inversion formulas
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21: 6.14 Integrals
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6.14.3
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22: 6.18 Methods of Computation
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►For large and , expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available.
…Also, other ranges of can be covered by use of the continuation formulas of §6.4.
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►For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998).
For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968).
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23: 16.15 Integral Representations and Integrals
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►For these and other formulas, including double Mellin–Barnes integrals, see Erdélyi et al. (1953a, §5.8).
…For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).
24: Bibliography C
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A quadrature formula for the Hankel transform.
Numer. Algorithms 9 (2), pp. 343–354.
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The inverse of the error function.
Pacific J. Math. 13 (2), pp. 459–470.
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Power series for inverse Jacobian elliptic functions.
Math. Comp. 77 (263), pp. 1615–1621.
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Inverse Acoustic and Electromagnetic Scattering Theory.
2nd edition, Applied Mathematical Sciences, Vol. 93, Springer-Verlag, Berlin.
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An encyclopaedia of cubature formulas.
J. Complexity 19 (3), pp. 445–453.
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25: Bibliography S
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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms.
Math. Tables Aids Comput. 9 (52), pp. 164–177.
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Calculation of the gamma function by Stirling’s formula.
Math. Comp. 25 (114), pp. 317–322.
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On the calculation of the inverse of the error function.
Math. Comp. 22 (101), pp. 144–158.
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Gaussian Quadrature Formulas.
Prentice-Hall Inc., Englewood Cliffs, N.J..
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26: 19.26 Addition Theorems
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§19.26(i) General Formulas
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19.26.11
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§19.26(iii) Duplication Formulas
… ►The equations inverse to and the two other equations obtained by permuting (see (19.26.19)) are … ►
19.26.25
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27: Bibliography
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Solitons, Nonlinear Evolution Equations and Inverse Scattering.
London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge.
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Solitons and the Inverse Scattering Transform.
SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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Special value of the hypergeometric function and connection formulae among asymptotic expansions.
J. Indian Math. Soc. (N.S.) 51, pp. 161–221.
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Formulas for higher derivatives of the Riemann zeta function.
Math. Comp. 44 (169), pp. 223–232.
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Bernoulli’s power-sum formulas revisited.
Math. Gaz. 90 (518), pp. 276–279.
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28: 1.2 Elementary Algebra
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►where is defined by the Leibniz formula
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The Inverse
►If det() , has a unique inverse, , such that … ►has a unique solution, . … ►Formula (1.2.77) is more generally valid for all square matrices , not necessarily non-defective, see Hall (2015, Thm 2.12).29: 22.20 Methods of Computation
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22.20.4
►and the inverse sine has its principal value (§4.23(ii)).
…This formula for becomes unstable near .
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