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Lagrange inversion theorem

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1: 4.37 Inverse Hyperbolic Functions
§4.37 Inverse Hyperbolic Functions
Inverse Hyperbolic Sine
Inverse Hyperbolic Cosine
Inverse Hyperbolic Tangent
Other Inverse Functions
2: 4.23 Inverse Trigonometric Functions
§4.23 Inverse Trigonometric Functions
Inverse Sine
Inverse Cosine
Inverse Tangent
Other Inverse Functions
3: 22.15 Inverse Functions
§22.15 Inverse Functions
§22.15(i) Definitions
Each of these inverse functions is multivalued. The principal values satisfy …
4: 3.3 Interpolation
§3.3 Interpolation
§3.3(i) Lagrange Interpolation
With an error term the Lagrange interpolation formula for f is given by …
§3.3(ii) Lagrange Interpolation with Equally-Spaced Nodes
§3.3(iii) Divided Differences
5: 1.10 Functions of a Complex Variable
Picard’s Theorem
§1.10(iv) Residue Theorem
Rouché’s Theorem
Lagrange Inversion Theorem
Extended Inversion Theorem
6: 2.2 Transcendental Equations
where F 0 = f 0 and s F s ( s 1 ) is the coefficient of x - 1 in the asymptotic expansion of ( f ( x ) ) s (Lagrange’s formula for the reversion of series). …
7: 28.27 Addition Theorems
§28.27 Addition Theorems
Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
8: Bibliography D
  • S. C. Dhar (1940) Note on the addition theorem of parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 29–30.
  • A. R. DiDonato and A. H. Morris (1986) Computation of the incomplete gamma function ratios and their inverses. ACM Trans. Math. Software 12 (4), pp. 377–393.
  • H. Ding, K. I. Gross, and D. St. P. Richards (1996) Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.
  • J. Dougall (1907) On Vandermonde’s theorem, and some more general expansions. Proc. Edinburgh Math. Soc. 25, pp. 114–132.
  • J. J. Duistermaat (1974) Oscillatory integrals, Lagrange immersions and unfolding of singularities. Comm. Pure Appl. Math. 27, pp. 207–281.
  • 9: 4.38 Inverse Hyperbolic Functions: Further Properties
    §4.38 Inverse Hyperbolic Functions: Further Properties
    §4.38(i) Power Series
    §4.38(ii) Derivatives
    §4.38(iii) Addition Formulas
    4.38.19 Arctanh u ± Arccoth v = Arctanh ( u v ± 1 v ± u ) = Arccoth ( v ± u u v ± 1 ) .
    10: 22.18 Mathematical Applications
    Lemniscate
    Inversely: … See Akhiezer (1990, Chapter 8) and McKean and Moll (1999, Chapter 2) for discussions of the inverse mapping. …
    §22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem