Orthogonal Polynomials - 18.15 Asymptotic Approximations

From testwiki
Revision as of 11:46, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.15.E6 ( sin ⁡ 1 2 ⁢ θ ) α + 1 2 ⁢ ( cos ⁡ 1 2 ⁢ θ ) β + 1 2 ⁢ P n ( α , β ) ⁡ ( cos ⁡ θ ) = Γ ⁡ ( n + α + 1 ) 2 1 2 ⁢ ρ α ⁢ n ! ⁢ ( θ 1 2 ⁢ J α ⁡ ( ρ ⁢ θ ) ⁢ ∑ m = 0 M A m ⁢ ( θ ) ρ 2 ⁢ m + θ 3 2 ⁢ J α + 1 ⁡ ( ρ ⁢ θ ) ⁢ ∑ m = 0 M - 1 B m ⁢ ( θ ) ρ 2 ⁢ m + 1 + ε M ⁢ ( ρ , θ ) ) superscript 1 2 𝜃 𝛼 1 2 superscript 1 2 𝜃 𝛽 1 2 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝜃 Euler-Gamma 𝑛 𝛼 1 superscript 2 1 2 superscript 𝜌 𝛼 𝑛 superscript 𝜃 1 2 Bessel-J 𝛼 𝜌 𝜃 superscript subscript 𝑚 0 𝑀 subscript 𝐴 𝑚 𝜃 superscript 𝜌 2 𝑚 superscript 𝜃 3 2 Bessel-J 𝛼 1 𝜌 𝜃 superscript subscript 𝑚 0 𝑀 1 subscript 𝐵 𝑚 𝜃 superscript 𝜌 2 𝑚 1 subscript 𝜀 𝑀 𝜌 𝜃 {\displaystyle{\displaystyle(\sin\tfrac{1}{2}\theta)^{\alpha+\frac{1}{2}}(\cos% \tfrac{1}{2}\theta)^{\beta+\frac{1}{2}}P^{(\alpha,\beta)}_{n}\left(\cos\theta% \right)=\frac{\Gamma\left(n+\alpha+1\right)}{2^{\frac{1}{2}}\rho^{\alpha}n!}\*% \left(\theta^{\frac{1}{2}}J_{\alpha}\left(\rho\theta\right)\sum_{m=0}^{M}% \dfrac{A_{m}(\theta)}{\rho^{2m}}+\theta^{\frac{3}{2}}J_{\alpha+1}\left(\rho% \theta\right)\sum_{m=0}^{M-1}\dfrac{B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M% }(\rho,\theta)\right)}}
(\sin@@{\tfrac{1}{2}\theta})^{\alpha+\frac{1}{2}}(\cos@@{\tfrac{1}{2}\theta})^{\beta+\frac{1}{2}}\JacobipolyP{\alpha}{\beta}{n}@{\cos@@{\theta}} = \frac{\EulerGamma@{n+\alpha+1}}{2^{\frac{1}{2}}\rho^{\alpha}n!}\*\left(\theta^{\frac{1}{2}}\BesselJ{\alpha}@{\rho\theta}\sum_{m=0}^{M}\dfrac{A_{m}(\theta)}{\rho^{2m}}+\theta^{\frac{3}{2}}\BesselJ{\alpha+1}@{\rho\theta}\sum_{m=0}^{M-1}\dfrac{B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M}(\rho,\theta)\right)		 ℜ ⁡ ( ( α ) + k + 1 ) > 0 , ℜ ⁡ ( ( α + 1 ) + k + 1 ) > 0 , ℜ ⁡ ( n + α + 1 ) > 0 formulae-sequence 𝛼 𝑘 1 0 formulae-sequence 𝛼 1 𝑘 1 0 𝑛 𝛼 1 0 {\displaystyle{\displaystyle\Re((\alpha)+k+1)>0,\Re((\alpha+1)+k+1)>0,\Re(n+% \alpha+1)>0}} 
(sin((1)/(2)*theta))^(alpha +(1)/(2))*(cos((1)/(2)*theta))^(beta +(1)/(2))* JacobiP(n, alpha, beta, cos(theta)) = (GAMMA(n + alpha + 1))/((2)^((1)/(2))*(n +(1)/(2)*(alpha + beta + 1))^(alpha)* factorial(n))*((theta)^((1)/(2))* BesselJ(alpha, (n +(1)/(2)*(alpha + beta + 1))*theta)*sum((A[m](theta))/((n +(1)/(2)*(alpha + beta + 1))^(2*m)), m = 0..M)+ (theta)^((3)/(2))* BesselJ(alpha + 1, (n +(1)/(2)*(alpha + beta + 1))*theta)*sum((B[m](theta))/((n +(1)/(2)*(alpha + beta + 1))^(2*m + 1)), m = 0..M - 1)+ varepsilon[M]((n +(1)/(2)*(alpha + beta + 1)), theta))

(Sin[Divide[1,2]*\[Theta]])^(\[Alpha]+Divide[1,2])*(Cos[Divide[1,2]*\[Theta]])^(\[Beta]+Divide[1,2])* JacobiP[n, \[Alpha], \[Beta], Cos[\[Theta]]] == Divide[Gamma[n + \[Alpha]+ 1],(2)^(Divide[1,2])*(n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))^\[Alpha]* (n)!]*(\[Theta]^(Divide[1,2])* BesselJ[\[Alpha], (n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))*\[Theta]]*Sum[Divide[Subscript[A, m][\[Theta]],(n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))^(2*m)], {m, 0, M}, GenerateConditions->None]+ \[Theta]^(Divide[3,2])* BesselJ[\[Alpha]+ 1, (n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))*\[Theta]]*Sum[Divide[Subscript[B, m][\[Theta]],(n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1))^(2*m + 1)], {m, 0, M - 1}, GenerateConditions->None]+ Subscript[\[CurlyEpsilon], M][(n +Divide[1,2]*(\[Alpha]+ \[Beta]+ 1)), \[Theta]])
Failure Failure Skipped - Because timed out Skipped - Because timed out
18.15.E24 μ = 2 ⁢ n + 1 𝜇 2 𝑛 1 {\displaystyle{\displaystyle\mu=2n+1}}
\mu = 2n+1		 

mu = 2*n + 1
 
\[Mu] == 2*n + 1
 Skipped - no semantic math Skipped - no semantic math - -
18.15.E28 H n ⁡ ( x ) = 2 1 4 ⁢ ( μ 2 - 1 ) ⁢ e 1 2 ⁢ μ 2 ⁢ t 2 ⁢ U ⁡ ( - 1 2 ⁢ μ 2 , μ ⁢ t ⁢ 2 ) Hermite-polynomial-H 𝑛 𝑥 superscript 2 1 4 superscript 𝜇 2 1 superscript 𝑒 1 2 superscript 𝜇 2 superscript 𝑡 2 parabolic-U 1 2 superscript 𝜇 2 𝜇 𝑡 2 {\displaystyle{\displaystyle H_{n}\left(x\right)=2^{\frac{1}{4}(\mu^{2}-1)}e^{% \frac{1}{2}\mu^{2}t^{2}}U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)}}
\HermitepolyH{n}@{x} = 2^{\frac{1}{4}(\mu^{2}-1)}e^{\frac{1}{2}\mu^{2}t^{2}}\paraU@{-\tfrac{1}{2}\mu^{2}}{\mu t\sqrt{2}}		 

HermiteH(n, x) = (2)^((1)/(4)*((mu)^(2)- 1))* exp((1)/(2)*(mu)^(2)* (t)^(2))*CylinderU(-(1)/(2)*(mu)^(2), mu*t*sqrt(2))

HermiteH[n, x] == (2)^(Divide[1,4]*(\[Mu]^(2)- 1))* Exp[Divide[1,2]*\[Mu]^(2)* (t)^(2)]*ParabolicCylinderD[- 1/2 -(-Divide[1,2]*\[Mu]^(2)), \[Mu]*t*Sqrt[2]]
Failure Failure
Failed [300 / 300]
Result: -1.440969060-2.714107233*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, n = 1}


Result: 2.559030940-2.714107233*I
Test Values: {mu = 1/2*3^(1/2)+1/2*I, t = -3/2, x = 3/2, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.440969055661161, -2.714107231302052]
Test Values: {Rule[n, 1], Rule[t, -1.5], Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.559030944338839, -2.714107231302052]
Test Values: {Rule[n, 2], Rule[t, -1.5], Rule[x, 1.5], Rule[μ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=18.15&oldid=58342"