Hypergeometric Function - 15.12 Asymptotic Approximations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
15.12.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{+} = \atan@{\frac{\phase@@{z}-\phase@{1-z}-\pi}{\ln@@{|1-z^{-1}|}}}}
\alpha_{+} = \atan@{\frac{\phase@@{z}-\phase@{1-z}-\pi}{\ln@@{|1-z^{-1}|}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[+] = arctan((argument(z)- argument(1 - z)- Pi)/(ln(abs(1 - (z)^(- 1)))))

Subscript[\[Alpha], +] == ArcTan[Divide[Arg[z]- Arg[1 - z]- Pi,Log[Abs[1 - (z)^(- 1)]]]]
Error Failure - Error
15.12.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha_{-} = \atan@{\frac{\phase@@{z}-\phase@{1-z}+\pi}{\ln@@{|1-z^{-1}|}}}}
\alpha_{-} = \atan@{\frac{\phase@@{z}-\phase@{1-z}+\pi}{\ln@@{|1-z^{-1}|}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha[-] = arctan((argument(z)- argument(1 - z)+ Pi)/(ln(abs(1 - (z)^(- 1)))))

Subscript[\[Alpha], -] == ArcTan[Divide[Arg[z]- Arg[1 - z]+ Pi,Log[Abs[1 - (z)^(- 1)]]]]
Error Failure - Error
15.12.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(\frac{e^{t}-1}{t}\right)^{b-1}e^{t(1-c)}\left(1-z+ze^{-t}\right)^{-a} = \sum_{s=0}^{\infty}q_{s}(z)t^{s}}
\left(\frac{e^{t}-1}{t}\right)^{b-1}e^{t(1-c)}\left(1-z+ze^{-t}\right)^{-a} = \sum_{s=0}^{\infty}q_{s}(z)t^{s}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
((exp(t)- 1)/(t))^(b - 1)* exp(t*(1 - c))*(1 - z + z*exp(- t))^(- a) = sum(q[s](z)* (t)^(s), s = 0..infinity)
 
(Divide[Exp[t]- 1,t])^(b - 1)* Exp[t*(1 - c)]*(1 - z + z*Exp[- t])^(- a) == Sum[Subscript[q, s][z]* (t)^(s), {s, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
15.12.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \zeta = \acosh@@{z}}
\zeta = \acosh@@{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
zeta = arccosh(z)

\[Zeta] == ArcCosh[z]
Failure Failure
Failed [70 / 70]
Result: .2075464554-.2853981632*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I}


Result: -1.158478949+.806272408e-1*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, zeta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[0.1612451656432845, -0.8901042143273741]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ζ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.5217675362489347, -0.7070915124351547]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ζ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
15.12.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha = \left(-2\ln@{1-\left(\frac{z-1}{z+1}\right)^{2}}\right)^{1/2}}
\alpha = \left(-2\ln@{1-\left(\frac{z-1}{z+1}\right)^{2}}\right)^{1/2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
alpha = (- 2*ln(1 -((z - 1)/(z + 1))^(2)))^(1/2)

\[Alpha] == (- 2*Log[1 -(Divide[z - 1,z + 1])^(2)])^(1/2)
Failure Failure
Failed [21 / 21]
Result: 1.500000000-.3723881428*I
Test Values: {alpha = 3/2, z = 1/2*3^(1/2)+1/2*I}


Result: 1.500000000-1.665109222*I
Test Values: {alpha = 3/2, z = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [21 / 21]
Result: Complex[1.0067817778628907, 0.36121951329018404]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[α, 1.5]}


Result: Complex[0.006781777862890637, 0.36121951329018404]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[α, 0.5]}

... skip entries to safe data
15.12.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \zeta = \acosh@{\tfrac{1}{4}z-1}}
\zeta = \acosh@{\tfrac{1}{4}z-1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
zeta = arccosh((1)/(4)*z - 1)

\[Zeta] == ArcCosh[Divide[1,4]*z - 1]
Failure Failure
Failed [70 / 70]
Result: .6717322583-1.947968998*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I}


Result: -.6942931457-1.581943594*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, zeta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [70 / 70]
Result: Complex[0.29977340809145847, -2.404910564859421]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ζ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-0.3832392938007607, -2.221897862967202]
Test Values: {Rule[z, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ζ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
15.12.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \beta = \left(-\frac{3}{2}\zeta+\frac{9}{4}\ln@{\frac{2+e^{\zeta}}{2+e^{-\zeta}}}\right)^{1/3}}
\beta = \left(-\frac{3}{2}\zeta+\frac{9}{4}\ln@{\frac{2+e^{\zeta}}{2+e^{-\zeta}}}\right)^{1/3}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
beta = (-(3)/(2)*zeta +(9)/(4)*ln((2 + exp(zeta))/(2 + exp(- zeta))))^(1/3)

\[Beta] == (-Divide[3,2]*\[Zeta]+Divide[9,4]*Log[Divide[2 + Exp[\[Zeta]],2 + Exp[- \[Zeta]]]])^(1/3)
Failure Failure
Failed [30 / 30]
Result: 1.169986459-.1804633349*I
Test Values: {beta = 3/2, zeta = 1/2*3^(1/2)+1/2*I}


Result: 1.113419726-.9637472295e-2*I
Test Values: {beta = 3/2, zeta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[1.3347889019926584, -0.09407633084828147]
Test Values: {Rule[β, 1.5], Rule[ζ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[1.308560321923405, -0.0011617388335202368]
Test Values: {Rule[β, 1.5], Rule[ζ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]]}

... skip entries to safe data
15.12#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a_{0}(\zeta) = \tfrac{1}{2}G_{0}(\beta)+\tfrac{1}{2}G_{0}(-\beta)}
a_{0}(\zeta) = \tfrac{1}{2}G_{0}(\beta)+\tfrac{1}{2}G_{0}(-\beta)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
a[0](zeta) = (1)/(2)*G[0](beta)+(1)/(2)*G[0](- beta)
 
Subscript[a, 0][\[Zeta]] == Divide[1,2]*Subscript[G, 0][\[Beta]]+Divide[1,2]*Subscript[G, 0][- \[Beta]]
 Skipped - no semantic math Skipped - no semantic math - -
15.12#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a_{1}(\zeta) = \left(\tfrac{1}{2}G_{0}(\beta)-\tfrac{1}{2}G_{0}(-\beta)\right)/\beta}
a_{1}(\zeta) = \left(\tfrac{1}{2}G_{0}(\beta)-\tfrac{1}{2}G_{0}(-\beta)\right)/\beta		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
a[1](zeta) = ((1)/(2)*G[0](beta)-(1)/(2)*G[0](- beta))/beta
 
Subscript[a, 1][\[Zeta]] == (Divide[1,2]*Subscript[G, 0][\[Beta]]-Divide[1,2]*Subscript[G, 0][- \[Beta]])/\[Beta]
 Skipped - no semantic math Skipped - no semantic math - -
15.12.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle G_{0}(+\beta) = \left(2+e^{+\zeta}\right)^{c-b-(\ifrac{1}{2})}\left(1+e^{+\zeta}\right)^{a-c+(\ifrac{1}{2})}\left(z-1-e^{+\zeta}\right)^{-a+(\ifrac{1}{2})}\sqrt{\frac{\beta}{e^{\zeta}-e^{-\zeta}}}}
G_{0}(+\beta) = \left(2+e^{+\zeta}\right)^{c-b-(\ifrac{1}{2})}\left(1+e^{+\zeta}\right)^{a-c+(\ifrac{1}{2})}\left(z-1-e^{+\zeta}\right)^{-a+(\ifrac{1}{2})}\sqrt{\frac{\beta}{e^{\zeta}-e^{-\zeta}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
G[0](+ beta) = (2 + exp(+ zeta))^(c - b -((1)/(2)))*(1 + exp(+ zeta))^(a - c +((1)/(2)))*(z - 1 - exp(+ zeta))^(- a +((1)/(2)))*sqrt((beta)/(exp(zeta)- exp(- zeta)))
 
Subscript[G, 0][+ \[Beta]] == (2 + Exp[+ \[Zeta]])^(c - b -(Divide[1,2]))*(1 + Exp[+ \[Zeta]])^(a - c +(Divide[1,2]))*(z - 1 - Exp[+ \[Zeta]])^(- a +(Divide[1,2]))*Sqrt[Divide[\[Beta],Exp[\[Zeta]]- Exp[- \[Zeta]]]]
 Skipped - no semantic math Skipped - no semantic math - -
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