Incomplete Gamma and Related Functions - 8.18 Asymptotic Expansions of

From testwiki
Revision as of 11:19, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision β†’ (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
8.18.E2 ΞΎ = - ln ⁑ x πœ‰ π‘₯ {\displaystyle{\displaystyle\xi=-\ln x}}
\xi = -\ln@@{x}		 

xi = - ln(x)

\[Xi] == - Log[x]
Failure Failure
Failed [30 / 30]
Result: 1.271490512+.5000000000*I
Test Values: {x = 1.5, xi = 1/2*3^(1/2)+1/2*I}


Result: -.945348919e-1+.8660254040*I
Test Values: {x = 1.5, xi = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[1.271490511892603, 0.49999999999999994]
Test Values: {Rule[x, 1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.0945348918918354, 0.8660254037844387]
Test Values: {Rule[x, 1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.18.E4 a ⁒ F k + 1 = ( k + b - a ⁒ ΞΎ ) ⁒ F k + k ⁒ ΞΎ ⁒ F k - 1 π‘Ž subscript 𝐹 π‘˜ 1 π‘˜ 𝑏 π‘Ž πœ‰ subscript 𝐹 π‘˜ π‘˜ πœ‰ subscript 𝐹 π‘˜ 1 {\displaystyle{\displaystyle aF_{k+1}=(k+b-a\xi)F_{k}+k\xi F_{k-1}}}
aF_{k+1} = (k+b-a\xi)F_{k}+k\xi F_{k-1}		 

a*F[k + 1] = (k + b - a*xi)*F[k]+ k*xi*F[k - 1]
 
a*Subscript[F, k + 1] == (k + b - a*\[Xi])*Subscript[F, k]+ k*\[Xi]*Subscript[F, k - 1]
 Skipped - no semantic math Skipped - no semantic math - -
8.18#Ex1 F 0 = a - b ⁒ Q ⁑ ( b , a ⁒ ΞΎ ) subscript 𝐹 0 superscript π‘Ž 𝑏 incomplete-gamma-Q 𝑏 π‘Ž πœ‰ {\displaystyle{\displaystyle F_{0}=a^{-b}Q\left(b,a\xi\right)}}
F_{0} = a^{-b}\normincGammaQ@{b}{a\xi}		 β„œ ⁑ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}} 
F[0] = (a)^(- b)* GAMMA(b, a*xi)/GAMMA(b)

Subscript[F, 0] == (a)^(- b)* GammaRegularized[b, a*\[Xi]]
Failure Failure
Failed [300 / 300]
Result: 1.253924788+1.407498490*I
Test Values: {a = -1.5, b = -1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = 1/2*3^(1/2)+1/2*I}


Result: -.1121006157+1.773523894*I
Test Values: {a = -1.5, b = -1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.2539247882576399, 1.4074984905445393]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.11210061552679867, 1.7735238943289782]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.18#Ex2 F 1 = b - a ⁒ ΞΎ a ⁒ F 0 + ΞΎ b ⁒ e - a ⁒ ΞΎ a ⁒ Ξ“ ⁑ ( b ) subscript 𝐹 1 𝑏 π‘Ž πœ‰ π‘Ž subscript 𝐹 0 superscript πœ‰ 𝑏 superscript 𝑒 π‘Ž πœ‰ π‘Ž Euler-Gamma 𝑏 {\displaystyle{\displaystyle F_{1}=\frac{b-a\xi}{a}F_{0}+\frac{\xi^{b}e^{-a\xi% }}{a\Gamma\left(b\right)}}}
F_{1} = \frac{b-a\xi}{a}F_{0}+\frac{\xi^{b}e^{-a\xi}}{a\EulerGamma@{b}}		 β„œ ⁑ b > 0 𝑏 0 {\displaystyle{\displaystyle\Re b>0}} 
F[1] = (b - a*xi)/(a)*F[0]+((xi)^(b)* exp(- a*xi))/(a*GAMMA(b))

Subscript[F, 1] == Divide[b - a*\[Xi],a]*Subscript[F, 0]+Divide[\[Xi]^(b)* Exp[- a*\[Xi]],a*Gamma[b]]
Failure Failure
Failed [300 / 300]
Result: 2.329643864+4.621882749*I
Test Values: {a = -1.5, b = 1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = 1/2*3^(1/2)+1/2*I, F[1] = 1/2*3^(1/2)+1/2*I}


Result: .9636184598+4.987908153*I
Test Values: {a = -1.5, b = 1.5, xi = 1/2*3^(1/2)+1/2*I, F[0] = 1/2*3^(1/2)+1/2*I, F[1] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[2.32964386182885, 4.621882746395113]
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.9636184580444114, 4.9879081501795515]
Test Values: {Rule[a, -1.5], Rule[b, 1.5], Rule[ΞΎ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[F, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.18.E6 ( 1 - e - t t ) b - 1 = βˆ‘ k = 0 ∞ d k ⁒ ( t - ΞΎ ) k superscript 1 superscript 𝑒 𝑑 𝑑 𝑏 1 superscript subscript π‘˜ 0 subscript 𝑑 π‘˜ superscript 𝑑 πœ‰ π‘˜ {\displaystyle{\displaystyle\left(\frac{1-e^{-t}}{t}\right)^{b-1}=\sum_{k=0}^{% \infty}d_{k}(t-\xi)^{k}}}
\left(\frac{1-e^{-t}}{t}\right)^{b-1} = \sum_{k=0}^{\infty}d_{k}(t-\xi)^{k}		 

((1 - exp(- t))/(t))^(b - 1) = sum(d[k]*(t - xi)^(k), k = 0..infinity)
 
(Divide[1 - Exp[- t],t])^(b - 1) == Sum[Subscript[d, k]*(t - \[Xi])^(k), {k, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
8.18#Ex3 d 0 = ( 1 - x ΞΎ ) b - 1 subscript 𝑑 0 superscript 1 π‘₯ πœ‰ 𝑏 1 {\displaystyle{\displaystyle d_{0}=\left(\frac{1-x}{\xi}\right)^{b-1}}}
d_{0} = \left(\frac{1-x}{\xi}\right)^{b-1}		 

d[0] = ((1 - x)/(xi))^(b - 1)
 
Subscript[d, 0] == (Divide[1 - x,\[Xi]])^(b - 1)
 Skipped - no semantic math Skipped - no semantic math - -
8.18#Ex4 d 1 = x ⁒ ΞΎ + x - 1 ( 1 - x ) ⁒ ΞΎ ⁒ ( b - 1 ) ⁒ d 0 subscript 𝑑 1 π‘₯ πœ‰ π‘₯ 1 1 π‘₯ πœ‰ 𝑏 1 subscript 𝑑 0 {\displaystyle{\displaystyle d_{1}=\frac{x\xi+x-1}{(1-x)\xi}(b-1)d_{0}}}
d_{1} = \frac{x\xi+x-1}{(1-x)\xi}(b-1)d_{0}		 

d[1] = (x*xi + x - 1)/((1 - x)*xi)*(b - 1)*d[0]
 
Subscript[d, 1] == Divide[x*\[Xi]+ x - 1,(1 - x)*\[Xi]]*(b - 1)*Subscript[d, 0]
 Skipped - no semantic math Skipped - no semantic math - -
8.18.E8 x 0 = a / ( a + b ) subscript π‘₯ 0 π‘Ž π‘Ž 𝑏 {\displaystyle{\displaystyle x_{0}=a/(a+b)}}
x_{0} = a/(a+b)		 

x[0] = a/(a + b)
 
Subscript[x, 0] == a/(a + b)
 Skipped - no semantic math Skipped - no semantic math - -
8.18.E10 - 1 2 ⁒ Ξ· 2 = x 0 ⁒ ln ⁑ ( x x 0 ) + ( 1 - x 0 ) ⁒ ln ⁑ ( 1 - x 1 - x 0 ) 1 2 superscript πœ‚ 2 subscript π‘₯ 0 π‘₯ subscript π‘₯ 0 1 subscript π‘₯ 0 1 π‘₯ 1 subscript π‘₯ 0 {\displaystyle{\displaystyle-\tfrac{1}{2}\eta^{2}=x_{0}\ln\left(\frac{x}{x_{0}% }\right)+(1-x_{0})\ln\left(\frac{1-x}{1-x_{0}}\right)}}
-\tfrac{1}{2}\eta^{2} = x_{0}\ln@{\frac{x}{x_{0}}}+(1-x_{0})\ln@{\frac{1-x}{1-x_{0}}}		 

-(1)/(2)*(eta)^(2) = x[0]*ln((x)/(x[0]))+(1 - x[0])*ln((1 - x)/(1 - x[0]))

-Divide[1,2]*\[Eta]^(2) == Subscript[x, 0]*Log[Divide[x,Subscript[x, 0]]]+(1 - Subscript[x, 0])*Log[Divide[1 - x,1 - Subscript[x, 0]]]
Failure Failure
Failed [300 / 300]
Result: .580000474e-1+.458917392e-1*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, x = 1.5, x[0] = 1/2*3^(1/2)+1/2*I}


Result: 2.269862383+1.019641337*I
Test Values: {eta = 1/2*3^(1/2)+1/2*I, x = 1.5, x[0] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.058000047924774145, 0.04589173995258988]
Test Values: {Rule[x, 1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.2698623824536366, 1.0196413375539057]
Test Values: {Rule[x, 1.5], Rule[Ξ·, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, 0], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.18.E11 c 0 ⁒ ( Ξ· ) = 1 Ξ· - x 0 ⁒ ( 1 - x 0 ) x - x 0 subscript 𝑐 0 πœ‚ 1 πœ‚ subscript π‘₯ 0 1 subscript π‘₯ 0 π‘₯ subscript π‘₯ 0 {\displaystyle{\displaystyle c_{0}(\eta)=\frac{1}{\eta}-\frac{\sqrt{x_{0}(1-x_% {0})}}{x-x_{0}}}}
c_{0}(\eta) = \frac{1}{\eta}-\frac{\sqrt{x_{0}(1-x_{0})}}{x-x_{0}}		 

c[0](eta) = (1)/(eta)-(sqrt(x[0]*(1 - x[0])))/(x - x[0])
 
Subscript[c, 0][\[Eta]] == Divide[1,\[Eta]]-Divide[Sqrt[Subscript[x, 0]*(1 - Subscript[x, 0])],x - Subscript[x, 0]]
 Skipped - no semantic math Skipped - no semantic math - -
8.18.E12 c 0 ⁒ ( 0 ) = 1 - 2 ⁒ x 0 3 ⁒ x 0 ⁒ ( 1 - x 0 ) subscript 𝑐 0 0 1 2 subscript π‘₯ 0 3 subscript π‘₯ 0 1 subscript π‘₯ 0 {\displaystyle{\displaystyle c_{0}(0)=\frac{1-2x_{0}}{3\sqrt{x_{0}(1-x_{0})}}}}
c_{0}(0) = \frac{1-2x_{0}}{3\sqrt{x_{0}(1-x_{0})}}		 

c[0](0) = (1 - 2*x[0])/(3*sqrt(x[0]*(1 - x[0])))
 
Subscript[c, 0][0] == Divide[1 - 2*Subscript[x, 0],3*Sqrt[Subscript[x, 0]*(1 - Subscript[x, 0])]]
 Skipped - no semantic math Skipped - no semantic math - -
8.18.E15 ΞΌ ⁒ ln ⁑ ΞΆ - ΞΆ = ln ⁑ x + ΞΌ ⁒ ln ⁑ ( 1 - x ) + ( 1 + ΞΌ ) ⁒ ln ⁑ ( 1 + ΞΌ ) - ΞΌ πœ‡ 𝜁 𝜁 π‘₯ πœ‡ 1 π‘₯ 1 πœ‡ 1 πœ‡ πœ‡ {\displaystyle{\displaystyle\mu\ln\zeta-\zeta=\ln x+\mu\ln\left(1-x\right)+(1+% \mu)\ln\left(1+\mu\right)-\mu}}
\mu\ln@@{\zeta}-\zeta = \ln@@{x}+\mu\ln@{1-x}+(1+\mu)\ln@{1+\mu}-\mu		 

mu*ln(zeta)- zeta = ln(x)+ mu*ln(1 - x)+(1 + mu)*ln(1 + mu)- mu

\[Mu]*Log[\[Zeta]]- \[Zeta] == Log[x]+ \[Mu]*Log[1 - x]+(1 + \[Mu])*Log[1 + \[Mu]]- \[Mu]
Failure Failure
Failed [299 / 300]
Result: .405976146-2.738439399*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, x = 1.5, zeta = 1/2*3^(1/2)+1/2*I}


Result: .9866033870-1.744115280*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, x = 1.5, zeta = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [299 / 300]
Result: Complex[0.4059761460255107, -2.7384393975724306]
Test Values: {Rule[x, 1.5], Rule[ΞΆ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.0560847852373059, 1.7517066341083583]
Test Values: {Rule[x, 1.5], Rule[ΞΆ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ΞΌ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
8.18.E16 h 0 ⁒ ( ΞΆ , ΞΌ ) = ΞΌ ⁒ ( 1 ΞΆ - ΞΌ - ( 1 + ΞΌ ) - 3 / 2 x 0 - x ) subscript β„Ž 0 𝜁 πœ‡ πœ‡ 1 𝜁 πœ‡ superscript 1 πœ‡ 3 2 subscript π‘₯ 0 π‘₯ {\displaystyle{\displaystyle h_{0}(\zeta,\mu)=\mu\left(\frac{1}{\zeta-\mu}-% \frac{(1+\mu)^{-3/2}}{x_{0}-x}\right)}}
h_{0}(\zeta,\mu) = \mu\left(\frac{1}{\zeta-\mu}-\frac{(1+\mu)^{-3/2}}{x_{0}-x}\right)		 

h[0](zeta , mu) = mu*((1)/(zeta - mu)-((1 + mu)^(- 3/2))/(x[0]- x))
 
Subscript[h, 0][\[Zeta], \[Mu]] == \[Mu]*(Divide[1,\[Zeta]- \[Mu]]-Divide[(1 + \[Mu])^(- 3/2),Subscript[x, 0]- x])
 Skipped - no semantic math Skipped - no semantic math - -
8.18.E17 h 0 ⁒ ( ΞΌ , ΞΌ ) = 1 3 ⁒ ( 1 - ΞΌ 1 + ΞΌ - 1 ) subscript β„Ž 0 πœ‡ πœ‡ 1 3 1 πœ‡ 1 πœ‡ 1 {\displaystyle{\displaystyle h_{0}(\mu,\mu)=\frac{1}{3}\left(\frac{1-\mu}{% \sqrt{1+\mu}}-1\right)}}
h_{0}(\mu,\mu) = \frac{1}{3}\left(\frac{1-\mu}{\sqrt{1+\mu}}-1\right)		 

h[0](mu , mu) = (1)/(3)*((1 - mu)/(sqrt(1 + mu))- 1)
 
Subscript[h, 0][\[Mu], \[Mu]] == Divide[1,3]*(Divide[1 - \[Mu],Sqrt[1 + \[Mu]]]- 1)
 Skipped - no semantic math Skipped - no semantic math - -
8.18.E18 I x ⁑ ( a , b ) = p IncI π‘₯ π‘Ž 𝑏 𝑝 {\displaystyle{\displaystyle I_{x}\left(a,b\right)=p}}
\normincBetaI{x}@{a}{b} = p		 0 ≀ p , p ≀ 1 , β„œ ⁑ a > 0 , β„œ ⁑ b > 0 , β„œ ⁑ ( a + b ) > 0 formulae-sequence 0 𝑝 formulae-sequence 𝑝 1 formulae-sequence π‘Ž 0 formulae-sequence 𝑏 0 π‘Ž 𝑏 0 {\displaystyle{\displaystyle 0\leq p,p\leq 1,\Re a>0,\Re b>0,\Re(a+b)>0}} 
Error

BetaRegularized[x, a, b] == p
Missing Macro Error Failure -
Failed [105 / 108]
Result: DirectedInfinity[]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[p, 0.5], Rule[x, 1.5]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[p, 0.5], Rule[x, 0.5]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=8.18&oldid=58166"