DLMF:29.5.E5
type: Item
Digital Library of Mathematical Functions ID: 29.5.E5
DLMF defining formula: <math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle{\displaystyle{\lim_{k\to 1-}\frac{\mathit{Ec}^{m}_{\nu}\left(z,% k^{2}\right)}{\mathit{Ec}^{m}_{\nu}\left(0,k^{2}\right)}=\lim_{k\to 1-}\frac{% \mathit{Es}^{m+1}_{\nu}\left(z,k^{2}\right)}{\mathit{Es}^{m+1}_{\nu}\left(0,k^% {2}\right)}}=\frac{1}{(\cosh z)^{\mu}}F\left({\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,% \tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\atop\tfrac{1}{2}};{\tanh^{2}}z% \right),}}" display="inline"> lim k → 1 - ⁡ 𝐸𝑐 ν m ⁡ ( z , k 2 ) 𝐸𝑐 ν m ⁡ ( 0 , k 2 ) = lim k → 1 - ⁡ 𝐸𝑠 ν m + 1 ⁡ ( z , k 2 ) 𝐸𝑠 ν m + 1 ⁡ ( 0 , k 2 ) = 1 ( cosh ⁡ z ) μ ⁢ F ⁡ ( 1 2 ⁢ μ - 1 2 ⁢ ν , 1 2 ⁢ μ + 1 2 ⁢ ν + 1 2 1 2 ; tanh 2 ⁡ z ) , <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b"> <apply id="p1.1.m1.1.12.2.cmml" xref="p1.1.m1.1.12"> <and id="p1.1.m1.1.12.2a.cmml" xref="p1.1.m1.1.12"/> <apply id="p1.1.m1.1.12.2b.cmml" xref="p1.1.m1.1.12"> <eq id="p1.1.m1.1.4.cmml" xref="p1.1.m1.1.4"/> <apply id="p1.1.m1.1.12.2.2.cmml" xref="p1.1.m1.1.12.2.2"> <apply id="p1.1.m1.1.12.2.2.1.cmml" xref="p1.1.m1.1.12.2.2.1"> <csymbol cd="ambiguous" id="p1.1.m1.1.12.2.2.1.1.cmml" xref="p1.1.m1.1.12.2.2.1">subscript</csymbol> <limit id="p1.1.m1.1.1.cmml" xref="p1.1.m1.1.1"/> <apply id="p1.1.m1.1.2.1.cmml" xref="p1.1.m1.1.2.1"> <ci id="p1.1.m1.1.2.1.2.cmml" xref="p1.1.m1.1.2.1.2">→</ci> <ci id="p1.1.m1.1.2.1.1.cmml" xref="p1.1.m1.1.2.1.1">𝑘</ci> <apply id="p1.1.m1.1.2.1.5.cmml" xref="p1.1.m1.1.2.1.5"> <csymbol cd="latexml" id="p1.1.m1.1.2.1.5.1.cmml" xref="p1.1.m1.1.2.1.5">limit-from</csymbol> <cn type="integer" id="p1.1.m1.1.2.1.3.cmml" xref="p1.1.m1.1.2.1.3">1</cn> <minus id="p1.1.m1.1.2.1.4.cmml" xref="p1.1.m1.1.2.1.4"/> </apply> </apply> </apply> <apply id="p1.1.m1.1.3.cmml" xref="p1.1.m1.1.3"> <divide id="p1.1.m1.1.3.1.cmml" xref="p1.1.m1.1.3"/> <apply id="p1.1.m1.1.3.2.1.1.cmml" xref="p1.1.m1.1.3.2.1.9.2"> <apply id="p1.1.m1.1.3.2.1.2.1.1.cmml" xref="p1.1.m1.1.3.2.1.2.1.6"> <csymbol cd="…
constraint: m <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b"> <ci id="p1.1.m1.1.1.cmml" xref="p1.1.m1.1.1">𝑚</ci> </annotation-xml> {\displaystyle{\displaystyle m}}
Symbols used: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
Symbols used: Q12365
Symbols used: Q12367
Symbols used: hyperbolic cosine function
Symbols used: hyperbolic tangent function
Symbols used: nonnegative integer
Symbols used: complex variable
Symbols used: real parameter
Symbols used: parameter
instance of: Digital Library of Mathematical Functions Formula
MaRDI profile type: MaRDI formula profile