View source for 29.6

{{DISPLAYTITLE:Lamé Functions - 29.6 Fourier Series}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/29.6.E5 29.6.E5] || <math qid="Q8631">\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{\infty}A_{2p}^{2} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tfrac{1}{2}A_{0}^{2}+\sum_{p=1}^{\infty}A_{2p}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1)/(2)*(A[0])^(2)+ sum((A[2*p])^(2), p = 1..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[1,2]*(Subscript[A, 0])^(2)+ Sum[(Subscript[A, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E6 29.6.E6] || <math qid="Q8632">\tfrac{1}{2}A_{0}+\sum_{p=1}^{\infty}A_{2p} > 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tfrac{1}{2}A_{0}+\sum_{p=1}^{\infty}A_{2p} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1)/(2)*A[0]+ sum(A[2*p], p = 1..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[1,2]*Subscript[A, 0]+ Sum[Subscript[A, 2*p], {p, 1, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E7 29.6.E7] || <math qid="Q8633">\lim_{p\to\infty}\frac{A_{2p+2}}{A_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{A_{2p+2}}{A_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n, \nu = 2n, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((A[2*p + 2])/(A[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[A, 2*p + 2],Subscript[A, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6#Ex2 29.6#Ex2] || <math qid="Q8638">\beta_{p} = 4p^{2}(2-k^{2})</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\beta_{p} = 4p^{2}(2-k^{2})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">beta[p] = 4*(p)^(2)*(2 - (k)^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Beta], p] == 4*(p)^(2)*(2 - (k)^(2))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6#Ex3 29.6#Ex3] || <math qid="Q8639">\gamma_{p} = \tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p + 1)*(nu + 2*p)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p + 1)*(\[Nu]+ 2*p)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E12 29.6.E12] || <math qid="Q8640">\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{\infty}C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{\infty}C_{2p}C_{2p+2} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(1-\tfrac{1}{2}k^{2}\right)\left(\tfrac{1}{2}C_{0}^{2}+\sum_{p=1}^{\infty}C_{2p}^{2}\right)-\tfrac{1}{2}k^{2}\sum_{p=0}^{\infty}C_{2p}C_{2p+2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -(1)/(2)*(k)^(2))*((1)/(2)*(C[0])^(2)+ sum((C[2*p])^(2), p = 1..infinity))-(1)/(2)*(k)^(2)* sum(C[2*p]*C[2*p + 2], p = 0..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -Divide[1,2]*(k)^(2))*(Divide[1,2]*(Subscript[C, 0])^(2)+ Sum[(Subscript[C, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None])-Divide[1,2]*(k)^(2)* Sum[Subscript[C, 2*p]*Subscript[C, 2*p + 2], {p, 0, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E13 29.6.E13] || <math qid="Q8641">\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p} > 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1)/(2)*C[0]+ sum(C[2*p], p = 1..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Divide[1,2]*Subscript[C, 0]+ Sum[Subscript[C, 2*p], {p, 1, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E14 29.6.E14] || <math qid="Q8642">\lim_{p\to\infty}\frac{C_{2p+2}}{C_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{C_{2p+2}}{C_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+1, \nu = 2n+1, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((C[2*p + 2])/(C[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[C, 2*p + 2],Subscript[C, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E20 29.6.E20] || <math qid="Q8648">\sum_{p=0}^{\infty}A_{2p+1}^{2} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}A_{2p+1}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((A[2*p + 1])^(2), p = 0..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(Subscript[A, 2*p + 1])^(2), {p, 0, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E21 29.6.E21] || <math qid="Q8649">\sum_{p=0}^{\infty}A_{2p+1} > 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}A_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum(A[2*p + 1], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[Subscript[A, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E22 29.6.E22] || <math qid="Q8650">\lim_{p\to\infty}\frac{A_{2p+1}}{A_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{A_{2p+1}}{A_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+1, \nu = 2n+1, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((A[2*p + 1])/(A[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[A, 2*p + 1],Subscript[A, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6#Ex4 29.6#Ex4] || <math qid="Q8654">\alpha_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[p] = (1)/(2)*(nu - 2*p - 1)*(nu + 2*p + 2)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 1)*(\[Nu]+ 2*p + 2)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6#Ex6 29.6#Ex6] || <math qid="Q8656">\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p)*(nu + 2*p + 1)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p)*(\[Nu]+ 2*p + 1)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E28 29.6.E28] || <math qid="Q8658">\sum_{p=0}^{\infty}C_{2p+1} > 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}C_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum(C[2*p + 1], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[Subscript[C, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E29 29.6.E29] || <math qid="Q8659">\lim_{p\to\infty}\frac{C_{2p+1}}{C_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{C_{2p+1}}{C_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+2, \nu = 2n+2, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((C[2*p + 1])/(C[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[C, 2*p + 1],Subscript[C, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E35 29.6.E35] || <math qid="Q8665">\sum_{p=0}^{\infty}B_{2p+1}^{2} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}B_{2p+1}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((B[2*p + 1])^(2), p = 0..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(Subscript[B, 2*p + 1])^(2), {p, 0, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E36 29.6.E36] || <math qid="Q8666">\sum_{p=0}^{\infty}(2p+1)B_{2p+1} > 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}(2p+1)B_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 1)*B[2*p + 1], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 1)*Subscript[B, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E37 29.6.E37] || <math qid="Q8667">\lim_{p\to\infty}\frac{B_{2p+1}}{B_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{B_{2p+1}}{B_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+1, \nu = 2n+1, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((B[2*p + 1])/(B[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[B, 2*p + 1],Subscript[B, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6#Ex7 29.6#Ex7] || <math qid="Q8671">\alpha_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[p] = (1)/(2)*(nu - 2*p - 1)*(nu + 2*p + 2)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 1)*(\[Nu]+ 2*p + 2)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6#Ex9 29.6#Ex9] || <math qid="Q8673">\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p)*(nu + 2*p + 1)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p)*(\[Nu]+ 2*p + 1)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E43 29.6.E43] || <math qid="Q8675">\sum_{p=0}^{\infty}(2p+1)D_{2p+1} > 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}(2p+1)D_{2p+1} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 1)*D[2*p + 1], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 1)*Subscript[D, 2*p + 1], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E44 29.6.E44] || <math qid="Q8676">\lim_{p\to\infty}\frac{D_{2p+1}}{D_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{D_{2p+1}}{D_{2p-1}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+2, \nu = 2n+2, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((D[2*p + 1])/(D[2*p - 1]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[D, 2*p + 1],Subscript[D, 2*p - 1]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E50 29.6.E50] || <math qid="Q8682">\sum_{p=1}^{\infty}B_{2p}^{2} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=1}^{\infty}B_{2p}^{2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((B[2*p])^(2), p = 1..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(Subscript[B, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E51 29.6.E51] || <math qid="Q8683">\sum_{p=0}^{\infty}(2p+2)B_{2p+2} > 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}(2p+2)B_{2p+2} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 2)*B[2*p + 2], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 2)*Subscript[B, 2*p + 2], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E52 29.6.E52] || <math qid="Q8684">\lim_{p\to\infty}\frac{B_{2p+2}}{B_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{B_{2p+2}}{B_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+2, \nu = 2n+2, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((B[2*p + 2])/(B[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[B, 2*p + 2],Subscript[B, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6#Ex10 29.6#Ex10] || <math qid="Q8688">\alpha_{p} = \tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\alpha_{p} = \tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">alpha[p] = (1)/(2)*(nu - 2*p - 2)*(nu + 2*p + 3)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Alpha], p] == Divide[1,2]*(\[Nu]- 2*p - 2)*(\[Nu]+ 2*p + 3)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6#Ex11 29.6#Ex11] || <math qid="Q8689">\beta_{p} = (2p+2)^{2}(2-k^{2})</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\beta_{p} = (2p+2)^{2}(2-k^{2})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">beta[p] = (2*p + 2)^(2)*(2 - (k)^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[\[Beta], p] == (2*p + 2)^(2)*(2 - (k)^(2))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6#Ex12 29.6#Ex12] || <math qid="Q8690">\gamma_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\gamma_{p} = \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((1)/(2)*(nu - 2*p + 2)*(nu + 2*p - 1)*(k)^(2)) = (1)/(2)*(nu - 2*p - 1)*(nu + 2*p + 2)*(k)^(2)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(Divide[1,2]*(\[Nu]- 2*p + 2)*(\[Nu]+ 2*p - 1)*(k)^(2)) == Divide[1,2]*(\[Nu]- 2*p - 1)*(\[Nu]+ 2*p + 2)*(k)^(2)</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E57 29.6.E57] || <math qid="Q8691">\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=1}^{\infty}D_{2p}^{2}-\tfrac{1}{2}k^{2}\sum_{p=1}^{\infty}D_{2p}D_{2p+2} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=1}^{\infty}D_{2p}^{2}-\tfrac{1}{2}k^{2}\sum_{p=1}^{\infty}D_{2p}D_{2p+2} = 1</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -(1)/(2)*(k)^(2))*sum((D[2*p])^(2), p = 1..infinity)-(1)/(2)*(k)^(2)* sum(D[2*p]*D[2*p + 2], p = 1..infinity) = 1</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">(1 -Divide[1,2]*(k)^(2))*Sum[(Subscript[D, 2*p])^(2), {p, 1, Infinity}, GenerateConditions->None]-Divide[1,2]*(k)^(2)* Sum[Subscript[D, 2*p]*Subscript[D, 2*p + 2], {p, 1, Infinity}, GenerateConditions->None] == 1</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E58 29.6.E58] || <math qid="Q8692">\sum_{p=0}^{\infty}(2p+2)D_{2p+2} > 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\sum_{p=0}^{\infty}(2p+2)D_{2p+2} > 0</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">sum((2*p + 2)*D[2*p + 2], p = 0..infinity) > 0</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Sum[(2*p + 2)*Subscript[D, 2*p + 2], {p, 0, Infinity}, GenerateConditions->None] > 0</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/29.6.E59 29.6.E59] || <math qid="Q8693">\lim_{p\to\infty}\frac{D_{2p+2}}{D_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\lim_{p\to\infty}\frac{D_{2p+2}}{D_{2p}} = \frac{k^{2}}{(1+k^{\prime})^{2}}</syntaxhighlight> || <math>\nu \neq 2n+3, \nu = 2n+3, m > n</math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">limit((D[2*p + 2])/(D[2*p]), p = infinity) = ((k)^(2))/((1 +sqrt(1 - (k)^(2)))^(2))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Limit[Divide[Subscript[D, 2*p + 2],Subscript[D, 2*p]], p -> Infinity, GenerateConditions->None] == Divide[(k)^(2),(1 +Sqrt[1 - (k)^(2)])^(2)]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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