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 y m = - 2 ⁢ π ⁢ ( 2 ⁢ m + 1 ) subscript 𝑦 𝑚 2 𝜋 2 𝑚 1 {\displaystyle{\displaystyle y_{m}=-\sqrt{2\pi(2m+1)}}}
y_{m} = -\sqrt{2\pi(2m+1)}		 

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 x m , n + = 2 - y m ⁢ ( 2 ⁢ n + 1 2 + ( - 1 ) m ⁢ 1 2 + 1 4 ) ⁢ π superscript subscript 𝑥 𝑚 𝑛 2 subscript 𝑦 𝑚 2 𝑛 1 2 superscript 1 𝑚 1 2 1 4 𝜋 {\displaystyle{\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		 

(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 x n = + ( 8 27 ) 1 / 2 ⁢ | y n | 3 / 2 ⁢ ( 1 + ξ n ) subscript 𝑥 𝑛 superscript 8 27 1 2 superscript subscript 𝑦 𝑛 3 2 1 subscript 𝜉 𝑛 {\displaystyle{\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})		 

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 y n = - ( 3 ⁢ π ⁢ ( 8 ⁢ n + 5 ) 9 + 8 ⁢ ξ n ) 1 / 2 subscript 𝑦 𝑛 superscript 3 𝜋 8 𝑛 5 9 8 subscript 𝜉 𝑛 1 2 {\displaystyle{\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}		 

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 3 ⁢ π ⁢ ( 8 ⁢ n + 5 ) 9 + 8 ⁢ ξ n ⁢ ξ n 3 / 2 = 27 16 ⁢ ( 3 2 ) 1 / 2 ⁢ ( ln ⁡ ( 1 ξ n ) + 3 ⁢ ln ⁡ ( 3 2 ) ) 3 𝜋 8 𝑛 5 9 8 subscript 𝜉 𝑛 superscript subscript 𝜉 𝑛 3 2 27 16 superscript 3 2 1 2 1 subscript 𝜉 𝑛 3 3 2 {\displaystyle{\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\left(\frac{1}{\xi_{n}}\right)+% 3\ln\left(\dfrac{3}{2}\right)\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)		 

(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)))

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]])
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 z n = + 3 ⁢ ( 1 4 ⁢ π ⁢ ( 2 ⁢ n - 1 2 ) ) 1 / 3 subscript 𝑧 𝑛 3 superscript 1 4 𝜋 2 𝑛 1 2 1 3 {\displaystyle{\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}\\		 

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 Δ ⁢ z = 9 ⁢ π 2 ⁢ z n 2 Δ 𝑧 9 𝜋 2 superscript subscript 𝑧 𝑛 2 {\displaystyle{\displaystyle\Delta z=\frac{9\pi}{2z_{n}^{2}}}}
\Delta z = \frac{9\pi}{2z_{n}^{2}}		 

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 Δ ⁢ x = 6 ⁢ π z n Δ 𝑥 6 𝜋 subscript 𝑧 𝑛 {\displaystyle{\displaystyle\Delta x=\frac{6\pi}{z_{n}}}}
\Delta x = \frac{6\pi}{z_{n}}		 

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 exp ⁡ ( - 2 ⁢ π ⁢ i ⁢ ( z - z n Δ ⁢ z + 2 ⁢ x Δ ⁢ x ) ) ⁢ ( 2 ⁢ exp ⁡ ( - 6 ⁢ π ⁢ i ⁢ x Δ ⁢ x ) ⁢ cos ⁡ ( 2 ⁢ 3 ⁢ π ⁢ y Δ ⁢ x ) + 1 ) = 3 2 𝜋 𝑖 𝑧 subscript 𝑧 𝑛 Δ 𝑧 2 𝑥 Δ 𝑥 2 6 𝜋 𝑖 𝑥 Δ 𝑥 2 3 𝜋 𝑦 Δ 𝑥 1 3 {\displaystyle{\displaystyle\exp\left(-2\pi i\left(\frac{z-z_{n}}{\Delta z}+% \frac{2x}{\Delta x}\right)\right)\*{\left(2\exp\left(\frac{-6\pi ix}{\Delta x}% \right)\cos\left(\frac{2\sqrt{3}\pi y}{\Delta x}\right)+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}		 

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)

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
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