Integrals with Coalescing Saddles - 36.7 Zeros
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 36.7#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y_{m} = -\sqrt{2\pi(2m+1)}}
y_{m} = -\sqrt{2\pi(2m+1)} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | y[m] = -sqrt(2*Pi*(2*m + 1)) |
Subscript[y, m] == -Sqrt[2*Pi*(2*m + 1)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.7#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x_{m,n}^{+} = \sqrt{\dfrac{2}{-y_{m}}}\left(2n+\tfrac{1}{2}+(-1)^{m}\tfrac{1}{2}+\tfrac{1}{4}\right)\pi}
x_{m,n}^{+} = \sqrt{\dfrac{2}{-y_{m}}}\left(2n+\tfrac{1}{2}+(-1)^{m}\tfrac{1}{2}+\tfrac{1}{4}\right)\pi |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (x[m , n])^(+) = sqrt((2)/(- y[m]))*(2*n +(1)/(2)+(- 1)^(m)*(1)/(2)+(1)/(4))*Pi |
(Subscript[x, m , n])^(+) == Sqrt[Divide[2,- Subscript[y, m]]]*(2*n +Divide[1,2]+(- 1)^(m)*Divide[1,2]+Divide[1,4])*Pi |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.7#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x_{n} = +\left(\dfrac{8}{27}\right)^{1/2}|y_{n}|^{3/2}(1+\xi_{n})}
x_{n} = +\left(\dfrac{8}{27}\right)^{1/2}|y_{n}|^{3/2}(1+\xi_{n}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | x[n] = +((8)/(27))^(1/2)*(abs(y[n]))^(3/2)*(1 + xi[n]) |
Subscript[x, n] == +(Divide[8,27])^(1/2)*(Abs[Subscript[y, n]])^(3/2)*(1 + Subscript[\[Xi], n]) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.7#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y_{n} = -\left(\frac{3\pi(8n+5)}{9+8\xi_{n}}\right)^{1/2}}
y_{n} = -\left(\frac{3\pi(8n+5)}{9+8\xi_{n}}\right)^{1/2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | y[n] = -((3*Pi*(8*n + 5))/(9 + 8*xi[n]))^(1/2) |
Subscript[y, n] == -(Divide[3*Pi*(8*n + 5),9 + 8*Subscript[\[Xi], n]])^(1/2) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.7.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{3\pi(8n+5)}{9+8\xi_{n}}\xi_{n}^{3/2} = \dfrac{27}{16}\left(\dfrac{3}{2}\right)^{1/2}\left(\ln@{\frac{1}{\xi_{n}}}+3\ln@{\dfrac{3}{2}}\right)}
\frac{3\pi(8n+5)}{9+8\xi_{n}}\xi_{n}^{3/2} = \dfrac{27}{16}\left(\dfrac{3}{2}\right)^{1/2}\left(\ln@{\frac{1}{\xi_{n}}}+3\ln@{\dfrac{3}{2}}\right) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (3*Pi*(8*n + 5))/(9 + 8*xi[n])*(xi[n])^(3/2) = (27)/(16)*((3)/(2))^(1/2)*(ln((1)/(xi[n]))+ 3*ln((3)/(2)))
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Divide[3*Pi*(8*n + 5),9 + 8*Subscript[\[Xi], n]]*(Subscript[\[Xi], n])^(3/2) == Divide[27,16]*(Divide[3,2])^(1/2)*(Log[Divide[1,Subscript[\[Xi], n]]]+ 3*Log[Divide[3,2]])
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Failure | Failure | Failed [300 / 300] Result: 3.887397376+4.913760852*I
Test Values: {xi = 1/2*3^(1/2)+1/2*I, xi[n] = 1/2*3^(1/2)+1/2*I, n = 1}
Result: 7.826714845+7.271674350*I
Test Values: {xi = 1/2*3^(1/2)+1/2*I, xi[n] = 1/2*3^(1/2)+1/2*I, n = 2}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[3.8873973728456934, 4.913760851775014]
Test Values: {Rule[n, 1], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ξ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[7.826714842534605, 7.271674348744825]
Test Values: {Rule[n, 2], Rule[ξ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ξ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |
| 36.7.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z_{n} = + 3(\tfrac{1}{4}\pi(2n-\tfrac{1}{2}))^{1/3}\\}
z_{n} = + 3(\tfrac{1}{4}\pi(2n-\tfrac{1}{2}))^{1/3}\\ |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | z[n] = + 3*((1)/(4)*Pi*(2*n -(1)/(2)))^(1/3) |
Subscript[z, n] == + 3*(Divide[1,4]*Pi*(2*n -Divide[1,2]))^(1/3) |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.7#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Delta z = \frac{9\pi}{2z_{n}^{2}}}
\Delta z = \frac{9\pi}{2z_{n}^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Delta*z = (9*Pi)/(2*(z[n])^(2)) |
\[CapitalDelta]*z == Divide[9*Pi,2*(Subscript[z, n])^(2)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.7#Ex6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Delta x = \frac{6\pi}{z_{n}}}
\Delta x = \frac{6\pi}{z_{n}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Delta*x = (6*Pi)/(x + y*I[n]) |
\[CapitalDelta]*x == Divide[6*Pi,Subscript[x + y*I, n]] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 36.7.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \exp@{-2\pi i\left(\frac{z-z_{n}}{\Delta z}+\frac{2x}{\Delta x}\right)}\*{\left(2\exp@{\frac{-6\pi ix}{\Delta x}}\cos@{\frac{2\sqrt{3}\pi y}{\Delta x}}+1\right)} = \sqrt{3}}
\exp@{-2\pi i\left(\frac{z-z_{n}}{\Delta z}+\frac{2x}{\Delta x}\right)}\*{\left(2\exp@{\frac{-6\pi ix}{\Delta x}}\cos@{\frac{2\sqrt{3}\pi y}{\Delta x}}+1\right)} = \sqrt{3} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | exp(- 2*Pi*I*(((x + y*I)-x + y*I[n])/(Delta*(x + y*I))+(2*x)/(Delta*x)))*(2*exp((- 6*Pi*I*x)/(Delta*x))*cos((2*sqrt(3)*Pi*y)/(Delta*x))+ 1) = sqrt(3)
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Exp[- 2*Pi*I*(Divide[(x + y*I)-Subscript[x + y*I, n],\[CapitalDelta]*(x + y*I)]+Divide[2*x,\[CapitalDelta]*x])]*(2*Exp[Divide[- 6*Pi*I*x,\[CapitalDelta]*x]]*Cos[Divide[2*Sqrt[3]*Pi*y,\[CapitalDelta]*x]]+ 1) == Sqrt[3]
|
Failure | Failure | Error | Failed [300 / 300]
Result: Plus[-1.7320508075688772, Times[Complex[1.0151974851172445, -0.010763380729874927], Power[2.718281828459045, Times[Complex[0.0, -6.283185307179586], Plus[Complex[1.7320508075688774, -0.9999999999999999], Times[Complex[0.4553418012614795, 0.12200846792814624], Plus[Complex[1.5, -1.5], Times[-1.0, Subscript[Complex[1.5, -1.5], 1]]]]]]]]]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[y, -1.5], Rule[Δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Plus[-1.7320508075688772, Times[Complex[1.0151974851172445, -0.010763380729874927], Power[2.718281828459045, Times[Complex[0.0, -6.283185307179586], Plus[Complex[1.7320508075688774, -0.9999999999999999], Times[Complex[0.4553418012614795, 0.12200846792814624], Plus[Complex[1.5, -1.5], Times[-1.0, Subscript[Complex[1.5, -1.5], 2]]]]]]]]]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[y, -1.5], Rule[Δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |