Weierstrass Elliptic and Modular Functions - 23.15 Definitions

From testwiki
Revision as of 12:01, 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
23.15.E1 q = exp ⁡ ( - π ⁢ K ′ ⁡ ( k ) K ⁡ ( k ) ) 𝑞 𝜋 complementary-complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle q=\exp\left(-\pi\frac{{K^{\prime}}\left(k\right)}% {K\left(k\right)}\right)}}
q = \exp@{-\pi\frac{\ccompellintKk@{k}}{\compellintKk@{k}}}		 

q = exp(- Pi*(EllipticCK(k))/(EllipticK(k)))

q == Exp[- Pi*Divide[EllipticK[1-(k)^2],EllipticK[(k)^2]]]
Failure Failure Error
Failed [30 / 30]
Result: Complex[-0.1339745962155613, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.9466424242240871, 0.7022944994770247]
Test Values: {Rule[k, 2], Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
23.15#Ex1 k = θ 2 2 ⁡ ( 0 , q ) θ 3 2 ⁡ ( 0 , q ) 𝑘 Jacobi-theta 2 2 0 𝑞 Jacobi-theta 3 2 0 𝑞 {\displaystyle{\displaystyle k=\frac{{\theta_{2}^{2}}\left(0,q\right)}{{\theta% _{3}^{2}}\left(0,q\right)}}}
k = \frac{\Jacobithetaq{2}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}		 

k = ((JacobiTheta2(0, q))^(2))/((JacobiTheta3(0, q))^(2))

k == Divide[(EllipticTheta[2, 0, q])^(2),(EllipticTheta[3, 0, q])^(2)]
Failure Failure Error
Failed [5 / 30]
Result: Complex[1.0, -308.9309168668012]
Test Values: {Rule[k, 1], Rule[q, -0.5]}


Result: Complex[2.0, -308.9309168668012]
Test Values: {Rule[k, 2], Rule[q, -0.5]}

... skip entries to safe data
23.15.E3 𝒜 ⁢ τ = a ⁢ τ + b c ⁢ τ + d 𝒜 𝜏 𝑎 𝜏 𝑏 𝑐 𝜏 𝑑 {\displaystyle{\displaystyle\mathcal{A}\tau=\frac{a\tau+b}{c\tau+d}}}
\mathcal{A}\tau = \frac{a\tau+b}{c\tau+d}		 

A*tau = (a*tau + b)/(c*tau + d)
 
A*\[Tau] == Divide[a*\[Tau]+ b,c*\[Tau]+ d]
 Skipped - no semantic math Skipped - no semantic math - -
23.15.E4 a ⁢ d - b ⁢ c = 1 𝑎 𝑑 𝑏 𝑐 1 {\displaystyle{\displaystyle ad-bc=1}}
ad-bc = 1		 

a*d - b*c = 1
 
a*d - b*c == 1
 Skipped - no semantic math Skipped - no semantic math - -
23.15.E6 λ ⁡ ( τ ) = θ 2 4 ⁡ ( 0 , q ) θ 3 4 ⁡ ( 0 , q ) modular-Lambda 𝜏 Jacobi-theta 2 4 0 𝑞 Jacobi-theta 3 4 0 𝑞 {\displaystyle{\displaystyle\lambda\left(\tau\right)=\frac{{\theta_{2}^{4}}% \left(0,q\right)}{{\theta_{3}^{4}}\left(0,q\right)}}}
\modularlambdatau@{\tau} = \frac{\Jacobithetaq{2}^{4}@{0}{q}}{\Jacobithetaq{3}^{4}@{0}{q}}		 

Error

ModularLambda[\[Tau]] == Divide[(EllipticTheta[2, 0, q])^(4),(EllipticTheta[3, 0, q])^(4)]
Missing Macro Error Failure -
Failed [4 / 100]
Result: Complex[95438.81139616246, 21.966995277463894]
Test Values: {Rule[q, -0.5], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[95438.81139616246, -0.8660254037844387]
Test Values: {Rule[q, -0.5], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
23.15.E7 J ⁡ ( τ ) = ( θ 2 8 ⁡ ( 0 , q ) + θ 3 8 ⁡ ( 0 , q ) + θ 4 8 ⁡ ( 0 , q ) ) 3 54 ⁢ ( θ 1 ′ ⁡ ( 0 , q ) ) 8 Kleins-invariant-modular-J 𝜏 superscript Jacobi-theta 2 8 0 𝑞 Jacobi-theta 3 8 0 𝑞 Jacobi-theta 4 8 0 𝑞 3 54 superscript diffop Jacobi-theta 1 1 0 𝑞 8 {\displaystyle{\displaystyle J\left(\tau\right)=\frac{\left({\theta_{2}^{8}}% \left(0,q\right)+{\theta_{3}^{8}}\left(0,q\right)+{\theta_{4}^{8}}\left(0,q% \right)\right)^{3}}{54\left(\theta_{1}'\left(0,q\right)\right)^{8}}}}
\KleincompinvarJtau@{\tau} = \frac{\left(\Jacobithetaq{2}^{8}@{0}{q}+\Jacobithetaq{3}^{8}@{0}{q}+\Jacobithetaq{4}^{8}@{0}{q}\right)^{3}}{54\left(\Jacobithetaq{1}'@{0}{q}\right)^{8}}		 

Error

KleinInvariantJ[\[Tau]] == Divide[((EllipticTheta[2, 0, q])^(8)+ (EllipticTheta[3, 0, q])^(8)+ (EllipticTheta[4, 0, q])^(8))^(3),54*(D[EllipticTheta[1, 0, q], {0, 1}])^(8)]
Missing Macro Error Failure -
Failed [100 / 100]
Result: Plus[Complex[-71.08223570333668, -2.1851275073468844*^-14], Times[-0.018518518518518517, Power[D[0.0
Test Values: {0.0, 1.0}], -8], Power[Plus[Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8]], 3]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Times[-0.018518518518518517, Power[D[0.0
Test Values: {0.0, 1.0}], -8], Power[Plus[Power[EllipticTheta[2, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[3, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8], Power[EllipticTheta[4, 0.0, Complex[0.8660254037844387, 0.49999999999999994]], 8]], 3]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
23.15.E9 η ⁡ ( τ ) = ( 1 2 ⁢ θ 1 ′ ⁡ ( 0 , q ) ) 1 / 3 Dedekind-modular-Eta 𝜏 superscript 1 2 diffop Jacobi-theta 1 1 0 𝑞 1 3 {\displaystyle{\displaystyle\eta\left(\tau\right)=\left(\tfrac{1}{2}\theta_{1}% '\left(0,q\right)\right)^{1/3}}}
\Dedekindeta@{\tau} = \left(\tfrac{1}{2}\Jacobithetaq{1}'@{0}{q}\right)^{1/3}		 

Error

DedekindEta[\[Tau]] == (Divide[1,2]*D[EllipticTheta[1, 0, q], {0, 1}])^(1/3)
Missing Macro Error Failure -
Failed [10 / 10]
Result: Plus[0.7682254223260567, Times[-0.7937005259840998, Power[D[0.0
Test Values: {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Complex[0, 1]]}


Result: Plus[0.7682254223260567, Times[-0.7937005259840998, Power[D[0.0
Test Values: {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]}

... skip entries to safe data
23.15.E9 ( 1 2 ⁢ θ 1 ′ ⁡ ( 0 , q ) ) 1 / 3 = e i ⁢ π ⁢ τ / 12 ⁢ θ 3 ⁡ ( 1 2 ⁢ π ⁢ ( 1 + τ ) | 3 ⁢ τ ) superscript 1 2 diffop Jacobi-theta 1 1 0 𝑞 1 3 superscript 𝑒 𝑖 𝜋 𝜏 12 Jacobi-theta-tau 3 1 2 𝜋 1 𝜏 3 𝜏 {\displaystyle{\displaystyle\left(\tfrac{1}{2}\theta_{1}'\left(0,q\right)% \right)^{1/3}=e^{i\pi\tau/12}\theta_{3}\left(\tfrac{1}{2}\pi(1+\tau)\middle|3% \tau\right)}}
\left(\tfrac{1}{2}\Jacobithetaq{1}'@{0}{q}\right)^{1/3} = e^{i\pi\tau/12}\Jacobithetatau{3}@{\tfrac{1}{2}\pi(1+\tau)}{3\tau}		 

((1)/(2)*diff( JacobiTheta1(0, q), 0$(1) ))^(1/3) = exp(I*Pi*tau/12)*JacobiTheta3((1)/(2)*Pi*(1 + tau),exp(I*Pi*3*tau))

(Divide[1,2]*D[EllipticTheta[1, 0, q], {0, 1}])^(1/3) == Exp[I*Pi*\[Tau]/12]*EllipticTheta[3, Divide[1,2]*Pi*(1 + \[Tau]), Exp[I*Pi*(3*\[Tau])]]
Error Failure -
Failed [10 / 10]
Result: Plus[Complex[-0.7682254223260567, 1.7569052324234997*^-19], Times[0.7937005259840998, Power[D[0.0
Test Values: {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Complex[0, 1]]}


Result: Plus[Complex[-0.7682254223260567, 1.7569052324234997*^-19], Times[0.7937005259840998, Power[D[0.0
Test Values: {0.0, 1.0}], Rational[1, 3]]]], {Rule[q, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[τ, Complex[0, 1]]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=23.15&oldid=58427"