Bessel Functions - 10.20 Uniform Asymptotic Expansions for Large Order
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 10.20.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \left(\deriv{\zeta}{z}\right)^{2} = \frac{1-z^{2}}{\zeta z^{2}}}
\left(\deriv{\zeta}{z}\right)^{2} = \frac{1-z^{2}}{\zeta z^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (diff(zeta, z))^(2) = (1 - (z)^(2))/(zeta*(z)^(2))
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(D[\[Zeta], z])^(2) == Divide[1 - (z)^(2),\[Zeta]*(z)^(2)]
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Failure | Failure | Failed [70 / 70] Result: .8660254030+.4999999994*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I}
Result: .4999999994-.8660254030*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[0.8660254037844386, 0.4999999999999999]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[0.4999999999999999, -0.8660254037844386]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 10.20.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2}{3}\zeta^{\frac{3}{2}} = \int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\diff{t}}
\frac{2}{3}\zeta^{\frac{3}{2}} = \int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < z, z \leq 1} | (2)/(3)*(zeta)^((3)/(2)) = int((sqrt(1 - (t)^(2)))/(t), t = z..1)
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Divide[2,3]*\[Zeta]^(Divide[3,2]) == Integrate[Divide[Sqrt[1 - (t)^(2)],t], {t, z, 1}, GenerateConditions->None]
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Error | Aborted | - | Skipped - Because timed out |
| 10.20.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\diff{t} = \ln@{\frac{1+\sqrt{1-z^{2}}}{z}}-\sqrt{1-z^{2}}}
\int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\diff{t} = \ln@{\frac{1+\sqrt{1-z^{2}}}{z}}-\sqrt{1-z^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 < z, z \leq 1} | int((sqrt(1 - (t)^(2)))/(t), t = z..1) = ln((1 +sqrt(1 - (z)^(2)))/(z))-sqrt(1 - (z)^(2))
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Integrate[Divide[Sqrt[1 - (t)^(2)],t], {t, z, 1}, GenerateConditions->None] == Log[Divide[1 +Sqrt[1 - (z)^(2)],z]]-Sqrt[1 - (z)^(2)]
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Error | Aborted | - | Skipped - Because timed out |
| 10.20.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2}{3}(-\zeta)^{\frac{3}{2}} = \int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\diff{t}}
\frac{2}{3}(-\zeta)^{\frac{3}{2}} = \int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1 \leq z, z < \infty} | (2)/(3)*(- zeta)^((3)/(2)) = int((sqrt((t)^(2)- 1))/(t), t = 1..z)
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Divide[2,3]*(- \[Zeta])^(Divide[3,2]) == Integrate[Divide[Sqrt[(t)^(2)- 1],t], {t, 1, z}, GenerateConditions->None]
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Failure | Aborted | Failed [20 / 20] Result: -.7483698391+.4714045210*I
Test Values: {z = 3/2, zeta = 1/2*3^(1/2)+1/2*I}
Result: -.2769653183-.6666666667*I
Test Values: {z = 3/2, zeta = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [20 / 20]
Result: Complex[-0.7483698389729962, 0.4714045207910317]
Test Values: {Rule[z, 1.5], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-0.27696531818196457, -0.6666666666666666]
Test Values: {Rule[z, 1.5], Rule[ζ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 10.20.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\diff{t} = \sqrt{z^{2}-1}-\asec@@{z}}
\int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\diff{t} = \sqrt{z^{2}-1}-\asec@@{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 1 \leq z, z < \infty} | int((sqrt((t)^(2)- 1))/(t), t = 1..z) = sqrt((z)^(2)- 1)- arcsec(z)
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Integrate[Divide[Sqrt[(t)^(2)- 1],t], {t, 1, z}, GenerateConditions->None] == Sqrt[(z)^(2)- 1]- ArcSec[z]
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Failure | Aborted | Successful [Tested: 2] | Successful [Tested: 2] |
| 10.20#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{0}(0) = 1}
A_{0}(0) = 1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | A[0](0) = 1 |
Subscript[A, 0][0] == 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.20#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{1}(0) = -\tfrac{1}{225}}
A_{1}(0) = -\tfrac{1}{225} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | A[1](0) = -(1)/(225) |
Subscript[A, 1][0] == -Divide[1,225] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.20#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{2}(0) = \tfrac{1\;51439}{2182\;95000}}
A_{2}(0) = \tfrac{1\;51439}{2182\;95000} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | A[2](0) = (151439)/(218295000) |
Subscript[A, 2][0] == Divide[151439,218295000] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.20#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle A_{3}(0) = -\tfrac{8872\;78009}{250\;49351\;25000}}
A_{3}(0) = -\tfrac{8872\;78009}{250\;49351\;25000} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | A[3](0) = -(887278009)/(2504935125000) |
Subscript[A, 3][0] == -Divide[887278009,2504935125000] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.20#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle B_{0}(0) = \tfrac{1}{70}2^{\frac{1}{3}}}
B_{0}(0) = \tfrac{1}{70}2^{\frac{1}{3}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | B[0](0) = (1)/(70)*(2)^((1)/(3)) |
Subscript[B, 0][0] == Divide[1,70]*(2)^(Divide[1,3]) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.20#Ex6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle B_{1}(0) = -\tfrac{1213}{10\;23750}2^{\frac{1}{3}}}
B_{1}(0) = -\tfrac{1213}{10\;23750}2^{\frac{1}{3}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | B[1](0) = -(1213)/(1023750)*(2)^((1)/(3)) |
Subscript[B, 1][0] == -Divide[1213,1023750]*(2)^(Divide[1,3]) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.20#Ex7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle B_{2}(0) = \tfrac{1\;65425\;37833}{3774\;32055\;00000}2^{\frac{1}{3}}}
B_{2}(0) = \tfrac{1\;65425\;37833}{3774\;32055\;00000}2^{\frac{1}{3}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | B[2](0) = (16542537833)/(37743205500000)*(2)^((1)/(3)) |
Subscript[B, 2][0] == Divide[16542537833,37743205500000]*(2)^(Divide[1,3]) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.20#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle B_{3}(0) = -\tfrac{959\;71711\;84603}{25\;47666\;37125\;00000}2^{\frac{1}{3}}}
B_{3}(0) = -\tfrac{959\;71711\;84603}{25\;47666\;37125\;00000}2^{\frac{1}{3}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | B[3](0) = -(9597171184603)/(25476663712500000)*(2)^((1)/(3)) |
Subscript[B, 3][0] == -Divide[9597171184603,25476663712500000]*(2)^(Divide[1,3]) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.20.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \zeta = (\tfrac{3}{2})^{\frac{2}{3}}(\tau- i\pi)^{\frac{2}{3}}}
\zeta = (\tfrac{3}{2})^{\frac{2}{3}}(\tau- i\pi)^{\frac{2}{3}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 \leq \tau, \tau < \infty} | zeta = ((3)/(2))^((2)/(3))*(tau - I*Pi)^((2)/(3)) |
\[Zeta] == (Divide[3,2])^(Divide[2,3])*(\[Tau]- I*Pi)^(Divide[2,3]) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.20.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \zeta = e^{- i\pi/3}\tau}
\zeta = e^{- i\pi/3}\tau |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 \leq \tau, \tau \leq (\tfrac{3}{2}\pi)^{\frac{2}{3}}} | zeta = exp(- I*Pi/3)*tau |
\[Zeta] == Exp[- I*Pi/3]*\[Tau] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 10.20.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z = +(\tau\coth@@{\tau}-\tau^{2})^{\frac{1}{2}}+\iunit(\tau^{2}-\tau\tanh@@{\tau})^{\frac{1}{2}}}
z = +(\tau\coth@@{\tau}-\tau^{2})^{\frac{1}{2}}+\iunit(\tau^{2}-\tau\tanh@@{\tau})^{\frac{1}{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 \leq \tau, \tau \leq \tau_{0}} | z = +(tau*coth(tau)- (tau)^(2))^((1)/(2))+ I*((tau)^(2)- tau*tanh(tau))^((1)/(2))
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z == +(\[Tau]*Coth[\[Tau]]- \[Tau]^(2))^(Divide[1,2])+ I*(\[Tau]^(2)- \[Tau]*Tanh[\[Tau]])^(Divide[1,2])
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Failure | Failure | Failed [21 / 21] Result: .8660254040-1.214547924*I
Test Values: {tau = 3/2, z = 1/2*3^(1/2)+1/2*I}
Result: -.5000000000-.8485225201*I
Test Values: {tau = 3/2, z = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Skip - No test values generated |
| 10.20.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z = -(\tau\coth@@{\tau}-\tau^{2})^{\frac{1}{2}}-\iunit(\tau^{2}-\tau\tanh@@{\tau})^{\frac{1}{2}}}
z = -(\tau\coth@@{\tau}-\tau^{2})^{\frac{1}{2}}-\iunit(\tau^{2}-\tau\tanh@@{\tau})^{\frac{1}{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 0 \leq \tau, \tau \leq \tau_{0}} | z = -(tau*coth(tau)- (tau)^(2))^((1)/(2))- I*((tau)^(2)- tau*tanh(tau))^((1)/(2))
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z == -(\[Tau]*Coth[\[Tau]]- \[Tau]^(2))^(Divide[1,2])- I*(\[Tau]^(2)- \[Tau]*Tanh[\[Tau]])^(Divide[1,2])
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Failure | Failure | Failed [21 / 21] Result: .8660254040+2.214547924*I
Test Values: {tau = 3/2, z = 1/2*3^(1/2)+1/2*I}
Result: -.5000000000+2.580573328*I
Test Values: {tau = 3/2, z = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Skip - No test values generated |