Asymptotic Approximations - 2.6 Distributional Methods

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
2.6.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle S(x) = \int_{0}^{\infty}\frac{1}{(1+t)^{1/3}(x+t)}\diff{t}}
S(x) = \int_{0}^{\infty}\frac{1}{(1+t)^{1/3}(x+t)}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
S(x) = int((1)/((1 + t)^(1/3)*(x + t)), t = 0..infinity)

S[x] == Integrate[Divide[1,(1 + t)^(1/3)*(x + t)], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [30 / 30]
Result: -1.405894671+.7500000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, x = 1.5}


Result: -3.111297626+.2500000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, x = .5}


Result: -.774895738+1.*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, x = 2}


Result: -3.454932777+1.299038106*I
Test Values: {S = -1/2+1/2*I*3^(1/2), x = 1.5}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[-1.4058946699058708, 0.7499999999999999]
Test Values: {Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 1.5]}


Result: Complex[-3.1112976262861083, 0.24999999999999997]
Test Values: {Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[x, 0.5]}

... skip entries to safe data
2.6.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (1+t)^{-1/3} = \sum_{s=0}^{\infty}\binom{-\frac{1}{3}}{s}t^{-s-(1/3)}}
(1+t)^{-1/3} = \sum_{s=0}^{\infty}\binom{-\frac{1}{3}}{s}t^{-s-(1/3)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(1 + t)^(- 1/3) = sum(binomial(-(1)/(3),s)*(t)^(- s -(1/3)), s = 0..infinity)

(1 + t)^(- 1/3) == Sum[Binomial[-Divide[1,3],s]*(t)^(- s -(1/3)), {s, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [1 / 6]
Result: Float(infinity)+Float(infinity)*I
Test Values: {t = -.5}

Failed [1 / 6]
Result: Complex[1.8898815748423095, 1.0911236359717216]
Test Values: {Rule[t, -0.5]}

2.6.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha+\beta}}\diff{t} = \frac{\EulerGamma@{\alpha}\EulerGamma@{\beta}}{\EulerGamma@{\alpha+\beta}}\frac{1}{x^{\beta}}}
\int_{0}^{\infty}\frac{t^{\alpha-1}}{(x+t)^{\alpha+\beta}}\diff{t} = \frac{\EulerGamma@{\alpha}\EulerGamma@{\beta}}{\EulerGamma@{\alpha+\beta}}\frac{1}{x^{\beta}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{\alpha} > 0, \realpart@@{\beta} > 0, \realpart@@{(\alpha)} > 0, \realpart@@{(\beta)} > 0, \realpart@@{(\alpha+\beta)} > 0} 
int(((t)^(alpha - 1))/((x + t)^(alpha + beta)), t = 0..infinity) = (GAMMA(alpha)*GAMMA(beta))/(GAMMA(alpha + beta))*(1)/((x)^(beta))

Integrate[Divide[(t)^(\[Alpha]- 1),(x + t)^(\[Alpha]+ \[Beta])], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[\[Alpha]]*Gamma[\[Beta]],Gamma[\[Alpha]+ \[Beta]]]*Divide[1,(x)^\[Beta]]
Failure Successful Successful [Tested: 27] Successful [Tested: 27]
2.6.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\diff{t} = \frac{2\pi}{\sqrt{3}}\frac{(-1)^{s}}{x^{s+(1/3)}}}
\int_{0}^{\infty}\frac{t^{-s-(1/3)}}{x+t}\diff{t} = \frac{2\pi}{\sqrt{3}}\frac{(-1)^{s}}{x^{s+(1/3)}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(((t)^(- s -(1/3)))/(x + t), t = 0..infinity) = (2*Pi)/(sqrt(3))*((- 1)^(s))/((x)^(s +(1/3)))

Integrate[Divide[(t)^(- s -(1/3)),x + t], {t, 0, Infinity}, GenerateConditions->None] == Divide[2*Pi,Sqrt[3]]*Divide[(- 1)^(s),(x)^(s +(1/3))]
Failure Failure Skipped - Because timed out Successful [Tested: 3]
2.6.E22 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi_{\varepsilon}(t) = \frac{e^{-\varepsilon t}}{t+z}}
\phi_{\varepsilon}(t) = \frac{e^{-\varepsilon t}}{t+z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
phi[varepsilon](t) = (exp(- varepsilon*t))/(t + z)
 
Subscript[\[Phi], \[CurlyEpsilon]][t] == Divide[Exp[- \[CurlyEpsilon]*t],t + z]
 Skipped - no semantic math Skipped - no semantic math - -
2.6.E30 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle R_{n}(z) = \frac{(-1)^{n}}{z^{n}}\int_{0}^{\infty}\frac{\tau^{n}f_{n}(\tau)}{\tau+z}\diff{\tau}}
R_{n}(z) = \frac{(-1)^{n}}{z^{n}}\int_{0}^{\infty}\frac{\tau^{n}f_{n}(\tau)}{\tau+z}\diff{\tau}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
R[n](z) = ((- 1)^(n))/((z)^(n))*int(((tau)^(n)* f[n](tau))/(tau + z), tau = 0..infinity)

Subscript[R, n][z] == Divide[(- 1)^(n),(z)^(n)]*Integrate[Divide[\[Tau]^(n)* Subscript[f, n][\[Tau]],\[Tau]+ z], {\[Tau], 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: Float(infinity)+Float(infinity)*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, R[n] = 1/2*3^(1/2)+1/2*I, f[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: DirectedInfinity[]
Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: DirectedInfinity[]
Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[R, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
2.6.E41 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle f = \sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{n}c_{s}\Diracdelta^{(s-1)}+f_{n}}
f = \sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{n}c_{s}\Diracdelta^{(s-1)}+f_{n}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
f = sum(a[s]*(t)^(- s - alpha), s = 0..n - 1)- sum(c[s]*subs( temp=+, diff( Dirac(temp), temp$(s - 1) ) )*f[n], s = 1..n)

f == Sum[Subscript[a, s]*(t)^(- s - \[Alpha]), {s, 0, n - 1}, GenerateConditions->None]- Sum[Subscript[c, s]*(D[DiracDelta[temp], {temp, s - 1}]/.temp-> +)*Subscript[f, n], {s, 1, n}, GenerateConditions->None]
Error Failure - Error
2.6.E42 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle f = \sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}\Diracdelta^{(s-1)}+f_{n}}
f = \sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}\Diracdelta^{(s-1)}+f_{n}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
f = sum(a[s]*(t)^(- s - 1), s = 0..n - 1)- sum(d[s]*subs( temp=+, diff( Dirac(temp), temp$(s - 1) ) )*f[n], s = 1..n)

f == Sum[Subscript[a, s]*(t)^(- s - 1), {s, 0, n - 1}, GenerateConditions->None]- Sum[Subscript[d, s]*(D[DiracDelta[temp], {temp, s - 1}]/.temp-> +)*Subscript[f, n], {s, 1, n}, GenerateConditions->None]
Error Failure - Error
2.6.E47 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \delta_{n}(x) = \sum_{j=0}^{n}\binom{n}{j}\frac{\EulerGamma@{\mu+1}}{\EulerGamma@{\mu+1-j}}I^{\mu}\left(t^{n-j}f_{n,j}\right)(x)}
\delta_{n}(x) = \sum_{j=0}^{n}\binom{n}{j}\frac{\EulerGamma@{\mu+1}}{\EulerGamma@{\mu+1-j}}I^{\mu}\left(t^{n-j}f_{n,j}\right)(x)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\mu+1)} > 0, \realpart@@{(\mu+1-j)} > 0} 
delta[n](x) = sum(binomial(n,j)*(GAMMA(mu + 1))/(GAMMA(mu + 1 - j))*(I((t)^(n - j)* f[n , j])*)^(mu)*(x), j = 0..n)

Subscript[\[Delta], n][x] == Sum[Binomial[n,j]*Divide[Gamma[\[Mu]+ 1],Gamma[\[Mu]+ 1 - j]]*(I[(t)^(n - j)* Subscript[f, n , j]]*)^\[Mu]*(x), {j, 0, n}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: 2.186980427+1.033699533*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .6751758732+2.165771578*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: -.429437374-4.142136088*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: 1.015338573+1.637942321*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, mu = 1/2*3^(1/2)+1/2*I, t = -1.5, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n,j] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
 Skipped - Because timed out
2.6.E50 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle f_{n}(t) = (-1)^{n}\frac{t^{1-n-\alpha}}{1+t}}
f_{n}(t) = (-1)^{n}\frac{t^{1-n-\alpha}}{1+t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
f[n](t) = (- 1)^(n)*((t)^(1 - n - alpha))/(1 + t)
 
Subscript[f, n][t] == (- 1)^(n)*Divide[(t)^(1 - n - \[Alpha]),1 + t]
 Skipped - no semantic math Skipped - no semantic math - -
2.6.E53 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\left|\delta_{n}(x)\right|} \leq \frac{\EulerGamma@{\mu+1}\EulerGamma@{1-\alpha}}{\EulerGamma@{\mu+1-\alpha}\EulerGamma@{n+\alpha}}\*\sum_{j=0}^{n}\dbinom{n}{j}\frac{\EulerGamma@{n+\alpha-j}}{\left|\EulerGamma@{\mu+1-j}\right|}x^{\mu-\alpha}}
{\left|\delta_{n}(x)\right|} \leq \frac{\EulerGamma@{\mu+1}\EulerGamma@{1-\alpha}}{\EulerGamma@{\mu+1-\alpha}\EulerGamma@{n+\alpha}}\*\sum_{j=0}^{n}\dbinom{n}{j}\frac{\EulerGamma@{n+\alpha-j}}{\left|\EulerGamma@{\mu+1-j}\right|}x^{\mu-\alpha}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(\mu+1)} > 0, \realpart@@{(1-\alpha)} > 0, \realpart@@{(\mu+1-\alpha)} > 0, \realpart@@{(n+\alpha)} > 0, \realpart@@{(n+\alpha-j)} > 0, \realpart@@{(\mu+1-j)} > 0} 
abs(delta[n](x)) <= (GAMMA(mu + 1)*GAMMA(1 - alpha))/(GAMMA(mu + 1 - alpha)*GAMMA(n + alpha))* sum(binomial(n,j)*(GAMMA(n + alpha - j))/(abs(GAMMA(mu + 1 - j)))*(x)^(mu - alpha), j = 0..n)

Abs[Subscript[\[Delta], n][x]] <= Divide[Gamma[\[Mu]+ 1]*Gamma[1 - \[Alpha]],Gamma[\[Mu]+ 1 - \[Alpha]]*Gamma[n + \[Alpha]]]* Sum[Binomial[n,j]*Divide[Gamma[n + \[Alpha]- j],Abs[Gamma[\[Mu]+ 1 - j]]]*(x)^(\[Mu]- \[Alpha]), {j, 0, n}, GenerateConditions->None]
Failure Failure Successful [Tested: 300] Skipped - Because timed out
2.6.E56 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle h(t) = \sum_{s=0}^{n-1}b_{s}t^{-s-\beta}+h_{n}(t)}
h(t) = \sum_{s=0}^{n-1}b_{s}t^{-s-\beta}+h_{n}(t)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
h(t) = sum(b[s]*(t)^(- s - beta), s = 0..n - 1)+ h[n](t)
 
h[t] == Sum[Subscript[b, s]*(t)^(- s - \[Beta]), {s, 0, n - 1}, GenerateConditions->None]+ Subscript[h, n][t]
 Skipped - no semantic math Skipped - no semantic math - -
2.6.E59 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}t^{\lambda}\diff{t} = 0}
\int_{0}^{\infty}t^{\lambda}\diff{t} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((t)^(lambda), t = 0..infinity) = 0

Integrate[(t)^\[Lambda], {t, 0, Infinity}, GenerateConditions->None] == 0
Failure Failure
Failed [10 / 10]
Result: Float(undefined)
Test Values: {lambda = 1/2*3^(1/2)+1/2*I, lambda = 1+I}


Result: Float(undefined)
Test Values: {lambda = -1/2+1/2*I*3^(1/2), lambda = 1+I}


Result: Float(undefined)
Test Values: {lambda = 1/2-1/2*I*3^(1/2), lambda = 1+I}


Result: Float(undefined)
Test Values: {lambda = -1/2*3^(1/2)-1/2*I, lambda = 1+I}

... skip entries to safe data
Failed [1 / 1]
Result: Complex[2.105124266860741235376093541450691432144791*^+55894, -3.724980817286574983657738842232337454559011*^+55894]
Test Values: {Rule[λ, Complex[1, 1]]}

2.6#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \delta_{n}(x) = \int_{0}^{\infty}f_{n}(t)h_{n}(xt)\diff{t}}
\delta_{n}(x) = \int_{0}^{\infty}f_{n}(t)h_{n}(xt)\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
delta[n](x) = int(f[n](t)* h[n](x*t), t = 0..infinity)

Subscript[\[Delta], n][x] == Integrate[Subscript[f, n][t]* Subscript[h, n][x*t], {t, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: Float(infinity)+Float(infinity)*I
Test Values: {delta = 1/2*3^(1/2)+1/2*I, x = 1.5, delta[n] = 1/2*3^(1/2)+1/2*I, f[n] = 1/2*3^(1/2)+1/2*I, h[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.59977280929447116972275470162594*^+83839, -2.77088778626521950864048398971341*^+83839]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[δ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.59977280929447116972275470162594*^+83839, -2.77088778626521950864048398971341*^+83839]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[h, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[δ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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