum_{k=1}^{\infty}(-1)^{k}J_{k}\left(z\right)J_{n+k}\left(z\right)=J% _{n}\left(2z\right)}}$ `\sum_{k=0}^{n}\BesselJ{k}@{z}\BesselJ{n-k}@{z}+2\sum_{k=1}^{\infty}(-1)^{k}\BesselJ{k}@{z}\BesselJ{n+k}@{z} = \BesselJ{n}@{2z}`	${\displaystyle{\displaystyle\Re(k+k+1)>0,\Re((n-k)+k+1)>0,\Re((n+k)+k+1)>0,\Re% (n+k+1)>0}}$	sum(BesselJ(k, z)BesselJ(n - k, z), k = 0..n)+ 2sum((- 1)^(k)* BesselJ(k, z)BesselJ(n + k, z), k = 1..infinity) = BesselJ(n, 2z)	Sum[BesselJ[k, z]BesselJ[n - k, z], {k, 0, n}, GenerateConditions->None]+ 2Sum[(- 1)^(k)* BesselJ[k, z]BesselJ[n + k, z], {k, 1, Infinity}, GenerateConditions->None] == BesselJ[n, 2z]	Aborted	Failure	Skipped - Because timed out	Failed [21 / 21] Result: Plus[Complex[0.024343533040476317, 0.10797471990649704], Times[2.0, NSum[Times[Power[-1, k], BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[1, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[-0.006069425709337772, 0.017711723121060452], Times[2.0, NSum[Times[Power[-1, k], BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[2, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]] Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
10.23#Ex1	${\displaystyle{\displaystyle w=\sqrt{u^{2}+v^{2}-2uv\cos\alpha}}}$ `w = \sqrt{u^{2}+v^{2}-2uv\cos@@{\alpha}}`		w = sqrt((u)^(2)+ (v)^(2)- 2uv*cos(alpha))	w == Sqrt[(u)^(2)+ (v)^(2)- 2uv*Cos[\[Alpha]]]	Failure	Failure	Failed [300 / 300] Result: -.3146075610-.1816387601I Test Values: {alpha = 3/2, u = 1/23^(1/2)+1/2I, v = 1/23^(1/2)+1/2I, w = 1/23^(1/2)+1/2I} Result: -1.680632965+.1843866439I Test Values: {alpha = 3/2, u = 1/23^(1/2)+1/2I, v = 1/23^(1/2)+1/2I, w = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.3146075609842255, -0.18163876002333418] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]} Result: Complex[0.4375091763619045, 0.252596040745477] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]} ... skip entries to safe data
10.23#Ex2	${\displaystyle{\displaystyle u-v\cos\alpha=w\cos\chi}}$ `u-v\cos@@{\alpha} = w\cos@@{\chi}`		u - vcos(alpha) = wcos(chi)	u - vCos[\[Alpha]] == wCos[\[Chi]]	Failure	Failure	Failed [300 / 300] Result: -.263783978e-1+.4431282844I Test Values: {alpha = 3/2, chi = 1/23^(1/2)+1/2I, u = 1/23^(1/2)+1/2I, v = 1/23^(1/2)+1/2I, w = 1/23^(1/2)+1/2I} Result: .8262683052-.3665121890I Test Values: {alpha = 3/2, chi = 1/23^(1/2)+1/2I, u = 1/23^(1/2)+1/2I, v = 1/23^(1/2)+1/2I, w = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.026378398027867456, 0.44312828415668515] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.023973249213014358, -0.5554825514041751] Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.23#Ex3	${\displaystyle{\displaystyle v\sin\alpha=w\sin\chi}}$ `v\sin@@{\alpha} = w\sin@@{\chi}`		vsin(alpha) = wsin(chi)	vSin[\[Alpha]] == wSin[\[Chi]]	Failure	Failure	Failed [300 / 300] Result: .2887554391-.2231097873I Test Values: {alpha = 3/2, chi = 1/23^(1/2)+1/2I, v = 1/23^(1/2)+1/2I, w = 1/23^(1/2)+1/2I} Result: 1.585713279-.763530664e-1I Test Values: {alpha = 3/2, chi = 1/23^(1/2)+1/2I, v = 1/23^(1/2)+1/2I, w = -1/2+1/2I3^(1/2)} ... skip entries to safe data	Failed [294 / 300] Result: Complex[0.2887554393029954, -0.22310978722682606] Test Values: {Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.8740447527972026, 0.09051196331992012] Test Values: {Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
10.23.E9	${\displaystyle{\displaystyle e^{iv\cos\alpha}=\frac{\Gamma\left(\nu\right)}{(% \tfrac{1}{2}v)^{\nu}}\\sum_{k=0}^{\infty}(\nu+k)i^{k}J_{\nu+k}\left(v\right)C% ^{(\nu)}_{k}\left(\cos\alpha\right)}}$ `e^{iv\cos@@{\alpha}} = \frac{\EulerGamma@{\nu}}{(\tfrac{1}{2}v)^{\nu}}\\sum_{k=0}^{\infty}(\nu+k)i^{k}\BesselJ{\nu+k}@{v}\ultrasphpoly{\nu}{k}@{\cos@@{\alpha}}`	${\displaystyle{\displaystyle\Re((\nu+k)+k+1)>0,\Re(\nu)>0}}$	exp(Ivcos(alpha)) = (GAMMA(nu))/(((1)/(2)v)^(nu)) sum((nu + k)(I)^(k) BesselJ(nu + k, v)*GegenbauerC(k, nu, cos(alpha)), k = 0..infinity)	Exp[IvCos[\[Alpha]]] == Divide[Gamma[\[Nu]],(Divide[1,2]v)^\[Nu]] Sum[(\[Nu]+ k)(I)^(k) BesselJ[\[Nu]+ k, v]*GegenbauerC[k, \[Nu], Cos[\[Alpha]]], {k, 0, Infinity}, GenerateConditions->None]	Aborted	Failure	Skipped - Because timed out	Skipped - Because timed out
10.23.E15	${\displaystyle{\displaystyle(\tfrac{1}{2}z)^{\nu}=\sum_{k=0}^{\infty}\frac{(% \nu+2k)\Gamma\left(\nu+k\right)}{k!}J_{\nu+2k}\left(z\right)}}$ `(\tfrac{1}{2}z)^{\nu} = \sum_{k=0}^{\infty}\frac{(\nu+2k)\EulerGamma@{\nu+k}}{k!}\BesselJ{\nu+2k}@{z}`	${\displaystyle{\displaystyle\Re((\nu+2k)+k+1)>0,\Re(\nu+k)>0}}$	((1)/(2)z)^(nu) = sum(((nu + 2k)GAMMA(nu + k))/(factorial(k))BesselJ(nu + 2*k, z), k = 0..infinity)	(Divide[1,2]z)^\[Nu] == Sum[Divide[(\[Nu]+ 2k)Gamma[\[Nu]+ k],(k)!]BesselJ[\[Nu]+ 2*k, z], {k, 0, Infinity}, GenerateConditions->None]	Aborted	Successful	Skipped - Because timed out	Successful [Tested: 7]
10.23.E16	${\displaystyle{\displaystyle Y_{0}\left(z\right)=\frac{2}{\pi}\left(\ln\left(% \tfrac{1}{2}z\right)+\gamma\right)J_{0}\left(z\right)-\frac{4}{\pi}\sum_{k=1}^% {\infty}(-1)^{k}\frac{J_{2k}\left(z\right)}{k}}}$ `\BesselY{0}@{z} = \frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\BesselJ{0}@{z}-\frac{4}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{\BesselJ{2k}@{z}}{k}`	${\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((2k)+k+1)>0,\Re((-0)+k+1)>0}}$	BesselY(0, z) = (2)/(Pi)(ln((1)/(2)z)+ gamma)BesselJ(0, z)-(4)/(Pi)sum((- 1)^(k)(BesselJ(2k, z))/(k), k = 1..infinity)	BesselY[0, z] == Divide[2,Pi](Log[Divide[1,2]z]+ EulerGamma)BesselJ[0, z]-Divide[4,Pi]Sum[(- 1)^(k)Divide[BesselJ[2k, z],k], {k, 1, Infinity}, GenerateConditions->None]	Aborted	Successful	Successful [Tested: 7]	Successful [Tested: 7]
10.23.E17	${\displaystyle{\displaystyle Y_{n}\left(z\right)=-\frac{n!(\tfrac{1}{2}z)^{-n}% }{\pi}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}J_{k}\left(z\right)}{k!(n-k)}+% \frac{2}{\pi}\left(\ln\left(\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)J_% {n}\left(z\right)-\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)J_{n+2k}% \left(z\right)}{k(n+k)}}}$ `\BesselY{n}@{z} = -\frac{n!(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}\BesselJ{k}@{z}}{k!(n-k)}+\frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}-\digamma@{n+1}\right)\BesselJ{n}@{z}-\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)\BesselJ{n+2k}@{z}}{k(n+k)}`	${\displaystyle{\displaystyle\Re(n+k+1)>0,\Re(k+k+1)>0,\Re((n+2k)+k+1)>0,\Re((-% n)+k+1)>0}}$	BesselY(n, z) = -(factorial(n)((1)/(2)z)^(- n))/(Pi)sum((((1)/(2)z)^(k)* BesselJ(k, z))/(factorial(k)(n - k)), k = 0..n - 1)+(2)/(Pi)(ln((1)/(2)z)- Psi(n + 1))BesselJ(n, z)-(2)/(Pi)sum((- 1)^(k)((n + 2k)BesselJ(n + 2k, z))/(k(n + k)), k = 1..infinity)	BesselY[n, z] == -Divide[(n)!(Divide[1,2]z)^(- n),Pi]Sum[Divide[(Divide[1,2]z)^(k)* BesselJ[k, z],(k)!(n - k)], {k, 0, n - 1}, GenerateConditions->None]+Divide[2,Pi](Log[Divide[1,2]z]- PolyGamma[n + 1])BesselJ[n, z]-Divide[2,Pi]Sum[(- 1)^(k)Divide[(n + 2k)BesselJ[n + 2k, z],k(n + k)], {k, 1, Infinity}, GenerateConditions->None]	Aborted	Failure	Manual Skip!	Failed [16 / 21] Result: Plus[Complex[-0.41373222494160333, 0.38808044477324316], Times[Complex[0.5513288954217921, -0.31830988618379064], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[Plus[Times[-1, ], 1], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], []], Times[Plus[4, Times[12, ], Times[12, Power[, 2]], Times[4, Power[, 3]], Times[-4, 1], Times[-8, , 1], Times[-4, Power[, 2], 1], Times[, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-1, 1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[1, ]]], Times[4, Plus[1, ], Plus[-5, Times[-6, ], Times[-2, Power[, 2]], Times[3, 1], Times[2, , 1]], [Plus[2, ]]], Times[-4, Plus[1, ], Plus[2, ], Plus[-2, Times[-1, ], 1], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[1, -1], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Plus[Times[Power[1, -1], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]<syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.6198631863998064, 5.383408526303685], Times[Complex[0.0, -15.278874536821952], DifferenceRoot[Function[{, } Test Values: {Equal[Plus[Times[-1, Power[-1, Rational[1, 3]], Plus[-3, ], []], Times[Plus[-8, Times[-3, Power[-1, Rational[1, 3]]], Times[-12, ], Times[Power[-1, Rational[1, 3]], ], Times[4, Power[, 3]]], [Plus[1, ]]], Times[-8, Plus[1, ], Plus[-2, Power[, 2]], [Plus[2, ]]], Times[4, Plus[-1, ], Plus[1, ], Plus[2, ], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Rational[1, 3], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Plus[Times[Rational[1, 3], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Rational[1, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], BesselJ[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][3.0]]], {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data

@@ Line 14: / Line 14: @@
 ! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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-| [https://dlmf.nist.gov/10.23.E3 10.23.E3] || [[Item:Q3455|<math>\BesselJ{0}^{2}@{z}+2\sum_{k=1}^{\infty}\BesselJ{k}^{2}@{z} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselJ{0}^{2}@{z}+2\sum_{k=1}^{\infty}\BesselJ{k}^{2}@{z} = 1</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{(k+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(BesselJ(0, z))^(2)+ 2*sum((BesselJ(k, z))^(2), k = 1..infinity) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>(BesselJ[0, z])^(2)+ 2*Sum[(BesselJ[k, z])^(2), {k, 1, Infinity}, GenerateConditions->None] == 1</syntaxhighlight> || Aborted || Successful || Successful [Tested: 7] || Successful [Tested: 7]
+| [https://dlmf.nist.gov/10.23.E3 10.23.E3] || <math qid="Q3455">\BesselJ{0}^{2}@{z}+2\sum_{k=1}^{\infty}\BesselJ{k}^{2}@{z} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselJ{0}^{2}@{z}+2\sum_{k=1}^{\infty}\BesselJ{k}^{2}@{z} = 1</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{(k+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(BesselJ(0, z))^(2)+ 2*sum((BesselJ(k, z))^(2), k = 1..infinity) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>(BesselJ[0, z])^(2)+ 2*Sum[(BesselJ[k, z])^(2), {k, 1, Infinity}, GenerateConditions->None] == 1</syntaxhighlight> || Aborted || Successful || Successful [Tested: 7] || Successful [Tested: 7]
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-| [https://dlmf.nist.gov/10.23.E4 10.23.E4] || [[Item:Q3456|<math>\sum_{k=0}^{2n}(-1)^{k}\BesselJ{k}@{z}\BesselJ{2n-k}@{z}\\ +2\sum_{k=1}^{\infty}\BesselJ{k}@{z}\BesselJ{2n+k}@{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{2n}(-1)^{k}\BesselJ{k}@{z}\BesselJ{2n-k}@{z}\\ +2\sum_{k=1}^{\infty}\BesselJ{k}@{z}\BesselJ{2n+k}@{z} = 0</syntaxhighlight> || <math>n \geq 1, \realpart@@{(k+k+1)} > 0, \realpart@@{((2n-k)+k+1)} > 0, \realpart@@{((2n+k)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sum((- 1)^(k)* BesselJ(k, z)*BesselJ(2*n - k, z)*; , k = 0..2*n)+ 2*sum(BesselJ(k, z)*BesselJ(2*n + k, z), k = 1..infinity) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(k)* BesselJ[k, z]*BesselJ[2*n - k, z]*, {k, 0, 2*n}, GenerateConditions->None]+ 2*Sum[BesselJ[k, z]*BesselJ[2*n + k, z], {k, 1, Infinity}, GenerateConditions->None] == 0</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.00727987412712798, -0.017853077134921347], Times[2.0, NSum[Times[BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[2, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
+| [https://dlmf.nist.gov/10.23.E4 10.23.E4] || <math qid="Q3456">\sum_{k=0}^{2n}(-1)^{k}\BesselJ{k}@{z}\BesselJ{2n-k}@{z}\\ +2\sum_{k=1}^{\infty}\BesselJ{k}@{z}\BesselJ{2n+k}@{z} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{2n}(-1)^{k}\BesselJ{k}@{z}\BesselJ{2n-k}@{z}\\ +2\sum_{k=1}^{\infty}\BesselJ{k}@{z}\BesselJ{2n+k}@{z} = 0</syntaxhighlight> || <math>n \geq 1, \realpart@@{(k+k+1)} > 0, \realpart@@{((2n-k)+k+1)} > 0, \realpart@@{((2n+k)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sum((- 1)^(k)* BesselJ(k, z)*BesselJ(2*n - k, z)*; , k = 0..2*n)+ 2*sum(BesselJ(k, z)*BesselJ(2*n + k, z), k = 1..infinity) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(k)* BesselJ[k, z]*BesselJ[2*n - k, z]*, {k, 0, 2*n}, GenerateConditions->None]+ 2*Sum[BesselJ[k, z]*BesselJ[2*n + k, z], {k, 1, Infinity}, GenerateConditions->None] == 0</syntaxhighlight> || Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.00727987412712798, -0.017853077134921347], Times[2.0, NSum[Times[BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[2, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
 Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[2.4034761502300195*^-4, -3.087748713313073*^-5], Times[2.0, NSum[Times[BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[4, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
 Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/10.23.E5 10.23.E5] || [[Item:Q3457|<math>\sum_{k=0}^{n}\BesselJ{k}@{z}\BesselJ{n-k}@{z}+2\sum_{k=1}^{\infty}(-1)^{k}\BesselJ{k}@{z}\BesselJ{n+k}@{z} = \BesselJ{n}@{2z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{n}\BesselJ{k}@{z}\BesselJ{n-k}@{z}+2\sum_{k=1}^{\infty}(-1)^{k}\BesselJ{k}@{z}\BesselJ{n+k}@{z} = \BesselJ{n}@{2z}</syntaxhighlight> || <math>\realpart@@{(k+k+1)} > 0, \realpart@@{((n-k)+k+1)} > 0, \realpart@@{((n+k)+k+1)} > 0, \realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sum(BesselJ(k, z)*BesselJ(n - k, z), k = 0..n)+ 2*sum((- 1)^(k)* BesselJ(k, z)*BesselJ(n + k, z), k = 1..infinity) = BesselJ(n, 2*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[BesselJ[k, z]*BesselJ[n - k, z], {k, 0, n}, GenerateConditions->None]+ 2*Sum[(- 1)^(k)* BesselJ[k, z]*BesselJ[n + k, z], {k, 1, Infinity}, GenerateConditions->None] == BesselJ[n, 2*z]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.024343533040476317, 0.10797471990649704], Times[2.0, NSum[Times[Power[-1, k], BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[1, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
+| [https://dlmf.nist.gov/10.23.E5 10.23.E5] || <math qid="Q3457">\sum_{k=0}^{n}\BesselJ{k}@{z}\BesselJ{n-k}@{z}+2\sum_{k=1}^{\infty}(-1)^{k}\BesselJ{k}@{z}\BesselJ{n+k}@{z} = \BesselJ{n}@{2z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{n}\BesselJ{k}@{z}\BesselJ{n-k}@{z}+2\sum_{k=1}^{\infty}(-1)^{k}\BesselJ{k}@{z}\BesselJ{n+k}@{z} = \BesselJ{n}@{2z}</syntaxhighlight> || <math>\realpart@@{(k+k+1)} > 0, \realpart@@{((n-k)+k+1)} > 0, \realpart@@{((n+k)+k+1)} > 0, \realpart@@{(n+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sum(BesselJ(k, z)*BesselJ(n - k, z), k = 0..n)+ 2*sum((- 1)^(k)* BesselJ(k, z)*BesselJ(n + k, z), k = 1..infinity) = BesselJ(n, 2*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[BesselJ[k, z]*BesselJ[n - k, z], {k, 0, n}, GenerateConditions->None]+ 2*Sum[(- 1)^(k)* BesselJ[k, z]*BesselJ[n + k, z], {k, 1, Infinity}, GenerateConditions->None] == BesselJ[n, 2*z]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [21 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[0.024343533040476317, 0.10797471990649704], Times[2.0, NSum[Times[Power[-1, k], BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[1, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
 Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.006069425709337772, 0.017711723121060452], Times[2.0, NSum[Times[Power[-1, k], BesselJ[k, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], BesselJ[Plus[2, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
 Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/10.23#Ex1 10.23#Ex1] || [[Item:Q3458|<math>w = \sqrt{u^{2}+v^{2}-2uv\cos@@{\alpha}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w = \sqrt{u^{2}+v^{2}-2uv\cos@@{\alpha}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>w = sqrt((u)^(2)+ (v)^(2)- 2*u*v*cos(alpha))</syntaxhighlight> || <syntaxhighlight lang=mathematica>w == Sqrt[(u)^(2)+ (v)^(2)- 2*u*v*Cos[\[Alpha]]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.3146075610-.1816387601*I
+| [https://dlmf.nist.gov/10.23#Ex1 10.23#Ex1] || <math qid="Q3458">w = \sqrt{u^{2}+v^{2}-2uv\cos@@{\alpha}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>w = \sqrt{u^{2}+v^{2}-2uv\cos@@{\alpha}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>w = sqrt((u)^(2)+ (v)^(2)- 2*u*v*cos(alpha))</syntaxhighlight> || <syntaxhighlight lang=mathematica>w == Sqrt[(u)^(2)+ (v)^(2)- 2*u*v*Cos[\[Alpha]]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.3146075610-.1816387601*I
 Test Values: {alpha = 3/2, u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.680632965+.1843866439*I
 Test Values: {alpha = 3/2, u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.3146075609842255, -0.18163876002333418]
@@ Line 30: / Line 30: @@
 Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/10.23#Ex2 10.23#Ex2] || [[Item:Q3459|<math>u-v\cos@@{\alpha} = w\cos@@{\chi}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>u-v\cos@@{\alpha} = w\cos@@{\chi}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>u - v*cos(alpha) = w*cos(chi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>u - v*Cos[\[Alpha]] == w*Cos[\[Chi]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.263783978e-1+.4431282844*I
+| [https://dlmf.nist.gov/10.23#Ex2 10.23#Ex2] || <math qid="Q3459">u-v\cos@@{\alpha} = w\cos@@{\chi}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>u-v\cos@@{\alpha} = w\cos@@{\chi}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>u - v*cos(alpha) = w*cos(chi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>u - v*Cos[\[Alpha]] == w*Cos[\[Chi]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.263783978e-1+.4431282844*I
 Test Values: {alpha = 3/2, chi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8262683052-.3665121890*I
 Test Values: {alpha = 3/2, chi = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.026378398027867456, 0.44312828415668515]
@@ Line 36: / Line 36: @@
 Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/10.23#Ex3 10.23#Ex3] || [[Item:Q3460|<math>v\sin@@{\alpha} = w\sin@@{\chi}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>v\sin@@{\alpha} = w\sin@@{\chi}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>v*sin(alpha) = w*sin(chi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>v*Sin[\[Alpha]] == w*Sin[\[Chi]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2887554391-.2231097873*I
+| [https://dlmf.nist.gov/10.23#Ex3 10.23#Ex3] || <math qid="Q3460">v\sin@@{\alpha} = w\sin@@{\chi}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>v\sin@@{\alpha} = w\sin@@{\chi}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>v*sin(alpha) = w*sin(chi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>v*Sin[\[Alpha]] == w*Sin[\[Chi]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .2887554391-.2231097873*I
 Test Values: {alpha = 3/2, chi = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.585713279-.763530664e-1*I
 Test Values: {alpha = 3/2, chi = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I, w = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [294 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.2887554393029954, -0.22310978722682606]
@@ Line 42: / Line 42: @@
 Test Values: {Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |-
-| [https://dlmf.nist.gov/10.23.E9 10.23.E9] || [[Item:Q3463|<math>e^{iv\cos@@{\alpha}} = \frac{\EulerGamma@{\nu}}{(\tfrac{1}{2}v)^{\nu}}\*\sum_{k=0}^{\infty}(\nu+k)i^{k}\BesselJ{\nu+k}@{v}\ultrasphpoly{\nu}{k}@{\cos@@{\alpha}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{iv\cos@@{\alpha}} = \frac{\EulerGamma@{\nu}}{(\tfrac{1}{2}v)^{\nu}}\*\sum_{k=0}^{\infty}(\nu+k)i^{k}\BesselJ{\nu+k}@{v}\ultrasphpoly{\nu}{k}@{\cos@@{\alpha}}</syntaxhighlight> || <math>\realpart@@{((\nu+k)+k+1)} > 0, \realpart@@{(\nu)} > 0</math> || <syntaxhighlight lang=mathematica>exp(I*v*cos(alpha)) = (GAMMA(nu))/(((1)/(2)*v)^(nu))* sum((nu + k)*(I)^(k)* BesselJ(nu + k, v)*GegenbauerC(k, nu, cos(alpha)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[I*v*Cos[\[Alpha]]] == Divide[Gamma[\[Nu]],(Divide[1,2]*v)^\[Nu]]* Sum[(\[Nu]+ k)*(I)^(k)* BesselJ[\[Nu]+ k, v]*GegenbauerC[k, \[Nu], Cos[\[Alpha]]], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || Skipped - Because timed out
+| [https://dlmf.nist.gov/10.23.E9 10.23.E9] || <math qid="Q3463">e^{iv\cos@@{\alpha}} = \frac{\EulerGamma@{\nu}}{(\tfrac{1}{2}v)^{\nu}}\*\sum_{k=0}^{\infty}(\nu+k)i^{k}\BesselJ{\nu+k}@{v}\ultrasphpoly{\nu}{k}@{\cos@@{\alpha}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{iv\cos@@{\alpha}} = \frac{\EulerGamma@{\nu}}{(\tfrac{1}{2}v)^{\nu}}\*\sum_{k=0}^{\infty}(\nu+k)i^{k}\BesselJ{\nu+k}@{v}\ultrasphpoly{\nu}{k}@{\cos@@{\alpha}}</syntaxhighlight> || <math>\realpart@@{((\nu+k)+k+1)} > 0, \realpart@@{(\nu)} > 0</math> || <syntaxhighlight lang=mathematica>exp(I*v*cos(alpha)) = (GAMMA(nu))/(((1)/(2)*v)^(nu))* sum((nu + k)*(I)^(k)* BesselJ(nu + k, v)*GegenbauerC(k, nu, cos(alpha)), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[I*v*Cos[\[Alpha]]] == Divide[Gamma[\[Nu]],(Divide[1,2]*v)^\[Nu]]* Sum[(\[Nu]+ k)*(I)^(k)* BesselJ[\[Nu]+ k, v]*GegenbauerC[k, \[Nu], Cos[\[Alpha]]], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || Skipped - Because timed out
 |-
-| [https://dlmf.nist.gov/10.23.E15 10.23.E15] || [[Item:Q3469|<math>(\tfrac{1}{2}z)^{\nu} = \sum_{k=0}^{\infty}\frac{(\nu+2k)\EulerGamma@{\nu+k}}{k!}\BesselJ{\nu+2k}@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(\tfrac{1}{2}z)^{\nu} = \sum_{k=0}^{\infty}\frac{(\nu+2k)\EulerGamma@{\nu+k}}{k!}\BesselJ{\nu+2k}@{z}</syntaxhighlight> || <math>\realpart@@{((\nu+2k)+k+1)} > 0, \realpart@@{(\nu+k)} > 0</math> || <syntaxhighlight lang=mathematica>((1)/(2)*z)^(nu) = sum(((nu + 2*k)*GAMMA(nu + k))/(factorial(k))*BesselJ(nu + 2*k, z), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Divide[1,2]*z)^\[Nu] == Sum[Divide[(\[Nu]+ 2*k)*Gamma[\[Nu]+ k],(k)!]*BesselJ[\[Nu]+ 2*k, z], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Successful || Skipped - Because timed out || Successful [Tested: 7]
+| [https://dlmf.nist.gov/10.23.E15 10.23.E15] || <math qid="Q3469">(\tfrac{1}{2}z)^{\nu} = \sum_{k=0}^{\infty}\frac{(\nu+2k)\EulerGamma@{\nu+k}}{k!}\BesselJ{\nu+2k}@{z}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(\tfrac{1}{2}z)^{\nu} = \sum_{k=0}^{\infty}\frac{(\nu+2k)\EulerGamma@{\nu+k}}{k!}\BesselJ{\nu+2k}@{z}</syntaxhighlight> || <math>\realpart@@{((\nu+2k)+k+1)} > 0, \realpart@@{(\nu+k)} > 0</math> || <syntaxhighlight lang=mathematica>((1)/(2)*z)^(nu) = sum(((nu + 2*k)*GAMMA(nu + k))/(factorial(k))*BesselJ(nu + 2*k, z), k = 0..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(Divide[1,2]*z)^\[Nu] == Sum[Divide[(\[Nu]+ 2*k)*Gamma[\[Nu]+ k],(k)!]*BesselJ[\[Nu]+ 2*k, z], {k, 0, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Successful || Skipped - Because timed out || Successful [Tested: 7]
 |-
-| [https://dlmf.nist.gov/10.23.E16 10.23.E16] || [[Item:Q3470|<math>\BesselY{0}@{z} = \frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\BesselJ{0}@{z}-\frac{4}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{\BesselJ{2k}@{z}}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselY{0}@{z} = \frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\BesselJ{0}@{z}-\frac{4}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{\BesselJ{2k}@{z}}{k}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{((2k)+k+1)} > 0, \realpart@@{((-0)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselY(0, z) = (2)/(Pi)*(ln((1)/(2)*z)+ gamma)*BesselJ(0, z)-(4)/(Pi)*sum((- 1)^(k)*(BesselJ(2*k, z))/(k), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselY[0, z] == Divide[2,Pi]*(Log[Divide[1,2]*z]+ EulerGamma)*BesselJ[0, z]-Divide[4,Pi]*Sum[(- 1)^(k)*Divide[BesselJ[2*k, z],k], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Successful || Successful [Tested: 7] || Successful [Tested: 7]
+| [https://dlmf.nist.gov/10.23.E16 10.23.E16] || <math qid="Q3470">\BesselY{0}@{z} = \frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\BesselJ{0}@{z}-\frac{4}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{\BesselJ{2k}@{z}}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselY{0}@{z} = \frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}+\EulerConstant\right)\BesselJ{0}@{z}-\frac{4}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{\BesselJ{2k}@{z}}{k}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{((2k)+k+1)} > 0, \realpart@@{((-0)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselY(0, z) = (2)/(Pi)*(ln((1)/(2)*z)+ gamma)*BesselJ(0, z)-(4)/(Pi)*sum((- 1)^(k)*(BesselJ(2*k, z))/(k), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselY[0, z] == Divide[2,Pi]*(Log[Divide[1,2]*z]+ EulerGamma)*BesselJ[0, z]-Divide[4,Pi]*Sum[(- 1)^(k)*Divide[BesselJ[2*k, z],k], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Successful || Successful [Tested: 7] || Successful [Tested: 7]
 |-
-| [https://dlmf.nist.gov/10.23.E17 10.23.E17] || [[Item:Q3471|<math>\BesselY{n}@{z} = -\frac{n!(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}\BesselJ{k}@{z}}{k!(n-k)}+\frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}-\digamma@{n+1}\right)\BesselJ{n}@{z}-\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)\BesselJ{n+2k}@{z}}{k(n+k)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselY{n}@{z} = -\frac{n!(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}\BesselJ{k}@{z}}{k!(n-k)}+\frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}-\digamma@{n+1}\right)\BesselJ{n}@{z}-\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)\BesselJ{n+2k}@{z}}{k(n+k)}</syntaxhighlight> || <math>\realpart@@{(n+k+1)} > 0, \realpart@@{(k+k+1)} > 0, \realpart@@{((n+2k)+k+1)} > 0, \realpart@@{((-n)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselY(n, z) = -(factorial(n)*((1)/(2)*z)^(- n))/(Pi)*sum((((1)/(2)*z)^(k)* BesselJ(k, z))/(factorial(k)*(n - k)), k = 0..n - 1)+(2)/(Pi)*(ln((1)/(2)*z)- Psi(n + 1))*BesselJ(n, z)-(2)/(Pi)*sum((- 1)^(k)*((n + 2*k)*BesselJ(n + 2*k, z))/(k*(n + k)), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselY[n, z] == -Divide[(n)!*(Divide[1,2]*z)^(- n),Pi]*Sum[Divide[(Divide[1,2]*z)^(k)* BesselJ[k, z],(k)!*(n - k)], {k, 0, n - 1}, GenerateConditions->None]+Divide[2,Pi]*(Log[Divide[1,2]*z]- PolyGamma[n + 1])*BesselJ[n, z]-Divide[2,Pi]*Sum[(- 1)^(k)*Divide[(n + 2*k)*BesselJ[n + 2*k, z],k*(n + k)], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Manual Skip! || <div class="toccolours mw-collapsible mw-collapsed">Failed [16 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.41373222494160333, 0.38808044477324316], Times[Complex[0.5513288954217921, -0.31830988618379064], DifferenceRoot[Function[{, }
+| [https://dlmf.nist.gov/10.23.E17 10.23.E17] || <math qid="Q3471">\BesselY{n}@{z} = -\frac{n!(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}\BesselJ{k}@{z}}{k!(n-k)}+\frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}-\digamma@{n+1}\right)\BesselJ{n}@{z}-\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)\BesselJ{n+2k}@{z}}{k(n+k)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselY{n}@{z} = -\frac{n!(\tfrac{1}{2}z)^{-n}}{\pi}\sum_{k=0}^{n-1}\frac{(\tfrac{1}{2}z)^{k}\BesselJ{k}@{z}}{k!(n-k)}+\frac{2}{\pi}\left(\ln@{\tfrac{1}{2}z}-\digamma@{n+1}\right)\BesselJ{n}@{z}-\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{k}\frac{(n+2k)\BesselJ{n+2k}@{z}}{k(n+k)}</syntaxhighlight> || <math>\realpart@@{(n+k+1)} > 0, \realpart@@{(k+k+1)} > 0, \realpart@@{((n+2k)+k+1)} > 0, \realpart@@{((-n)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>BesselY(n, z) = -(factorial(n)*((1)/(2)*z)^(- n))/(Pi)*sum((((1)/(2)*z)^(k)* BesselJ(k, z))/(factorial(k)*(n - k)), k = 0..n - 1)+(2)/(Pi)*(ln((1)/(2)*z)- Psi(n + 1))*BesselJ(n, z)-(2)/(Pi)*sum((- 1)^(k)*((n + 2*k)*BesselJ(n + 2*k, z))/(k*(n + k)), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BesselY[n, z] == -Divide[(n)!*(Divide[1,2]*z)^(- n),Pi]*Sum[Divide[(Divide[1,2]*z)^(k)* BesselJ[k, z],(k)!*(n - k)], {k, 0, n - 1}, GenerateConditions->None]+Divide[2,Pi]*(Log[Divide[1,2]*z]- PolyGamma[n + 1])*BesselJ[n, z]-Divide[2,Pi]*Sum[(- 1)^(k)*Divide[(n + 2*k)*BesselJ[n + 2*k, z],k*(n + k)], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Manual Skip! || <div class="toccolours mw-collapsible mw-collapsed">Failed [16 / 21]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.41373222494160333, 0.38808044477324316], Times[Complex[0.5513288954217921, -0.31830988618379064], DifferenceRoot[Function[{, }
 Test Values: {Equal[Plus[Times[Plus[Times[-1, ], 1], Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2], []], Times[Plus[4, Times[12, ], Times[12, Power[, 2]], Times[4, Power[, 3]], Times[-4, 1], Times[-8, , 1], Times[-4, Power[, 2], 1], Times[, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]], Times[-1, 1, Power[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], 2]]], [Plus[1, ]]], Times[4, Plus[1, ], Plus[-5, Times[-6, ], Times[-2, Power[, 2]], Times[3, 1], Times[2, , 1]], [Plus[2, ]]], Times[-4, Plus[1, ], Plus[2, ], Plus[-2, Times[-1, ], 1], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Power[1, -1], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Plus[Times[Power[1, -1], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]<syntaxhighlight lang=mathematica>Result: Plus[Complex[-0.6198631863998064, 5.383408526303685], Times[Complex[0.0, -15.278874536821952], DifferenceRoot[Function[{, }
 Test Values: {Equal[Plus[Times[-1, Power[-1, Rational[1, 3]], Plus[-3, ], []], Times[Plus[-8, Times[-3, Power[-1, Rational[1, 3]]], Times[-12, ], Times[Power[-1, Rational[1, 3]], ], Times[4, Power[, 3]]], [Plus[1, ]]], Times[-8, Plus[1, ], Plus[-2, Power[, 2]], [Plus[2, ]]], Times[4, Plus[-1, ], Plus[1, ], Plus[2, ], [Plus[3, ]]]], 0], Equal[[0], 0], Equal[[1], Times[Rational[1, 3], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]], Equal[[2], Plus[Times[Rational[1, 3], BesselJ[0, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Times[Rational[1, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]], BesselJ[1, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]]]}]][3.0]]], {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
 |}
 </div>

10.23: Difference between revisions

Latest revision as of 11:24, 28 June 2021

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