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A general group element for the fundamental representation of SU(3) is expressed as a second order polynomial in the hermitian generating matrix H, with coefficients consisting of elementary trigonometric functions dependent on the sole invariant det(H), in addition to the group parameter.
Y Lehrer, “On functions of matrices” Rendiconti del Circolo Matematico di Palermo 6 (1957) 103-108 ; Y Lehrer–Ilamed, “On the direct calculations of the representations of the three-dimensional pure rotation group” Proc.Camb.Phil.Soc. 60 (1964) 61–66 (especially see Remark (1), Eqn(10))
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M Gell-Mann and Y Ne’eman, The Eightfold Way, W A Benjamin (1964) . Also see https://en.wikipedia.org/wiki/Gell-Mann_matrices
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A J MacFarlane, A Sudbery, and P H Weisz, “On Gell-Mann’s λ \lambda -Matrices, d d - and f f -Tensors, Octets, and Parametrizations of SU(3)” Commun.Math.Phys. 11 (1968) 77-90
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S P Rosen, “Finite Transformations in Various Representations of SU(3)” J.Math.Phys. 12 (1971) 673-681
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D Kusnezov, “Exact matrix expansions for group elements of SU(N)” J.Math.Phys. 36 (1995) 898-906
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A Laufer, “The exponential map of GL(N)” J.Phys.A:Math.Gen. 30 (1997) 5455
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R W D Nickalls, “Viète, Descartes and the cubic equation” Mathematical Gazette 90 (2006) 203–208 . Also see https://en.wikipedia.org/wiki/Cubic_function#Three_real_roots
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The Collected Mathematical Papers of Arthur Cayley
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The general method to express any analytic matrix function of a finite, diagonalizable matrix as a polynomial in the matrix, through the use of projection matrices, is due to J J Sylvester, Phil.Mag. 16 (1883) 267-269 . Also see https://en.wikipedia.org/wiki/Sylvester’s_formula
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M X He and P E Ricci, “On Taylor’s formula for the resolvent of a complex matrix” Computers and Mathematics with Applications 56 (2008) 2285–2288
2008
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T L Curtright, “More on Rotations as Spin Matrix Polynomials” e-Print: arXiv:1506.04648 [math-ph]
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https://en.wikipedia.org/wiki/Laplace_transform#Inverse_Laplace_transform
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