19.34: 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/19.34.E1 19.34.E1] || [[Item:Q6711|<math>ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos@@{\theta})^{-1/2}\cos@@{\theta}\diff{\theta} = 2ab\int_{-1}^{1}\frac{t\diff{t}}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos@@{\theta})^{-1/2}\cos@@{\theta}\diff{\theta} = 2ab\int_{-1}^{1}\frac{t\diff{t}}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>a*b*int(((h)^(2)+ (a)^(2)+ (b)^(2)- 2*a*b*cos(theta))^(- 1/2)* cos(theta), theta = 0..2*Pi) = 2*a*b*int((t)/(sqrt((1 + t)*(1 - t)*(a[3]- 2*a*b*t))), t = - 1..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>a*b*Integrate[((h)^(2)+ (a)^(2)+ (b)^(2)- 2*a*b*Cos[\[Theta]])^(- 1/2)* Cos[\[Theta]], {\[Theta], 0, 2*Pi}, GenerateConditions->None] == 2*a*b*Integrate[Divide[t,Sqrt[(1 + t)*(1 - t)*(Subscript[a, 3]- 2*a*b*t)]], {t, - 1, 1}, GenerateConditions->None]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
| [https://dlmf.nist.gov/19.34.E1 19.34.E1] || <math qid="Q6711">ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos@@{\theta})^{-1/2}\cos@@{\theta}\diff{\theta} = 2ab\int_{-1}^{1}\frac{t\diff{t}}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos@@{\theta})^{-1/2}\cos@@{\theta}\diff{\theta} = 2ab\int_{-1}^{1}\frac{t\diff{t}}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>a*b*int(((h)^(2)+ (a)^(2)+ (b)^(2)- 2*a*b*cos(theta))^(- 1/2)* cos(theta), theta = 0..2*Pi) = 2*a*b*int((t)/(sqrt((1 + t)*(1 - t)*(a[3]- 2*a*b*t))), t = - 1..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>a*b*Integrate[((h)^(2)+ (a)^(2)+ (b)^(2)- 2*a*b*Cos[\[Theta]])^(- 1/2)* Cos[\[Theta]], {\[Theta], 0, 2*Pi}, GenerateConditions->None] == 2*a*b*Integrate[Divide[t,Sqrt[(1 + t)*(1 - t)*(Subscript[a, 3]- 2*a*b*t)]], {t, - 1, 1}, GenerateConditions->None]</syntaxhighlight> || Aborted || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| [https://dlmf.nist.gov/19.34.E1 19.34.E1] || [[Item:Q6711|<math>2ab\int_{-1}^{1}\frac{t\diff{t}}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}} = 2abI(\mathbf{e}_{5})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2ab\int_{-1}^{1}\frac{t\diff{t}}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}} = 2abI(\mathbf{e}_{5})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>2*a*b*int((t)/(sqrt((1 + t)*(1 - t)*(a[3]- 2*a*b*t))), t = - 1..1) = 2*abI(e[5])</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*a*b*Integrate[Divide[t,Sqrt[(1 + t)*(1 - t)*(Subscript[a, 3]- 2*a*b*t)]], {t, - 1, 1}, GenerateConditions->None] == 2*abI[Subscript[e, 5]]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -3.959693187-6.593729744*I
| [https://dlmf.nist.gov/19.34.E1 19.34.E1] || <math qid="Q6711">2ab\int_{-1}^{1}\frac{t\diff{t}}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}} = 2abI(\mathbf{e}_{5})</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>2ab\int_{-1}^{1}\frac{t\diff{t}}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}} = 2abI(\mathbf{e}_{5})</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>2*a*b*int((t)/(sqrt((1 + t)*(1 - t)*(a[3]- 2*a*b*t))), t = - 1..1) = 2*abI(e[5])</syntaxhighlight> || <syntaxhighlight lang=mathematica>2*a*b*Integrate[Divide[t,Sqrt[(1 + t)*(1 - t)*(Subscript[a, 3]- 2*a*b*t)]], {t, - 1, 1}, GenerateConditions->None] == 2*abI[Subscript[e, 5]]</syntaxhighlight> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -3.959693187-6.593729744*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, a[3] = 1/2*3^(1/2)+1/2*I, e[5] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.187421133-4.946615428*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, a[3] = 1/2*3^(1/2)+1/2*I, e[5] = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 2.187421133-4.946615428*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, a[3] = 1/2*3^(1/2)+1/2*I, e[5] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
Test Values: {I = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, a[3] = 1/2*3^(1/2)+1/2*I, e[5] = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br>... skip entries to safe data</div></div> || Skipped - Because timed out
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| [https://dlmf.nist.gov/19.34#Ex1 19.34#Ex1] || [[Item:Q6712|<math>a_{3} = h^{2}+a^{2}+b^{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{3} = h^{2}+a^{2}+b^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[3] = (h)^(2)+ (a)^(2)+ (b)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 3] == (h)^(2)+ (a)^(2)+ (b)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/19.34#Ex1 19.34#Ex1] || <math qid="Q6712">a_{3} = h^{2}+a^{2}+b^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{3} = h^{2}+a^{2}+b^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[3] = (h)^(2)+ (a)^(2)+ (b)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 3] == (h)^(2)+ (a)^(2)+ (b)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/19.34#Ex2 19.34#Ex2] || [[Item:Q6713|<math>a_{5} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{5} = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[5] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 5] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/19.34#Ex2 19.34#Ex2] || <math qid="Q6713">a_{5} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{5} = 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[5] = 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, 5] == 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
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| [https://dlmf.nist.gov/19.34#Ex3 19.34#Ex3] || [[Item:Q6714|<math>b_{5} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>b_{5} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">b[5] = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[b, 5] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/19.34#Ex3 19.34#Ex3] || <math qid="Q6714">b_{5} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>b_{5} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">b[5] = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[b, 5] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/19.34.E3 19.34.E3] || [[Item:Q6715|<math>2abI(\mathbf{e}_{5}) = a_{3}I(\boldsymbol{{0}})-I(\mathbf{e}_{3})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>2abI(\mathbf{e}_{5}) = a_{3}I(\boldsymbol{{0}})-I(\mathbf{e}_{3})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">2*abI(e[5]) = a[3]*I(0)- I(e[3])</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">2*abI[Subscript[e, 5]] == Subscript[a, 3]*I[0]- I[Subscript[e, 3]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/19.34.E3 19.34.E3] || <math qid="Q6715">2abI(\mathbf{e}_{5}) = a_{3}I(\boldsymbol{{0}})-I(\mathbf{e}_{3})</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>2abI(\mathbf{e}_{5}) = a_{3}I(\boldsymbol{{0}})-I(\mathbf{e}_{3})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">2*abI(e[5]) = a[3]*I(0)- I(e[3])</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">2*abI[Subscript[e, 5]] == Subscript[a, 3]*I[0]- I[Subscript[e, 3]]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/19.34.E4 19.34.E4] || [[Item:Q6716|<math>r_{+}^{2} = a_{3}+ 2ab</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>r_{+}^{2} = a_{3}+ 2ab</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(r[+])^(2) = a[3]+ 2*a*b</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Subscript[r, +])^(2) == Subscript[a, 3]+ 2*a*b</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/19.34.E4 19.34.E4] || <math qid="Q6716">r_{+}^{2} = a_{3}+ 2ab</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>r_{+}^{2} = a_{3}+ 2ab</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(r[+])^(2) = a[3]+ 2*a*b</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Subscript[r, +])^(2) == Subscript[a, 3]+ 2*a*b</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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Latest revision as of 11:55, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.34.E1 a ⁢ b ⁢ ∫ 0 2 ⁢ π ( h 2 + a 2 + b 2 - 2 ⁢ a ⁢ b ⁢ cos ⁡ θ ) - 1 / 2 ⁢ cos ⁡ θ ⁢ d θ = 2 ⁢ a ⁢ b ⁢ ∫ - 1 1 t ⁢ d t ( 1 + t ) ⁢ ( 1 - t ) ⁢ ( a 3 - 2 ⁢ a ⁢ b ⁢ t ) 𝑎 𝑏 superscript subscript 0 2 𝜋 superscript superscript ℎ 2 superscript 𝑎 2 superscript 𝑏 2 2 𝑎 𝑏 𝜃 1 2 𝜃 𝜃 2 𝑎 𝑏 superscript subscript 1 1 𝑡 𝑡 1 𝑡 1 𝑡 subscript 𝑎 3 2 𝑎 𝑏 𝑡 {\displaystyle{\displaystyle ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos\theta% )^{-1/2}\cos\theta\mathrm{d}\theta=2ab\int_{-1}^{1}\frac{t\mathrm{d}t}{\sqrt{(% 1+t)(1-t)(a_{3}-2abt)}}}}
ab\int_{0}^{2\pi}(h^{2}+a^{2}+b^{2}-2ab\cos@@{\theta})^{-1/2}\cos@@{\theta}\diff{\theta} = 2ab\int_{-1}^{1}\frac{t\diff{t}}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}}		 

a*b*int(((h)^(2)+ (a)^(2)+ (b)^(2)- 2*a*b*cos(theta))^(- 1/2)* cos(theta), theta = 0..2*Pi) = 2*a*b*int((t)/(sqrt((1 + t)*(1 - t)*(a[3]- 2*a*b*t))), t = - 1..1)

a*b*Integrate[((h)^(2)+ (a)^(2)+ (b)^(2)- 2*a*b*Cos[\[Theta]])^(- 1/2)* Cos[\[Theta]], {\[Theta], 0, 2*Pi}, GenerateConditions->None] == 2*a*b*Integrate[Divide[t,Sqrt[(1 + t)*(1 - t)*(Subscript[a, 3]- 2*a*b*t)]], {t, - 1, 1}, GenerateConditions->None]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.34.E1 2 ⁢ a ⁢ b ⁢ ∫ - 1 1 t ⁢ d t ( 1 + t ) ⁢ ( 1 - t ) ⁢ ( a 3 - 2 ⁢ a ⁢ b ⁢ t ) = 2 ⁢ a ⁢ b ⁢ I ⁢ ( 𝐞 5 ) 2 𝑎 𝑏 superscript subscript 1 1 𝑡 𝑡 1 𝑡 1 𝑡 subscript 𝑎 3 2 𝑎 𝑏 𝑡 2 𝑎 𝑏 𝐼 subscript 𝐞 5 {\displaystyle{\displaystyle 2ab\int_{-1}^{1}\frac{t\mathrm{d}t}{\sqrt{(1+t)(1% -t)(a_{3}-2abt)}}=2abI(\mathbf{e}_{5})}}
2ab\int_{-1}^{1}\frac{t\diff{t}}{\sqrt{(1+t)(1-t)(a_{3}-2abt)}} = 2abI(\mathbf{e}_{5})		 

2*a*b*int((t)/(sqrt((1 + t)*(1 - t)*(a[3]- 2*a*b*t))), t = - 1..1) = 2*abI(e[5])

2*a*b*Integrate[Divide[t,Sqrt[(1 + t)*(1 - t)*(Subscript[a, 3]- 2*a*b*t)]], {t, - 1, 1}, GenerateConditions->None] == 2*abI[Subscript[e, 5]]
Failure Aborted
Failed [300 / 300]
Result: -3.959693187-6.593729744*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, a[3] = 1/2*3^(1/2)+1/2*I, e[5] = 1/2*3^(1/2)+1/2*I}


Result: 2.187421133-4.946615428*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, a = -3/2, b = -3/2, a[3] = 1/2*3^(1/2)+1/2*I, e[5] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
19.34#Ex1 a 3 = h 2 + a 2 + b 2 subscript 𝑎 3 superscript ℎ 2 superscript 𝑎 2 superscript 𝑏 2 {\displaystyle{\displaystyle a_{3}=h^{2}+a^{2}+b^{2}}}
a_{3} = h^{2}+a^{2}+b^{2}		 

a[3] = (h)^(2)+ (a)^(2)+ (b)^(2)
 
Subscript[a, 3] == (h)^(2)+ (a)^(2)+ (b)^(2)
 Skipped - no semantic math Skipped - no semantic math - -
19.34#Ex2 a 5 = 0 subscript 𝑎 5 0 {\displaystyle{\displaystyle a_{5}=0}}
a_{5} = 0		 

a[5] = 0
 
Subscript[a, 5] == 0
 Skipped - no semantic math Skipped - no semantic math - -
19.34#Ex3 b 5 = 1 subscript 𝑏 5 1 {\displaystyle{\displaystyle b_{5}=1}}
b_{5} = 1		 

b[5] = 1
 
Subscript[b, 5] == 1
 Skipped - no semantic math Skipped - no semantic math - -
19.34.E3 2 ⁢ a ⁢ b ⁢ I ⁢ ( 𝐞 5 ) = a 3 ⁢ I ⁢ ( 𝟎 ) - I ⁢ ( 𝐞 3 ) 2 𝑎 𝑏 𝐼 subscript 𝐞 5 subscript 𝑎 3 𝐼 0 𝐼 subscript 𝐞 3 {\displaystyle{\displaystyle 2abI(\mathbf{e}_{5})=a_{3}I(\boldsymbol{{0}})-I(% \mathbf{e}_{3})}}
2abI(\mathbf{e}_{5}) = a_{3}I(\boldsymbol{{0}})-I(\mathbf{e}_{3})		 

2*abI(e[5]) = a[3]*I(0)- I(e[3])
 
2*abI[Subscript[e, 5]] == Subscript[a, 3]*I[0]- I[Subscript[e, 3]]
 Skipped - no semantic math Skipped - no semantic math - -
19.34.E4 r + 2 = a 3 + 2 ⁢ a ⁢ b superscript subscript 𝑟 2 subscript 𝑎 3 2 𝑎 𝑏 {\displaystyle{\displaystyle r_{+}^{2}=a_{3}+2ab}}
r_{+}^{2} = a_{3}+ 2ab		 

(r[+])^(2) = a[3]+ 2*a*b
 
(Subscript[r, +])^(2) == Subscript[a, 3]+ 2*a*b
 Skipped - no semantic math Skipped - no semantic math - -
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