Asymptotic Approximations - 2.11 Remainder Terms; Stokes Phenomenon

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
2.11.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle I(m) = \int_{0}^{\pi}\frac{\cos@{mt}}{t^{2}+1}\diff{t}}
I(m) = \int_{0}^{\pi}\frac{\cos@{mt}}{t^{2}+1}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
I(m) = int((cos(m*t))/((t)^(2)+ 1), t = 0..Pi)

I[m] == Integrate[Divide[Cos[m*t],(t)^(2)+ 1], {t, 0, Pi}, GenerateConditions->None]
Failure Failure
Failed [30 / 30]
Result: 2.012164326+2.811364624*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, m = 1}


Result: 3.764776118+6.449767277*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, m = 2}


Result: 10.44871992+12.82571836*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, m = 3}


Result: -.5451540752-.6604650959*I
Test Values: {I = -1/2+1/2*I*3^(1/2), m = 1}

... skip entries to safe data
Failed [9 / 9]
Result: Complex[3.1301272053762923, 2.7021954356714506]
Test Values: {Rule[Complex[0, 1], 1], Rule[m, 1]}


Result: Complex[7.946986696458338, 4.871470912282225]
Test Values: {Rule[Complex[0, 1], 1], Rule[m, 2]}

... skip entries to safe data
2.11#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle q_{1}(t) = -\frac{2t}{(t^{2}+1)^{2}}}
q_{1}(t) = -\frac{2t}{(t^{2}+1)^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
q[1](t) = -(2*t)/(((t)^(2)+ 1)^(2))
 
Subscript[q, 1][t] == -Divide[2*t,((t)^(2)+ 1)^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle q_{2}(t) = \frac{24(t^{3}-t)}{(t^{2}+1)^{4}}}
q_{2}(t) = \frac{24(t^{3}-t)}{(t^{2}+1)^{4}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
q[2](t) = (24*((t)^(3)- t))/(((t)^(2)+ 1)^(4))
 
Subscript[q, 2][t] == Divide[24*((t)^(3)- t),((t)^(2)+ 1)^(4)]
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle q_{3}(t) = -\frac{240(3t^{5}-10t^{3}+3t)}{(t^{2}+1)^{6}}}
q_{3}(t) = -\frac{240(3t^{5}-10t^{3}+3t)}{(t^{2}+1)^{6}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
q[3](t) = -(240*(3*(t)^(5)- 10*(t)^(3)+ 3*t))/(((t)^(2)+ 1)^(6))
 
Subscript[q, 3][t] == -Divide[240*(3*(t)^(5)- 10*(t)^(3)+ 3*t),((t)^(2)+ 1)^(6)]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \genexpintE{p}@{z} = \frac{e^{-z}z^{p-1}}{\EulerGamma@{p}}\int_{0}^{\infty}\frac{e^{-zt}t^{p-1}}{1+t}\diff{t}}
\genexpintE{p}@{z} = \frac{e^{-z}z^{p-1}}{\EulerGamma@{p}}\int_{0}^{\infty}\frac{e^{-zt}t^{p-1}}{1+t}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{p} > 0} 
Ei(p, z) = (exp(- z)*(z)^(p - 1))/(GAMMA(p))*int((exp(- z*t)*(t)^(p - 1))/(1 + t), t = 0..infinity)

ExpIntegralE[p, z] == Divide[Exp[- z]*(z)^(p - 1),Gamma[p]]*Integrate[Divide[Exp[- z*t]*(t)^(p - 1),1 + t], {t, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 35]
2.11.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n = \rho-p+\alpha}
n = \rho-p+\alpha		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
n = rho - p + alpha
 
n == \[Rho]- p + \[Alpha]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{1+t} = \sum_{s=0}^{n-1}(-1)^{s}t^{s}+(-1)^{n}\frac{t^{n}}{1+t}}
\frac{1}{1+t} = \sum_{s=0}^{n-1}(-1)^{s}t^{s}+(-1)^{n}\frac{t^{n}}{1+t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(1)/(1 + t) = sum((- 1)^(s)* (t)^(s), s = 0..n - 1)+(- 1)^(n)*((t)^(n))/(1 + t)
 
Divide[1,1 + t] == Sum[(- 1)^(s)* (t)^(s), {s, 0, n - 1}, GenerateConditions->None]+(- 1)^(n)*Divide[(t)^(n),1 + t]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{e^{-z}}{2\pi}\int_{0}^{\infty}\frac{e^{-zt}t^{n+p-1}}{1+t}\diff{t} = \frac{\EulerGamma@{n+p}}{2\pi}\frac{\genexpintE{n+p}@{z}}{z^{n+p-1}}}
\frac{e^{-z}}{2\pi}\int_{0}^{\infty}\frac{e^{-zt}t^{n+p-1}}{1+t}\diff{t} = \frac{\EulerGamma@{n+p}}{2\pi}\frac{\genexpintE{n+p}@{z}}{z^{n+p-1}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{(n+p)} > 0} 
(exp(- z))/(2*Pi)*int((exp(- z*t)*(t)^(n + p - 1))/(1 + t), t = 0..infinity) = (GAMMA(n + p))/(2*Pi)*(Ei(n + p, z))/((z)^(n + p - 1))

Divide[Exp[- z],2*Pi]*Integrate[Divide[Exp[- z*t]*(t)^(n + p - 1),1 + t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[n + p],2*Pi]*Divide[ExpIntegralE[n + p, z],(z)^(n + p - 1)]
Successful Aborted - Successful [Tested: 189]
2.11.E14 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a_{2}(\theta,\alpha) = \frac{1}{12}(6\alpha^{2}-6\alpha+1)-\frac{\alpha}{1+e^{i\theta}}+\frac{1}{(1+e^{i\theta})^{2}}}
a_{2}(\theta,\alpha) = \frac{1}{12}(6\alpha^{2}-6\alpha+1)-\frac{\alpha}{1+e^{i\theta}}+\frac{1}{(1+e^{i\theta})^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
a[2](theta , alpha) = (1)/(12)*(6*(alpha)^(2)- 6*alpha + 1)-(alpha)/(1 + exp(I*theta))+(1)/((1 + exp(I*theta))^(2))
 
Subscript[a, 2][\[Theta], \[Alpha]] == Divide[1,12]*(6*\[Alpha]^(2)- 6*\[Alpha]+ 1)-Divide[\[Alpha],1 + Exp[I*\[Theta]]]+Divide[1,(1 + Exp[I*\[Theta]])^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E16 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c(\theta) = \sqrt{2(1+e^{i\theta}+i(\theta-\pi))}}
c(\theta) = \sqrt{2(1+e^{i\theta}+i(\theta-\pi))}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c(theta) = sqrt(2*(1 + exp(I*theta)+ I*(theta - Pi)))
 
c[\[Theta]] == Sqrt[2*(1 + Exp[I*\[Theta]]+ I*(\[Theta]- Pi))]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E17 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle h_{2s}(\theta,\alpha) = \frac{e^{i\alpha(\pi-\theta)}}{1+e^{-i\theta}}a_{2s}(\theta,\alpha)+(-1)^{s-1}i\frac{1\cdot 3\cdot 5\cdot\cdot\cdot(2s-1)}{(c(\theta))^{2s+1}}}
h_{2s}(\theta,\alpha) = \frac{e^{i\alpha(\pi-\theta)}}{1+e^{-i\theta}}a_{2s}(\theta,\alpha)+(-1)^{s-1}i\frac{1\cdot 3\cdot 5\cdot\cdot\cdot(2s-1)}{(c(\theta))^{2s+1}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
h[2*s](theta , alpha) = (exp(I*alpha*(Pi - theta)))/(1 + exp(- I*theta))*a[2*s](theta , alpha)+(- 1)^(s - 1)* I*(1 * 3 * 5 * * *(2*s - 1))/((c(theta))^(2*s + 1))
 
Subscript[h, 2*s][\[Theta], \[Alpha]] == Divide[Exp[I*\[Alpha]*(Pi - \[Theta])],1 + Exp[- I*\[Theta]]]*Subscript[a, 2*s][\[Theta], \[Alpha]]+(- 1)^(s - 1)* I*Divide[1 * 3 * 5 * * *(2*s - 1),(c[\[Theta]])^(2*s + 1)]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle h_{0}(\theta,\alpha) = \frac{e^{i\alpha(\pi-\theta)}}{1+e^{-i\theta}}-\frac{i}{c(\theta)}}
h_{0}(\theta,\alpha) = \frac{e^{i\alpha(\pi-\theta)}}{1+e^{-i\theta}}-\frac{i}{c(\theta)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
h[0](theta , alpha) = (exp(I*alpha*(Pi - theta)))/(1 + exp(- I*theta))-(I)/(c(theta))
 
Subscript[h, 0][\[Theta], \[Alpha]] == Divide[Exp[I*\[Alpha]*(Pi - \[Theta])],1 + Exp[- I*\[Theta]]]-Divide[I,c[\[Theta]]]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E19 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w_{j}(z) = e^{\lambda_{j}z}z^{\mu_{j}}\sum_{s=0}^{n-1}\frac{a_{s,j}}{z^{s}}+R_{n}^{(j)}(z)}
w_{j}(z) = e^{\lambda_{j}z}z^{\mu_{j}}\sum_{s=0}^{n-1}\frac{a_{s,j}}{z^{s}}+R_{n}^{(j)}(z)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
w[j](z) = exp(lambda[j]*z)*(z)^(mu[j])* sum((a[s , j])/((z)^(s)), s = 0..n - 1)+ (R[n])^(j)(z)
 
Subscript[w, j][z] == Exp[Subscript[\[Lambda], j]*z]*(z)^(Subscript[\[Mu], j])* Sum[Divide[Subscript[a, s , j],(z)^(s)], {s, 0, n - 1}, GenerateConditions->None]+ (Subscript[R, n])^(j)[z]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E26 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{5}\expintE@{5} = 0.17042\dots}
e^{5}\expintE@{5} = 0.17042\dots		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
exp(5)*Ei(5) = 0.17042

Exp[5]*ExpIntegralE[1, 5] == 0.17042
Failure Failure Skip - No test values generated Successful [Tested: 1]
2.11#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Delta^{0} = 0.00768}
\Delta^{0} = 0.00768		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(Delta)^(0) = 0.00768
 
\[CapitalDelta]^(0) == 0.00768
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Delta^{1} = 0.00154}
\Delta^{1} = 0.00154		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(Delta)^(1) = 0.00154
 
\[CapitalDelta]^(1) == 0.00154
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Delta^{2} = 0.00214}
\Delta^{2} = 0.00214		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(Delta)^(2) = 0.00214
 
\[CapitalDelta]^(2) == 0.00214
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Delta^{3} = 0.00192}
\Delta^{3} = 0.00192		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(Delta)^(3) = 0.00192
 
\[CapitalDelta]^(3) == 0.00192
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Delta^{4} = 0.00280}
\Delta^{4} = 0.00280		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(Delta)^(4) = 0.00280
 
\[CapitalDelta]^(4) == 0.00280
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Delta^{5} = 0.00434}
\Delta^{5} = 0.00434		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(Delta)^(5) = 0.00434
 
\[CapitalDelta]^(5) == 0.00434
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E28 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0.00384-0.00038+0.00027-0.00012+0.00009-0.00007 = 0.00363}
0.00384-0.00038+0.00027-0.00012+0.00009-0.00007 = 0.00363		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
0.00384 - 0.00038 + 0.00027 - 0.00012 + 0.00009 - 0.00007 = 0.00363
 
0.00384 - 0.00038 + 0.00027 - 0.00012 + 0.00009 - 0.00007 == 0.00363
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E30 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a_{n} = \frac{e^{-z/2}}{z^{n-\kappa}n!}\left(\mu^{2}-(\kappa-\tfrac{1}{2})^{2}\right)\*\left(\mu^{2}-(\kappa-\tfrac{3}{2})^{2}\right)\*\cdot\cdot\cdot\left(\mu^{2}-(\kappa-n+\tfrac{1}{2})^{2}\right)}
a_{n} = \frac{e^{-z/2}}{z^{n-\kappa}n!}\left(\mu^{2}-(\kappa-\tfrac{1}{2})^{2}\right)\*\left(\mu^{2}-(\kappa-\tfrac{3}{2})^{2}\right)\*\cdot\cdot\cdot\left(\mu^{2}-(\kappa-n+\tfrac{1}{2})^{2}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
a[n] = (exp(- z/2))/((z)^(n - kappa)* factorial(n))*((mu)^(2)-(kappa -(1)/(2))^(2))*((mu)^(2)-(kappa -(3)/(2))^(2))* * * *((mu)^(2)-(kappa - n +(1)/(2))^(2))
 
Subscript[a, n] == Divide[Exp[- z/2],(z)^(n - \[Kappa])* (n)!]*(\[Mu]^(2)-(\[Kappa]-Divide[1,2])^(2))*(\[Mu]^(2)-(\[Kappa]-Divide[3,2])^(2))* * * *(\[Mu]^(2)-(\[Kappa]- n +Divide[1,2])^(2))
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E31 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \WhittakerconfhyperW{2.3}{0.5}@{1.0} = -0.83299\;50268\;27526\;\cdots}
\WhittakerconfhyperW{2.3}{0.5}@{1.0} = -0.83299\;50268\;27526\;\cdots		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
WhittakerW(2.3, 0.5, 1.0) = - 0.832995026827526

WhittakerW[2.3, 0.5, 1.0] == - 0.832995026827526
Successful Failure - Successful [Tested: 1]
2.11.E32 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle d_{n} = \frac{\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}(j+1)^{n-1}\frac{s_{j}}{a_{j+1}}}{\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}(j+1)^{n-1}\frac{1}{a_{j+1}}}}
d_{n} = \frac{\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}(j+1)^{n-1}\frac{s_{j}}{a_{j+1}}}{\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}(j+1)^{n-1}\frac{1}{a_{j+1}}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
d[n] = (sum((- 1)^(j)*binomial(n,j)*(j + 1)^(n - 1)*(s[j])/(a[j + 1]), j = 0..n))/(sum((- 1)^(j)*binomial(n,j)*(j + 1)^(n - 1)*(1)/(a[j + 1]), j = 0..n))

Subscript[d, n] == Divide[Sum[(- 1)^(j)*Binomial[n,j]*(j + 1)^(n - 1)*Divide[Subscript[s, j],Subscript[a, j + 1]], {j, 0, n}, GenerateConditions->None],Sum[(- 1)^(j)*Binomial[n,j]*(j + 1)^(n - 1)*Divide[1,Subscript[a, j + 1]], {j, 0, n}, GenerateConditions->None]]
Failure Failure Error Skipped - Because timed out
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