In[]:=
θtest=RandomVariate[UniformDistribution[],10];Join[{{"θ (approx.)","{a, b}","{α, β, γ}","OP integral","closed form"}},Table[results[θ],{θ,θtest}]]//TableForm
Out[]//TableForm=
θ (approx.) | {a, b} | {α, β, γ} | OP integral | closed form | |||||
281π 1000 |
|
| 0.16188 | 0.16188+0. | |||||
287π 1000 |
|
| 0.171829 | 0.171829+0. | |||||
133π 1000 |
|
| 0.0440731 | 0.0440731+0. | |||||
39π 200 |
|
| 0.0677738 | 0.0677738+0. | |||||
11π 500 |
|
| 0.00137273 | 0.00137273+0. | |||||
77π 250 |
|
| 0.310676 | 0.310676+0. | |||||
63π 1000 |
|
| 0.00777194 | 0.00777194+0. | |||||
259π 1000 |
|
| 0.311701 | 0.311701+0. | |||||
103π 1000 |
|
| 0.0232981 | 0.0232981+0. | |||||
12π 125 |
|
| 0.0181041 | 0.0181041+0. |
In[]:=
θtest2=RandomVariate[UniformDistribution[],10];Table[results2[θ],{θ,Evaluate[θtest+π/2]}]//TableForm
Out[]//TableForm=
0.55879 | 0.55879 |
1.134 | 1.134 |
0.567512 | 0.567512 |
0.364606 | 0.364606 |
0.0387528 | 0.0387528 |
1.02902 | 1.02902 |
0.169472 | 0.169472 |
1.10111 | 1.10111 |
0.147161 | 0.147161 |
0.167568 | 0.167568 |
In[]:=
results[θ_]:=Module[{a=RandomReal[{2,10}],b,α,β,γ},b=RandomReal[{2,a}];α=a-b;β=a+b;γ=(β-α)/α; {Rationalize[Round[θ/π,.001]]π,{a,b},{α,β,γ},NIntegrate[Sin[x]/Sqrt[a-bSin[x]],{x,0,θ}], (2Sqrt[2])/Sqrt[-b] (EllipticE[ArcSin[Sqrt[b/(b-a)]],(b-a)/(2b)]-EllipticE[ArcSin[Sqrt[b/(b-a)(1-Sin[θ])]],(b-a)/(2b)])-2/Sqrt[a-b](EllipticF[π/4,(2b)/(b-a)]-EllipticF[π/4-θ/2,(2b)/(b-a)]) }]
results2[θ_]:=Module[{a=RandomReal[{2,10}],b,α,β,γ},b=RandomReal[{2,a}];α=a-b;β=a+b;γ=(β-α)/α; { NIntegrate[Sin[x]/Sqrt[a-bSin[x]],{x,π/2,θ}], 2/Sqrt[α]NIntegrate[2Sqrt[1-x^2]/Sqrt[1+γx^2]-1/(Sqrt[1-x^2]Sqrt[1+γx^2]),{x,0,Sqrt[(1-Sin[θ])/2]}] }]