Heun Functions - 31.10 Integral Equations and Representations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
31.10.E6 p ⁢ ( t ) = t γ ⁢ ( t - 1 ) δ ⁢ ( t - a ) ϵ 𝑝 𝑡 superscript 𝑡 𝛾 superscript 𝑡 1 𝛿 superscript 𝑡 𝑎 italic-ϵ {\displaystyle{\displaystyle p(t)=t^{\gamma}(t-1)^{\delta}(t-a)^{\epsilon}}}
p(t) = t^{\gamma}(t-1)^{\delta}(t-a)^{\epsilon}		 

p(t) = (t)^(gamma)*(t - 1)^(delta)*(t - a)^(epsilon)
 
p[t] == (t)^\[Gamma]*(t - 1)^\[Delta]*(t - a)^\[Epsilon]
 Skipped - no semantic math Skipped - no semantic math - -
31.10.E8 sin 2 ⁡ θ ⁢ ( ∂ 2 ⁡ 𝒦 ∂ ⁡ θ 2 + ( ( 1 - 2 ⁢ γ ) ⁢ tan ⁡ θ + 2 ⁢ ( δ + ϵ - 1 2 ) ⁢ cot ⁡ θ ) ⁢ ∂ ⁡ 𝒦 ∂ ⁡ θ - 4 ⁢ α ⁢ β ⁢ 𝒦 ) + ∂ 2 ⁡ 𝒦 ∂ ⁡ ϕ 2 + ( ( 1 - 2 ⁢ δ ) ⁢ cot ⁡ ϕ - ( 1 - 2 ⁢ ϵ ) ⁢ tan ⁡ ϕ ) ⁢ ∂ ⁡ 𝒦 ∂ ⁡ ϕ = 0 2 𝜃 partial-derivative 𝒦 𝜃 2 1 2 𝛾 𝜃 2 𝛿 italic-ϵ 1 2 𝜃 partial-derivative 𝒦 𝜃 4 𝛼 𝛽 𝒦 partial-derivative 𝒦 italic-ϕ 2 1 2 𝛿 italic-ϕ 1 2 italic-ϵ italic-ϕ partial-derivative 𝒦 italic-ϕ 0 {\displaystyle{\displaystyle{\sin^{2}}\theta\left(\frac{{\partial}^{2}\mathcal% {K}}{{\partial\theta}^{2}}+\left((1-2\gamma)\tan\theta+2(\delta+\epsilon-% \tfrac{1}{2})\cot\theta\right)\frac{\partial\mathcal{K}}{\partial\theta}-4% \alpha\beta\mathcal{K}\right)+\frac{{\partial}^{2}\mathcal{K}}{{\partial\phi}^% {2}}+\left((1-2\delta)\cot\phi-(1-2\epsilon)\tan\phi\right)\frac{\partial% \mathcal{K}}{\partial\phi}=0}}
\sin^{2}@@{\theta}\left(\pderiv[2]{\mathcal{K}}{\theta}+\left((1-2\gamma)\tan@@{\theta}+2(\delta+\epsilon-\tfrac{1}{2})\cot@@{\theta}\right)\pderiv{\mathcal{K}}{\theta}-4\alpha\beta\mathcal{K}\right)+\pderiv[2]{\mathcal{K}}{\phi}+\left((1-2\delta)\cot@@{\phi}-(1-2\epsilon)\tan@@{\phi}\right)\pderiv{\mathcal{K}}{\phi} = 0		 

(sin(theta))^(2)*(diff(K, [theta$(2)])+((1 - 2*gamma)*tan(theta)+ 2*(delta + epsilon -(1)/(2))*cot(theta))*diff(K, theta)- 4*alpha*beta*K)+ diff(K, [phi$(2)])+((1 - 2*delta)*cot(phi)-(1 - 2*epsilon)*tan(phi))*diff(K, phi) = 0

(Sin[\[Theta]])^(2)*(D[K, {\[Theta], 2}]+((1 - 2*\[Gamma])*Tan[\[Theta]]+ 2*(\[Delta]+ \[Epsilon]-Divide[1,2])*Cot[\[Theta]])*D[K, \[Theta]]- 4*\[Alpha]*\[Beta]*K)+ D[K, {\[Phi], 2}]+((1 - 2*\[Delta])*Cot[\[Phi]]-(1 - 2*\[Epsilon])*Tan[\[Phi]])*D[K, \[Phi]] == 0
Failure Failure
Failed [300 / 300]
Result: -2.252732458-7.327918109*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, epsilon = 1}


Result: -2.252732458-7.327918109*I
Test Values: {K = 1/2*3^(1/2)+1/2*I, alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, phi = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, epsilon = 2}

... skip entries to safe data
 Skipped - Because timed out
31.10.E10 𝒦 ⁢ ( z , t ) = ( z ⁢ t - a ) 1 2 - δ - σ ⁢ F 1 2 ⁡ ( 1 2 - δ - σ + α , 1 2 - δ - σ + β γ ; z ⁢ t a ) ⁢ F 1 2 ⁡ ( - 1 2 + δ + σ , - 1 2 + ϵ - σ δ ; a ⁢ ( z - 1 ) ⁢ ( t - 1 ) ( a - 1 ) ⁢ ( z ⁢ t - a ) ) 𝒦 𝑧 𝑡 superscript 𝑧 𝑡 𝑎 1 2 𝛿 𝜎 Gauss-hypergeometric-F-as-2F1 1 2 𝛿 𝜎 𝛼 1 2 𝛿 𝜎 𝛽 𝛾 𝑧 𝑡 𝑎 Gauss-hypergeometric-F-as-2F1 1 2 𝛿 𝜎 1 2 italic-ϵ 𝜎 𝛿 𝑎 𝑧 1 𝑡 1 𝑎 1 𝑧 𝑡 𝑎 {\displaystyle{\displaystyle\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma% }\*{{}_{2}F_{1}}\left({\frac{1}{2}-\delta-\sigma+\alpha,\frac{1}{2}-\delta-% \sigma+\beta\atop\gamma};\frac{zt}{a}\right)\*{{}_{2}F_{1}}\left({-\frac{1}{2}% +\delta+\sigma,-\frac{1}{2}+\epsilon-\sigma\atop\delta};\frac{a(z-1)(t-1)}{(a-% 1)(zt-a)}\right)}}
\mathcal{K}(z,t) = (zt-a)^{\frac{1}{2}-\delta-\sigma}\*\genhyperF{2}{1}@@{\frac{1}{2}-\delta-\sigma+\alpha,\frac{1}{2}-\delta-\sigma+\beta}{\gamma}{\frac{zt}{a}}\*\genhyperF{2}{1}@@{-\frac{1}{2}+\delta+\sigma,-\frac{1}{2}+\epsilon-\sigma}{\delta}{\frac{a(z-1)(t-1)}{(a-1)(zt-a)}}		 

K(z , t) = (z*t - a)^((1)/(2)- delta - sigma)* hypergeom([(1)/(2)- delta - sigma + alpha ,(1)/(2)- delta - sigma + beta], [gamma], (z*t)/(a))* hypergeom([-(1)/(2)+ delta + sigma , -(1)/(2)+ epsilon - sigma], [delta], (a*(z - 1)*(t - 1))/((a - 1)*(z*t - a)))

K[z , t] == (z*t - a)^(Divide[1,2]- \[Delta]- \[Sigma])* HypergeometricPFQ[{Divide[1,2]- \[Delta]- \[Sigma]+ \[Alpha],Divide[1,2]- \[Delta]- \[Sigma]+ \[Beta]}, {\[Gamma]}, Divide[z*t,a]]* HypergeometricPFQ[{-Divide[1,2]+ \[Delta]+ \[Sigma], -Divide[1,2]+ \[Epsilon]- \[Sigma]}, {\[Delta]}, Divide[a*(z - 1)*(t - 1),(a - 1)*(z*t - a)]]
Failure Failure Error Skipped - Because timed out
31.10.E18 ∂ 2 ⁡ 𝒦 ∂ ⁡ u 2 + ∂ 2 ⁡ 𝒦 ∂ ⁡ v 2 + ∂ 2 ⁡ 𝒦 ∂ ⁡ w 2 + 2 ⁢ γ - 1 u ⁢ ∂ ⁡ 𝒦 ∂ ⁡ u + 2 ⁢ δ - 1 v ⁢ ∂ ⁡ 𝒦 ∂ ⁡ v + 2 ⁢ ϵ - 1 w ⁢ ∂ ⁡ 𝒦 ∂ ⁡ w = 0 partial-derivative 𝒦 𝑢 2 partial-derivative 𝒦 𝑣 2 partial-derivative 𝒦 𝑤 2 2 𝛾 1 𝑢 partial-derivative 𝒦 𝑢 2 𝛿 1 𝑣 partial-derivative 𝒦 𝑣 2 italic-ϵ 1 𝑤 partial-derivative 𝒦 𝑤 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}\mathcal{K}}{{\partial u}^{2}}% +\frac{{\partial}^{2}\mathcal{K}}{{\partial v}^{2}}+\frac{{\partial}^{2}% \mathcal{K}}{{\partial w}^{2}}+\frac{2\gamma-1}{u}\frac{\partial\mathcal{K}}{% \partial u}+\frac{2\delta-1}{v}\frac{\partial\mathcal{K}}{\partial v}+\frac{2% \epsilon-1}{w}\frac{\partial\mathcal{K}}{\partial w}=0}}
\pderiv[2]{\mathcal{K}}{u}+\pderiv[2]{\mathcal{K}}{v}+\pderiv[2]{\mathcal{K}}{w}+\frac{2\gamma-1}{u}\pderiv{\mathcal{K}}{u}+\frac{2\delta-1}{v}\pderiv{\mathcal{K}}{v}+\frac{2\epsilon-1}{w}\pderiv{\mathcal{K}}{w} = 0		 

diff(K, [u$(2)])+ diff(K, [v$(2)])+ subs( temp=(I*(((s - a)*(t - a)*(z - a))/(a*(1 - a)))^(1/2)), diff( K, temp$(2) ) )+(2*gamma - 1)/(u)*diff(K, u)+(2*delta - 1)/(v)*diff(K, v)+(2*epsilon - 1)/(I*(((s - a)*(t - a)*(z - a))/(a*(1 - a)))^(1/2))*subs( temp=(I*(((s - a)*(t - a)*(z - a))/(a*(1 - a)))^(1/2)), diff( K, temp$(1) ) ) = 0

D[K, {u, 2}]+ D[K, {v, 2}]+ (D[K, {temp, 2}]/.temp-> (I*(Divide[(s - a)*(t - a)*(z - a),a*(1 - a)])^(1/2)))+Divide[2*\[Gamma]- 1,u]*D[K, u]+Divide[2*\[Delta]- 1,v]*D[K, v]+Divide[2*\[Epsilon]- 1,I*(Divide[(s - a)*(t - a)*(z - a),a*(1 - a)])^(1/2)]*(D[K, {temp, 1}]/.temp-> (I*(Divide[(s - a)*(t - a)*(z - a),a*(1 - a)])^(1/2))) == 0
Successful Successful - -
31.10#Ex7 u = r ⁢ cos ⁡ θ 𝑢 𝑟 𝜃 {\displaystyle{\displaystyle u=r\cos\theta}}
u = r\cos@@{\theta}		 

u = r*cos(theta)

u == r*Cos[\[Theta]]
Failure Failure
Failed [300 / 300]
Result: 1.961839932-.954243254e-1*I
Test Values: {r = -3/2, theta = 1/2*3^(1/2)+1/2*I, u = 1/2*3^(1/2)+1/2*I}


Result: .5958145280+.2706010786*I
Test Values: {r = -3/2, theta = 1/2*3^(1/2)+1/2*I, u = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.9618399323702764, -0.09542432534354878]
Test Values: {Rule[r, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[2.7076736790806044, 1.2036130644027554]
Test Values: {Rule[r, -1.5], Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
31.10#Ex8 v = r ⁢ sin ⁡ θ ⁢ sin ⁡ ϕ 𝑣 𝑟 𝜃 italic-ϕ {\displaystyle{\displaystyle v=r\sin\theta\sin\phi}}
v = r\sin@@{\theta}\sin@@{\phi}		 

v = r*sin(theta)*sin(phi)

v == r*Sin[\[Theta]]*Sin[\[Phi]]
Failure Failure
Failed [300 / 300]
Result: 1.801839169+1.369966168*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -3/2, theta = 1/2*3^(1/2)+1/2*I, v = 1/2*3^(1/2)+1/2*I}


Result: .4358137648+1.735991572*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -3/2, theta = 1/2*3^(1/2)+1/2*I, v = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[1.801839167885118, 1.3699661685131752]
Test Values: {Rule[r, -1.5], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.4330012446224153, 1.2666732793219693]
Test Values: {Rule[r, -1.5], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
31.10#Ex9 w = r ⁢ sin ⁡ θ ⁢ cos ⁡ ϕ 𝑤 𝑟 𝜃 italic-ϕ {\displaystyle{\displaystyle w=r\sin\theta\cos\phi}}
w = r\sin@@{\theta}\cos@@{\phi}		 

(I*(((s - a)*(t - a)*(z - a))/(a*(1 - a)))^(1/2)) = r*sin(theta)*cos(phi)

(I*(Divide[(s - a)*(t - a)*(z - a),a*(1 - a)])^(1/2)) == r*Sin[\[Theta]]*Cos[\[Phi]]
Failure Failure Manual Skip! Skipped - Because timed out
31.10.E21 ∂ 2 ⁡ 𝒦 ∂ ⁡ r 2 + 2 ⁢ ( γ + δ + ϵ ) - 1 r ⁢ ∂ ⁡ 𝒦 ∂ ⁡ r + 1 r 2 ⁢ ∂ 2 ⁡ 𝒦 ∂ ⁡ θ 2 + ( 2 ⁢ ( δ + ϵ ) - 1 ) ⁢ cot ⁡ θ - ( 2 ⁢ γ - 1 ) ⁢ tan ⁡ θ r 2 ⁢ ∂ ⁡ 𝒦 ∂ ⁡ θ + 1 r 2 ⁢ sin 2 ⁡ θ ⁢ ∂ 2 ⁡ 𝒦 ∂ ⁡ ϕ 2 + ( 2 ⁢ δ - 1 ) ⁢ cot ⁡ ϕ - ( 2 ⁢ ϵ - 1 ) ⁢ tan ⁡ ϕ r 2 ⁢ sin 2 ⁡ θ ⁢ ∂ ⁡ 𝒦 ∂ ⁡ ϕ = 0 partial-derivative 𝒦 𝑟 2 2 𝛾 𝛿 italic-ϵ 1 𝑟 partial-derivative 𝒦 𝑟 1 superscript 𝑟 2 partial-derivative 𝒦 𝜃 2 2 𝛿 italic-ϵ 1 𝜃 2 𝛾 1 𝜃 superscript 𝑟 2 partial-derivative 𝒦 𝜃 1 superscript 𝑟 2 2 𝜃 partial-derivative 𝒦 italic-ϕ 2 2 𝛿 1 italic-ϕ 2 italic-ϵ 1 italic-ϕ superscript 𝑟 2 2 𝜃 partial-derivative 𝒦 italic-ϕ 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}\mathcal{K}}{{\partial r}^{2}}% +\frac{2(\gamma+\delta+\epsilon)-1}{r}\frac{\partial\mathcal{K}}{\partial r}+% \frac{1}{r^{2}}\frac{{\partial}^{2}\mathcal{K}}{{\partial\theta}^{2}}+\frac{(2% (\delta+\epsilon)-1)\cot\theta-(2\gamma-1)\tan\theta}{r^{2}}\frac{\partial% \mathcal{K}}{\partial\theta}+\frac{1}{r^{2}{\sin^{2}}\theta}\frac{{\partial}^{% 2}\mathcal{K}}{{\partial\phi}^{2}}+\frac{(2\delta-1)\cot\phi-(2\epsilon-1)\tan% \phi}{r^{2}{\sin^{2}}\theta}\frac{\partial\mathcal{K}}{\partial\phi}=0}}
\pderiv[2]{\mathcal{K}}{r}+\frac{2(\gamma+\delta+\epsilon)-1}{r}\pderiv{\mathcal{K}}{r}+\frac{1}{r^{2}}\pderiv[2]{\mathcal{K}}{\theta}+\frac{(2(\delta+\epsilon)-1)\cot@@{\theta}-(2\gamma-1)\tan@@{\theta}}{r^{2}}\pderiv{\mathcal{K}}{\theta}+\frac{1}{r^{2}\sin^{2}@@{\theta}}\pderiv[2]{\mathcal{K}}{\phi}+\frac{(2\delta-1)\cot@@{\phi}-(2\epsilon-1)\tan@@{\phi}}{r^{2}\sin^{2}@@{\theta}}\pderiv{\mathcal{K}}{\phi} = 0		 

diff(K, [r$(2)])+(2*(gamma + delta + epsilon)- 1)/(r)*diff(K, r)+(1)/((r)^(2))*diff(K, [theta$(2)])+((2*(delta + epsilon)- 1)*cot(theta)-(2*gamma - 1)*tan(theta))/((r)^(2))*diff(K, theta)+(1)/((r)^(2)* (sin(theta))^(2))*diff(K, [phi$(2)])+((2*delta - 1)*cot(phi)-(2*epsilon - 1)*tan(phi))/((r)^(2)* (sin(theta))^(2))*diff(K, phi) = 0

D[K, {r, 2}]+Divide[2*(\[Gamma]+ \[Delta]+ \[Epsilon])- 1,r]*D[K, r]+Divide[1,(r)^(2)]*D[K, {\[Theta], 2}]+Divide[(2*(\[Delta]+ \[Epsilon])- 1)*Cot[\[Theta]]-(2*\[Gamma]- 1)*Tan[\[Theta]],(r)^(2)]*D[K, \[Theta]]+Divide[1,(r)^(2)* (Sin[\[Theta]])^(2)]*D[K, {\[Phi], 2}]+Divide[(2*\[Delta]- 1)*Cot[\[Phi]]-(2*\[Epsilon]- 1)*Tan[\[Phi]],(r)^(2)* (Sin[\[Theta]])^(2)]*D[K, \[Phi]] == 0
Successful Successful - Successful [Tested: 300]
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