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Question: Better implementation of numeric integration containing HeunG functions

Question:Better implementation of numeric integration containing HeunG functions

digerdiga 390 Maple 16
algebra integral infinity student
February 01 2013
0

I have added a .mw file showing the problem. I wish to calculate the integral at the bottom of the post. unfortunately the function im using defined at he beginning of the file doesnt behave well for values k larger than 25/30...

The integral I need to evaluate therefore cant be evaluated if one wants to integrate from 0 to Infinity.

Also just integrating till 20 or so is not accurate enough...

Has anybody an idea about better implementation of the problem.mapleprimes.mw

yg := Determinant(subs(c = (1+s)/(1-s), q = 2*k^2*s/(1-s)+k*(1+9*s)/(2*(1-s))+k^2+2*k, a = k, b = k+1/2, g = 1/2, d = 2*k+3, s = 1/2, k = -1+I*sqrt((1/4)*x-1), t = 1/2, sqrt(x-4) = k, Wronskian([(1-t)^(I*sqrt((1/4)*x-1))*HeunG(c, q, a, b, g, d, t), (1-t)^(I*sqrt((1/4)*x-1))*exp(-I*ln(2)*sqrt(x-4))*HeunG(1-c, -q+a*b, a, b, d, g, 1-t)], t)))

-((1/2)^(((1/2)*I)*k))^2*HeunG(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)*exp(-I*ln(2)*k)*HeunGPrime(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)-((1/2)^(((1/2)*I)*k))^2*exp(-I*ln(2)*k)*HeunG(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)*HeunGPrime(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)

 (1)

deltag := argument(yg)

argument(-((1/2)^(((1/2)*I)*k))^2*HeunG(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)*exp(-I*ln(2)*k)*HeunGPrime(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)-((1/2)^(((1/2)*I)*k))^2*exp(-I*ln(2)*k)*HeunG(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)*HeunGPrime(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2))

 (2)

plot(deltag, k = -35 .. 35)

  

diffdeltag := Im(diff(ln(yg), k))

Im(((2*I)*((1/2)^(((1/2)*I)*k))^2*HeunG(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)*ln(2)*exp(-I*ln(2)*k)*HeunGPrime(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)-((1/2)^(((1/2)*I)*k))^2*(((3/4)*I-(3/2)*k)*(D[2](HeunG))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+((1/2)*I)*(D[3](HeunG))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+((1/2)*I)*(D[4](HeunG))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+I*(D[6](HeunG))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2))*exp(-I*ln(2)*k)*HeunGPrime(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)-((1/2)^(((1/2)*I)*k))^2*HeunG(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)*exp(-I*ln(2)*k)*((-(3/2)*I+k)*(D[2](HeunGPrime))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+((1/2)*I)*(D[3](HeunGPrime))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+((1/2)*I)*(D[4](HeunGPrime))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+I*(D[5](HeunGPrime))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2))+(2*I)*((1/2)^(((1/2)*I)*k))^2*ln(2)*exp(-I*ln(2)*k)*HeunG(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)*HeunGPrime(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)-((1/2)^(((1/2)*I)*k))^2*exp(-I*ln(2)*k)*((-(3/2)*I+k)*(D[2](HeunG))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+((1/2)*I)*(D[3](HeunG))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+((1/2)*I)*(D[4](HeunG))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)+I*(D[5](HeunG))(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2))*HeunGPrime(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)-((1/2)^(((1/2)*I)*k))^2*exp(-I*ln(2)*k)*HeunG(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)*(((3/4)*I-(3/2)*k)*(D[2](HeunGPrime))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+((1/2)*I)*(D[3](HeunGPrime))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+((1/2)*I)*(D[4](HeunGPrime))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)+I*(D[6](HeunGPrime))(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)))/(-((1/2)^(((1/2)*I)*k))^2*HeunG(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)*exp(-I*ln(2)*k)*HeunGPrime(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)-((1/2)^(((1/2)*I)*k))^2*exp(-I*ln(2)*k)*HeunG(-2, 5-((3/2)*I)*k+(1/2)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1+I*k, 1/2, 1/2)*HeunGPrime(3, -9/2+((3/4)*I)*k-(3/4)*k^2, -1+((1/2)*I)*k, -1/2+((1/2)*I)*k, 1/2, 1+I*k, 1/2)))

 (3)

plot(diffdeltag, k = -25 .. 25)

  

 

 

I wish to calculate the integral

 

 

int(VectorCalculus:-`*`(diffdeltag, ln(VectorCalculus:-`+`(k^2, 4))), k = 0 .. infinity)

Warning,  computation interrupted

  

 

 

or maybe more maple conform after partial integrating and resumming (delta+Pi/2) in the first and second term in order to be finite

 

int(VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`*`(VectorCalculus:-`+`(deltag, VectorCalculus:-`*`(Pi, 1/2)), 2), k), 1/VectorCalculus:-`+`(k^2, 4)), k = 0 .. infinity)

Warning,  computation interrupted

  

 

 

 

 

neglecting the first term of partial integration

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

with(VectorCalculus)

[`&x`, `*`, `+`, `-`, `.`, `<,>`, `<|>`, About, AddCoordinates, ArcLength, BasisFormat, Binormal, Compatibility, ConvertVector, CrossProduct, Curl, Curvature, D, Del, DirectionalDiff, Divergence, DotProduct, Flux, GetCoordinateParameters, GetCoordinates, GetNames, GetPVDescription, GetRootPoint, GetSpace, Gradient, Hessian, IsPositionVector, IsRootedVector, IsVectorField, Jacobian, Laplacian, LineInt, MapToBasis, Nabla, Norm, Normalize, PathInt, PlotPositionVector, PlotVector, PositionVector, PrincipalNormal, RadiusOfCurvature, RootedVector, ScalarPotential, SetCoordinateParameters, SetCoordinates, SpaceCurve, SurfaceInt, TNBFrame, Tangent, TangentLine, TangentPlane, TangentVector, Torsion, Vector, VectorField, VectorPotential, VectorSpace, Wronskian, diff, eval, evalVF, int, limit, series]

 (4)

with(LinearAlgebra)

[`&x`, Add, Adjoint, BackwardSubstitute, BandMatrix, Basis, BezoutMatrix, BidiagonalForm, BilinearForm, CARE, CharacteristicMatrix, CharacteristicPolynomial, Column, ColumnDimension, ColumnOperation, ColumnSpace, CompanionMatrix, ConditionNumber, ConstantMatrix, ConstantVector, Copy, CreatePermutation, CrossProduct, DARE, DeleteColumn, DeleteRow, Determinant, Diagonal, DiagonalMatrix, Dimension, Dimensions, DotProduct, EigenConditionNumbers, Eigenvalues, Eigenvectors, Equal, ForwardSubstitute, FrobeniusForm, GaussianElimination, GenerateEquations, GenerateMatrix, Generic, GetResultDataType, GetResultShape, GivensRotationMatrix, GramSchmidt, HankelMatrix, HermiteForm, HermitianTranspose, HessenbergForm, HilbertMatrix, HouseholderMatrix, IdentityMatrix, IntersectionBasis, IsDefinite, IsOrthogonal, IsSimilar, IsUnitary, JordanBlockMatrix, JordanForm, KroneckerProduct, LA_Main, LUDecomposition, LeastSquares, LinearSolve, LyapunovSolve, Map, Map2, MatrixAdd, MatrixExponential, MatrixFunction, MatrixInverse, MatrixMatrixMultiply, MatrixNorm, MatrixPower, MatrixScalarMultiply, MatrixVectorMultiply, MinimalPolynomial, Minor, Modular, Multiply, NoUserValue, Norm, Normalize, NullSpace, OuterProductMatrix, Permanent, Pivot, PopovForm, QRDecomposition, RandomMatrix, RandomVector, Rank, RationalCanonicalForm, ReducedRowEchelonForm, Row, RowDimension, RowOperation, RowSpace, ScalarMatrix, ScalarMultiply, ScalarVector, SchurForm, SingularValues, SmithForm, StronglyConnectedBlocks, SubMatrix, SubVector, SumBasis, SylvesterMatrix, SylvesterSolve, ToeplitzMatrix, Trace, Transpose, TridiagonalForm, UnitVector, VandermondeMatrix, VectorAdd, VectorAngle, VectorMatrixMultiply, VectorNorm, VectorScalarMultiply, ZeroMatrix, ZeroVector, Zip]

 (5)

``


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