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 ( d ζ d z ) 2 = 1 - z 2 ζ ⁢ z 2 superscript derivative 𝜁 𝑧 2 1 superscript 𝑧 2 𝜁 superscript 𝑧 2 {\displaystyle{\displaystyle\left(\frac{\mathrm{d}\zeta}{\mathrm{d}z}\right)^{% 2}=\frac{1-z^{2}}{\zeta z^{2}}}}
\left(\deriv{\zeta}{z}\right)^{2} = \frac{1-z^{2}}{\zeta z^{2}}		 

(diff(zeta, z))^(2) = (1 - (z)^(2))/(zeta*(z)^(2))

(D[\[Zeta], z])^(2) == Divide[1 - (z)^(2),\[Zeta]*(z)^(2)]
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 2 3 ⁢ ζ 3 2 = ∫ z 1 1 - t 2 t ⁢ d t 2 3 superscript 𝜁 3 2 superscript subscript 𝑧 1 1 superscript 𝑡 2 𝑡 𝑡 {\displaystyle{\displaystyle\frac{2}{3}\zeta^{\frac{3}{2}}=\int_{z}^{1}\frac{% \sqrt{1-t^{2}}}{t}\mathrm{d}t}}
\frac{2}{3}\zeta^{\frac{3}{2}} = \int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\diff{t}		 0 < z , z ≤ 1 formulae-sequence 0 𝑧 𝑧 1 {\displaystyle{\displaystyle 0<z,z\leq 1}} 
(2)/(3)*(zeta)^((3)/(2)) = int((sqrt(1 - (t)^(2)))/(t), t = z..1)

Divide[2,3]*\[Zeta]^(Divide[3,2]) == Integrate[Divide[Sqrt[1 - (t)^(2)],t], {t, z, 1}, GenerateConditions->None]
Error Aborted - Skipped - Because timed out
10.20.E2 ∫ z 1 1 - t 2 t ⁢ d t = ln ⁡ ( 1 + 1 - z 2 z ) - 1 - z 2 superscript subscript 𝑧 1 1 superscript 𝑡 2 𝑡 𝑡 1 1 superscript 𝑧 2 𝑧 1 superscript 𝑧 2 {\displaystyle{\displaystyle\int_{z}^{1}\frac{\sqrt{1-t^{2}}}{t}\mathrm{d}t=% \ln\left(\frac{1+\sqrt{1-z^{2}}}{z}\right)-\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}}		 0 < z , z ≤ 1 formulae-sequence 0 𝑧 𝑧 1 {\displaystyle{\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))

Integrate[Divide[Sqrt[1 - (t)^(2)],t], {t, z, 1}, GenerateConditions->None] == Log[Divide[1 +Sqrt[1 - (z)^(2)],z]]-Sqrt[1 - (z)^(2)]
Error Aborted - Skipped - Because timed out
10.20.E3 2 3 ⁢ ( - ζ ) 3 2 = ∫ 1 z t 2 - 1 t ⁢ d t 2 3 superscript 𝜁 3 2 superscript subscript 1 𝑧 superscript 𝑡 2 1 𝑡 𝑡 {\displaystyle{\displaystyle\frac{2}{3}(-\zeta)^{\frac{3}{2}}=\int_{1}^{z}% \frac{\sqrt{t^{2}-1}}{t}\mathrm{d}t}}
\frac{2}{3}(-\zeta)^{\frac{3}{2}} = \int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\diff{t}		 1 ≤ z , z < ∞ formulae-sequence 1 𝑧 𝑧 {\displaystyle{\displaystyle 1\leq z,z<\infty}} 
(2)/(3)*(- zeta)^((3)/(2)) = int((sqrt((t)^(2)- 1))/(t), t = 1..z)

Divide[2,3]*(- \[Zeta])^(Divide[3,2]) == Integrate[Divide[Sqrt[(t)^(2)- 1],t], {t, 1, z}, GenerateConditions->None]
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 ∫ 1 z t 2 - 1 t ⁢ d t = z 2 - 1 - arcsec ⁡ z superscript subscript 1 𝑧 superscript 𝑡 2 1 𝑡 𝑡 superscript 𝑧 2 1 𝑧 {\displaystyle{\displaystyle\int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\mathrm{d}t=% \sqrt{z^{2}-1}-\operatorname{arcsec}z}}
\int_{1}^{z}\frac{\sqrt{t^{2}-1}}{t}\diff{t} = \sqrt{z^{2}-1}-\asec@@{z}		 1 ≤ z , z < ∞ formulae-sequence 1 𝑧 𝑧 {\displaystyle{\displaystyle 1\leq z,z<\infty}} 
int((sqrt((t)^(2)- 1))/(t), t = 1..z) = sqrt((z)^(2)- 1)- arcsec(z)

Integrate[Divide[Sqrt[(t)^(2)- 1],t], {t, 1, z}, GenerateConditions->None] == Sqrt[(z)^(2)- 1]- ArcSec[z]
Failure Aborted Successful [Tested: 2] Successful [Tested: 2]
10.20#Ex1 A 0 ⁢ ( 0 ) = 1 subscript 𝐴 0 0 1 {\displaystyle{\displaystyle A_{0}(0)=1}}
A_{0}(0) = 1		 

A[0](0) = 1
 
Subscript[A, 0][0] == 1
 Skipped - no semantic math Skipped - no semantic math - -
10.20#Ex2 A 1 ⁢ ( 0 ) = - 1 225 subscript 𝐴 1 0 1 225 {\displaystyle{\displaystyle A_{1}(0)=-\tfrac{1}{225}}}
A_{1}(0) = -\tfrac{1}{225}		 

A[1](0) = -(1)/(225)
 
Subscript[A, 1][0] == -Divide[1,225]
 Skipped - no semantic math Skipped - no semantic math - -
10.20#Ex3 A 2 ⁢ ( 0 ) = 1 51439 2182 95000 subscript 𝐴 2 0 1 51439 2182 95000 {\displaystyle{\displaystyle A_{2}(0)=\tfrac{1\;51439}{2182\;95000}}}
A_{2}(0) = \tfrac{1\;51439}{2182\;95000}		 

A[2](0) = (151439)/(218295000)
 
Subscript[A, 2][0] == Divide[151439,218295000]
 Skipped - no semantic math Skipped - no semantic math - -
10.20#Ex4 A 3 ⁢ ( 0 ) = - 8872 78009 250 49351 25000 subscript 𝐴 3 0 8872 78009 250 49351 25000 {\displaystyle{\displaystyle A_{3}(0)=-\tfrac{8872\;78009}{250\;49351\;25000}}}
A_{3}(0) = -\tfrac{8872\;78009}{250\;49351\;25000}		 

A[3](0) = -(887278009)/(2504935125000)
 
Subscript[A, 3][0] == -Divide[887278009,2504935125000]
 Skipped - no semantic math Skipped - no semantic math - -
10.20#Ex5 B 0 ⁢ ( 0 ) = 1 70 ⁢ 2 1 3 subscript 𝐵 0 0 1 70 superscript 2 1 3 {\displaystyle{\displaystyle B_{0}(0)=\tfrac{1}{70}2^{\frac{1}{3}}}}
B_{0}(0) = \tfrac{1}{70}2^{\frac{1}{3}}		 

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 B 1 ⁢ ( 0 ) = - 1213 10 23750 ⁢ 2 1 3 subscript 𝐵 1 0 1213 10 23750 superscript 2 1 3 {\displaystyle{\displaystyle B_{1}(0)=-\tfrac{1213}{10\;23750}2^{\frac{1}{3}}}}
B_{1}(0) = -\tfrac{1213}{10\;23750}2^{\frac{1}{3}}		 

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 B 2 ⁢ ( 0 ) = 1 65425 37833 3774 32055 00000 ⁢ 2 1 3 subscript 𝐵 2 0 1 65425 37833 3774 32055 00000 superscript 2 1 3 {\displaystyle{\displaystyle B_{2}(0)=\tfrac{1\;65425\;37833}{3774\;32055\;000% 00}2^{\frac{1}{3}}}}
B_{2}(0) = \tfrac{1\;65425\;37833}{3774\;32055\;00000}2^{\frac{1}{3}}		 

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 B 3 ⁢ ( 0 ) = - 959 71711 84603 25 47666 37125 00000 ⁢ 2 1 3 subscript 𝐵 3 0 959 71711 84603 25 47666 37125 00000 superscript 2 1 3 {\displaystyle{\displaystyle B_{3}(0)=-\tfrac{959\;71711\;84603}{25\;47666\;37% 125\;00000}2^{\frac{1}{3}}}}
B_{3}(0) = -\tfrac{959\;71711\;84603}{25\;47666\;37125\;00000}2^{\frac{1}{3}}		 

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 ζ = ( 3 2 ) 2 3 ⁢ ( τ - i ⁢ π ) 2 3 𝜁 superscript 3 2 2 3 superscript 𝜏 𝑖 𝜋 2 3 {\displaystyle{\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}}		 0 ≤ τ , τ < ∞ formulae-sequence 0 𝜏 𝜏 {\displaystyle{\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 ζ = e - i ⁢ π / 3 ⁢ τ 𝜁 superscript 𝑒 𝑖 𝜋 3 𝜏 {\displaystyle{\displaystyle\zeta=e^{-i\pi/3}\tau}}
\zeta = e^{- i\pi/3}\tau		 0 ≤ τ , τ ≤ ( 3 2 ⁢ π ) 2 3 formulae-sequence 0 𝜏 𝜏 superscript 3 2 𝜋 2 3 {\displaystyle{\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 z = + ( τ ⁢ coth ⁡ τ - τ 2 ) 1 2 + i ⁢ ( τ 2 - τ ⁢ tanh ⁡ τ ) 1 2 𝑧 superscript 𝜏 hyperbolic-cotangent 𝜏 superscript 𝜏 2 1 2 imaginary-unit superscript superscript 𝜏 2 𝜏 𝜏 1 2 {\displaystyle{\displaystyle z=+(\tau\coth\tau-\tau^{2})^{\frac{1}{2}}+\mathrm% {i}(\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}}		 0 ≤ τ , τ ≤ τ 0 formulae-sequence 0 𝜏 𝜏 subscript 𝜏 0 {\displaystyle{\displaystyle 0\leq\tau,\tau\leq\tau_{0}}} 
z = +(tau*coth(tau)- (tau)^(2))^((1)/(2))+ I*((tau)^(2)- tau*tanh(tau))^((1)/(2))

z == +(\[Tau]*Coth[\[Tau]]- \[Tau]^(2))^(Divide[1,2])+ I*(\[Tau]^(2)- \[Tau]*Tanh[\[Tau]])^(Divide[1,2])
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 z = - ( τ ⁢ coth ⁡ τ - τ 2 ) 1 2 - i ⁢ ( τ 2 - τ ⁢ tanh ⁡ τ ) 1 2 𝑧 superscript 𝜏 hyperbolic-cotangent 𝜏 superscript 𝜏 2 1 2 imaginary-unit superscript superscript 𝜏 2 𝜏 𝜏 1 2 {\displaystyle{\displaystyle z=-(\tau\coth\tau-\tau^{2})^{\frac{1}{2}}-\mathrm% {i}(\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}}		 0 ≤ τ , τ ≤ τ 0 formulae-sequence 0 𝜏 𝜏 subscript 𝜏 0 {\displaystyle{\displaystyle 0\leq\tau,\tau\leq\tau_{0}}} 
z = -(tau*coth(tau)- (tau)^(2))^((1)/(2))- I*((tau)^(2)- tau*tanh(tau))^((1)/(2))

z == -(\[Tau]*Coth[\[Tau]]- \[Tau]^(2))^(Divide[1,2])- I*(\[Tau]^(2)- \[Tau]*Tanh[\[Tau]])^(Divide[1,2])
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
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