10.24: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/10.24.E1 10.24.E1] || [[Item:Q3476|<math>x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((x)^(2)+ (nu)^(2))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+((x)^(2)+ \[Nu]^(2))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.948557159+2.125000000*I
| [https://dlmf.nist.gov/10.24.E1 10.24.E1] || <math qid="Q3476">x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((x)^(2)+ (nu)^(2))*w = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+((x)^(2)+ \[Nu]^(2))*w == 0</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.948557159+2.125000000*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2165063513+1.125000001*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2165063513+1.125000001*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 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[1.9485571585149875, 2.125]
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 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[1.9485571585149875, 2.125]
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Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.24#Ex1 10.24#Ex1] || [[Item:Q3477|<math>\BesselJimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselJ{i\nu}@{x}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselJimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselJ{i\nu}@{x}}</syntaxhighlight> || <math>\realpart@@{((\iunit \nu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselJ(I*nu, x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselJ[I*\[Nu], x]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 30]
| [https://dlmf.nist.gov/10.24#Ex1 10.24#Ex1] || <math qid="Q3477">\BesselJimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselJ{i\nu}@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselJimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselJ{i\nu}@{x}}</syntaxhighlight> || <math>\realpart@@{((\iunit \nu)+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselJ(I*nu, x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselJ[I*\[Nu], x]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 30]
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| [https://dlmf.nist.gov/10.24#Ex2 10.24#Ex2] || [[Item:Q3478|<math>\BesselYimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselY{i\nu}@{x}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselYimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselY{i\nu}@{x}}</syntaxhighlight> || <math>\realpart@@{((\iunit \nu)+k+1)} > 0, \realpart@@{((-(\iunit \nu))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselY(I*nu, x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselY[I*\[Nu], x]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 30]
| [https://dlmf.nist.gov/10.24#Ex2 10.24#Ex2] || <math qid="Q3478">\BesselYimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselY{i\nu}@{x}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselYimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselY{i\nu}@{x}}</syntaxhighlight> || <math>\realpart@@{((\iunit \nu)+k+1)} > 0, \realpart@@{((-(\iunit \nu))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselY(I*nu, x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselY[I*\[Nu], x]]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 30]
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| [https://dlmf.nist.gov/10.24.E3 10.24.E3] || [[Item:Q3479|<math>\EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}</syntaxhighlight> || <math>\realpart@@{(1+\iunit \nu)} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(1 + I*nu) = ((Pi*nu)/(sinh(Pi*nu)))^((1)/(2))* exp(I*gamma[nu])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1 + I*\[Nu]] == (Divide[Pi*\[Nu],Sinh[Pi*\[Nu]]])^(Divide[1,2])* Exp[I*Subscript[\[Gamma], \[Nu]]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .131682196e-1-.6479738907*I
| [https://dlmf.nist.gov/10.24.E3 10.24.E3] || <math qid="Q3479">\EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}</syntaxhighlight> || <math>\realpart@@{(1+\iunit \nu)} > 0</math> || <syntaxhighlight lang=mathematica>GAMMA(1 + I*nu) = ((Pi*nu)/(sinh(Pi*nu)))^((1)/(2))* exp(I*gamma[nu])</syntaxhighlight> || <syntaxhighlight lang=mathematica>Gamma[1 + I*\[Nu]] == (Divide[Pi*\[Nu],Sinh[Pi*\[Nu]]])^(Divide[1,2])* Exp[I*Subscript[\[Gamma], \[Nu]]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .131682196e-1-.6479738907*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2393622021-.2867640040*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .2393622021-.2867640040*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = -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[0.013168219691258531, -0.6479738909120968]
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = -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[0.013168219691258531, -0.6479738909120968]
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Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, ν], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, ν], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.24#Ex3 10.24#Ex3] || [[Item:Q3480|<math>\BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sech((1/2)*Pi*(- nu))*Re(BesselJ(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sech[1/2 Pi - \[Nu]] Re[BesselJ[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1765981285-.1547836875*I
| [https://dlmf.nist.gov/10.24#Ex3 10.24#Ex3] || <math qid="Q3480">\BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sech((1/2)*Pi*(- nu))*Re(BesselJ(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sech[1/2 Pi - \[Nu]] Re[BesselJ[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1765981285-.1547836875*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.059084556+.9282601935*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.059084556+.9282601935*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.6353785354467336, 0.04153700144653363]
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.6353785354467336, 0.04153700144653363]
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Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.24#Ex4 10.24#Ex4] || [[Item:Q3481|<math>\BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x}</syntaxhighlight> || <math>\realpart@@{((\iunit (-\nu))+k+1)} > 0, \realpart@@{((\iunit \nu)+k+1)} > 0, \realpart@@{((-(\iunit (-\nu)))+k+1)} > 0, \realpart@@{((-(\iunit \nu))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sech((1/2)*Pi*(- nu))*Re(BesselY(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sech[1/2 Pi - \[Nu]] Re[BesselY[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.6730010946+.5898680353*I
| [https://dlmf.nist.gov/10.24#Ex4 10.24#Ex4] || <math qid="Q3481">\BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x}</syntaxhighlight> || <math>\realpart@@{((\iunit (-\nu))+k+1)} > 0, \realpart@@{((\iunit \nu)+k+1)} > 0, \realpart@@{((-(\iunit (-\nu)))+k+1)} > 0, \realpart@@{((-(\iunit \nu))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sech((1/2)*Pi*(- nu))*Re(BesselY(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sech[1/2 Pi - \[Nu]] Re[BesselY[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.6730010946+.5898680353*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.1980888923+.1736197856*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.1980888923+.1736197856*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.16541121369118172, 0.7534126929509344]
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.16541121369118172, 0.7534126929509344]
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Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.24.E5 10.24.E5] || [[Item:Q3482|<math>\Wronskian@{\BesselJimag{\nu}@{x},\BesselYimag{\nu}@{x}} = 2/(\pi x)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\BesselJimag{\nu}@{x},\BesselYimag{\nu}@{x}} = 2/(\pi x)</syntaxhighlight> || <math>\realpart@@{((\iunit \nu)+k+1)} > 0, \realpart@@{((-(\iunit \nu))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)))*diff(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)), x)-diff(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)), x)*(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))) = 2/(Pi*x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]], Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]}, x] == 2/(Pi*x)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.3214564733-.7786157192*I
| [https://dlmf.nist.gov/10.24.E5 10.24.E5] || <math qid="Q3482">\Wronskian@{\BesselJimag{\nu}@{x},\BesselYimag{\nu}@{x}} = 2/(\pi x)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Wronskian@{\BesselJimag{\nu}@{x},\BesselYimag{\nu}@{x}} = 2/(\pi x)</syntaxhighlight> || <math>\realpart@@{((\iunit \nu)+k+1)} > 0, \realpart@@{((-(\iunit \nu))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)))*diff(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)), x)-diff(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)), x)*(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))) = 2/(Pi*x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Wronskian[{Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]], Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]}, x] == 2/(Pi*x)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.3214564733-.7786157192*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.6431025084-4.765445687*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.6431025084-4.765445687*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.4244131815783876, Times[Complex[0.017184424665049866, -0.12995814793225188], Plus[Times[Complex[5.94457417937745, -0.08806734388290616], Derivative[1][Re][Complex[0.5424102683642863, 1.3820413572565333]]], Times[Complex[0.04670634387761448, 2.0064149502593187], Derivative[1][Re][Complex[1.5013396639532606, -0.5145465005058608]]]]]]
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[-0.4244131815783876, Times[Complex[0.017184424665049866, -0.12995814793225188], Plus[Times[Complex[5.94457417937745, -0.08806734388290616], Derivative[1][Re][Complex[0.5424102683642863, 1.3820413572565333]]], Times[Complex[0.04670634387761448, 2.0064149502593187], Derivative[1][Re][Complex[1.5013396639532606, -0.5145465005058608]]]]]]
Line 48: Line 48:
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/10.24.E9 10.24.E9] || [[Item:Q3487|<math>\BesselYimag{0}@{x} = \BesselY{0}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselYimag{0}@{x} = \BesselY{0}@{x}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{((-0)+k+1)} > 0, \realpart@@{((\iunit 0)+k+1)} > 0, \realpart@@{((-(\iunit 0))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sech((1/2)*Pi*(0))*Re(BesselY(I*(0), x)) = BesselY(0, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sech[1/2 Pi 0] Re[BesselY[I 0, x]] == BesselY[0, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| [https://dlmf.nist.gov/10.24.E9 10.24.E9] || <math qid="Q3487">\BesselYimag{0}@{x} = \BesselY{0}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BesselYimag{0}@{x} = \BesselY{0}@{x}</syntaxhighlight> || <math>\realpart@@{(0+k+1)} > 0, \realpart@@{((-0)+k+1)} > 0, \realpart@@{((\iunit 0)+k+1)} > 0, \realpart@@{((-(\iunit 0))+k+1)} > 0</math> || <syntaxhighlight lang=mathematica>sech((1/2)*Pi*(0))*Re(BesselY(I*(0), x)) = BesselY(0, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sech[1/2 Pi 0] Re[BesselY[I 0, x]] == BesselY[0, x]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|}
|}
</div>
</div>

Latest revision as of 11:24, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple
Symbolic
Mathematica
Numeric
Maple
Numeric
Mathematica
10.24.E1 x 2 d 2 w d x 2 + x d w d x + ( x 2 + ν 2 ) w = 0 superscript 𝑥 2 derivative 𝑤 𝑥 2 𝑥 derivative 𝑤 𝑥 superscript 𝑥 2 superscript 𝜈 2 𝑤 0 {\displaystyle{\displaystyle x^{2}\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}+% x\frac{\mathrm{d}w}{\mathrm{d}x}+(x^{2}+\nu^{2})w=0}}
x^{2}\deriv[2]{w}{x}+x\deriv{w}{x}+(x^{2}+\nu^{2})w = 0

(x)^(2)* diff(w, [x$(2)])+ x*diff(w, x)+((x)^(2)+ (nu)^(2))*w = 0
(x)^(2)* D[w, {x, 2}]+ x*D[w, x]+((x)^(2)+ \[Nu]^(2))*w == 0
Failure Failure
Failed [300 / 300]
Result: 1.948557159+2.125000000*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 3/2}

Result: .2165063513+1.125000001*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, w = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.9485571585149875, 2.125]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[1.948557158514987, 0.12499999999999989]
Test Values: {Rule[w, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24#Ex1 J ~ ν ( x ) = sech ( 1 2 π ν ) ( J i ν ( x ) ) Bessel-J-imaginary-order 𝜈 𝑥 1 2 𝜋 𝜈 Bessel-J 𝑖 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{J}_{\nu}\left(x\right)=\operatorname{% sech}\left(\tfrac{1}{2}\pi\nu\right)\Re\left(J_{i\nu}\left(x\right)\right)}}
\BesselJimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselJ{i\nu}@{x}}
( ( i ν ) + k + 1 ) > 0 imaginary-unit 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0}}
sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselJ(I*nu, x))
Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselJ[I*\[Nu], x]]
Successful Successful - Successful [Tested: 30]
10.24#Ex2 Y ~ ν ( x ) = sech ( 1 2 π ν ) ( Y i ν ( x ) ) Bessel-Y-Weber-imaginary-order 𝜈 𝑥 1 2 𝜋 𝜈 Bessel-Y-Weber 𝑖 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{Y}_{\nu}\left(x\right)=\operatorname{% sech}\left(\tfrac{1}{2}\pi\nu\right)\Re\left(Y_{i\nu}\left(x\right)\right)}}
\BesselYimag{\nu}@{x} = \sech@{\tfrac{1}{2}\pi\nu}\realpart@{\BesselY{i\nu}@{x}}
( ( i ν ) + k + 1 ) > 0 , ( ( - ( i ν ) ) + k + 1 ) > 0 formulae-sequence imaginary-unit 𝜈 𝑘 1 0 imaginary-unit 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0,\Re((-(\mathrm{i}\nu))+% k+1)>0}}
sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)) = sech((1)/(2)*Pi*nu)*Re(BesselY(I*nu, x))
Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]] == Sech[Divide[1,2]*Pi*\[Nu]]*Re[BesselY[I*\[Nu], x]]
Successful Successful - Successful [Tested: 30]
10.24.E3 Γ ( 1 + i ν ) = ( π ν sinh ( π ν ) ) 1 2 e i γ ν Euler-Gamma 1 𝑖 𝜈 superscript 𝜋 𝜈 𝜋 𝜈 1 2 superscript 𝑒 𝑖 subscript 𝛾 𝜈 {\displaystyle{\displaystyle\Gamma\left(1+i\nu\right)=\left(\frac{\pi\nu}{% \sinh\left(\pi\nu\right)}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}}}
\EulerGamma@{1+i\nu} = \left(\frac{\pi\nu}{\sinh@{\pi\nu}}\right)^{\frac{1}{2}}e^{i\gamma_{\nu}}
( 1 + i ν ) > 0 1 imaginary-unit 𝜈 0 {\displaystyle{\displaystyle\Re(1+\mathrm{i}\nu)>0}}
GAMMA(1 + I*nu) = ((Pi*nu)/(sinh(Pi*nu)))^((1)/(2))* exp(I*gamma[nu])
Gamma[1 + I*\[Nu]] == (Divide[Pi*\[Nu],Sinh[Pi*\[Nu]]])^(Divide[1,2])* Exp[I*Subscript[\[Gamma], \[Nu]]]
Failure Failure
Failed [300 / 300]
Result: .131682196e-1-.6479738907*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = 1/2*3^(1/2)+1/2*I}

Result: .2393622021-.2867640040*I
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, nu = 1/2*3^(1/2)+1/2*I, gamma[nu] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.013168219691258531, -0.6479738909120968]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, ν], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.23936220222535412, -0.28676400411697583]
Test Values: {Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[γ, ν], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24#Ex3 J ~ - ν ( x ) = J ~ ν ( x ) Bessel-J-imaginary-order 𝜈 𝑥 Bessel-J-imaginary-order 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{J}_{-\nu}\left(x\right)=\widetilde{J}_{% \nu}\left(x\right)}}
\BesselJimag{-\nu}@{x} = \BesselJimag{\nu}@{x}

sech((1/2)*Pi*(- nu))*Re(BesselJ(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x))
Sech[1/2 Pi - \[Nu]] Re[BesselJ[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]]
Failure Failure
Failed [12 / 30]
Result: .1765981285-.1547836875*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}

Result: -1.059084556+.9282601935*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-0.6353785354467336, 0.04153700144653363]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[0.2910880978413849, 0.681683596996288]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24#Ex4 Y ~ - ν ( x ) = Y ~ ν ( x ) Bessel-Y-Weber-imaginary-order 𝜈 𝑥 Bessel-Y-Weber-imaginary-order 𝜈 𝑥 {\displaystyle{\displaystyle\widetilde{Y}_{-\nu}\left(x\right)=\widetilde{Y}_{% \nu}\left(x\right)}}
\BesselYimag{-\nu}@{x} = \BesselYimag{\nu}@{x}
( ( i ( - ν ) ) + k + 1 ) > 0 , ( ( i ν ) + k + 1 ) > 0 , ( ( - ( i ( - ν ) ) ) + k + 1 ) > 0 , ( ( - ( i ν ) ) + k + 1 ) > 0 formulae-sequence imaginary-unit 𝜈 𝑘 1 0 formulae-sequence imaginary-unit 𝜈 𝑘 1 0 formulae-sequence imaginary-unit 𝜈 𝑘 1 0 imaginary-unit 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}(-\nu))+k+1)>0,\Re((\mathrm{i}\nu)+% k+1)>0,\Re((-(\mathrm{i}(-\nu)))+k+1)>0,\Re((-(\mathrm{i}\nu))+k+1)>0}}
sech((1/2)*Pi*(- nu))*Re(BesselY(I*(- nu), x)) = sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))
Sech[1/2 Pi - \[Nu]] Re[BesselY[I - \[Nu], x]] == Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]
Failure Failure
Failed [12 / 30]
Result: -.6730010946+.5898680353*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}

Result: -.1980888923+.1736197856*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.16541121369118172, 0.7534126929509344]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-0.3242468905843751, -0.9796849117084342]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24.E5 𝒲 { J ~ ν ( x ) , Y ~ ν ( x ) } = 2 / ( π x ) Wronskian Bessel-J-imaginary-order 𝜈 𝑥 Bessel-Y-Weber-imaginary-order 𝜈 𝑥 2 𝜋 𝑥 {\displaystyle{\displaystyle\mathscr{W}\left\{\widetilde{J}_{\nu}\left(x\right% ),\widetilde{Y}_{\nu}\left(x\right)\right\}=2/(\pi x)}}
\Wronskian@{\BesselJimag{\nu}@{x},\BesselYimag{\nu}@{x}} = 2/(\pi x)
( ( i ν ) + k + 1 ) > 0 , ( ( - ( i ν ) ) + k + 1 ) > 0 formulae-sequence imaginary-unit 𝜈 𝑘 1 0 imaginary-unit 𝜈 𝑘 1 0 {\displaystyle{\displaystyle\Re((\mathrm{i}\nu)+k+1)>0,\Re((-(\mathrm{i}\nu))+% k+1)>0}}
(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)))*diff(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x)), x)-diff(sech((1/2)*Pi*(nu))*Re(BesselJ(I*(nu), x)), x)*(sech((1/2)*Pi*(nu))*Re(BesselY(I*(nu), x))) = 2/(Pi*x)
Wronskian[{Sech[1/2 Pi \[Nu]] Re[BesselJ[I \[Nu], x]], Sech[1/2 Pi \[Nu]] Re[BesselY[I \[Nu], x]]}, x] == 2/(Pi*x)
Failure Failure
Failed [12 / 30]
Result: -.3214564733-.7786157192*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 3/2}

Result: -.6431025084-4.765445687*I
Test Values: {nu = 1/2*3^(1/2)+1/2*I, x = 1/2}

... skip entries to safe data
Failed [30 / 30]
Result: Plus[-0.4244131815783876, Times[Complex[0.017184424665049866, -0.12995814793225188], Plus[Times[Complex[5.94457417937745, -0.08806734388290616], Derivative[1][Re][Complex[0.5424102683642863, 1.3820413572565333]]], Times[Complex[0.04670634387761448, 2.0064149502593187], Derivative[1][Re][Complex[1.5013396639532606, -0.5145465005058608]]]]]]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Plus[-0.4244131815783876, Times[Complex[-0.5062208144169521, 0.3689208146583662], Plus[Times[Complex[1.2690034139339206, -1.428145592425075], Derivative[1][Re][Complex[-0.5230512553281585, -0.7250724679588263]]], Times[Complex[0.9907135967899046, 0.5862869255257461], Derivative[1][Re][Complex[0.9118063408652576, -0.381897212811936]]]]]]
Test Values: {Rule[x, 1.5], Rule[ν, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
10.24.E9 Y ~ 0 ( x ) = Y 0 ( x ) Bessel-Y-Weber-imaginary-order 0 𝑥 Bessel-Y-Weber 0 𝑥 {\displaystyle{\displaystyle\widetilde{Y}_{0}\left(x\right)=Y_{0}\left(x\right% )}}
\BesselYimag{0}@{x} = \BesselY{0}@{x}
( 0 + k + 1 ) > 0 , ( ( - 0 ) + k + 1 ) > 0 , ( ( i 0 ) + k + 1 ) > 0 , ( ( - ( i 0 ) ) + k + 1 ) > 0 formulae-sequence 0 𝑘 1 0 formulae-sequence 0 𝑘 1 0 formulae-sequence imaginary-unit 0 𝑘 1 0 imaginary-unit 0 𝑘 1 0 {\displaystyle{\displaystyle\Re(0+k+1)>0,\Re((-0)+k+1)>0,\Re((\mathrm{i}0)+k+1% )>0,\Re((-(\mathrm{i}0))+k+1)>0}}
sech((1/2)*Pi*(0))*Re(BesselY(I*(0), x)) = BesselY(0, x)
Sech[1/2 Pi 0] Re[BesselY[I 0, x]] == BesselY[0, x]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]