relation to Whittaker equation

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31—32 of 32 matching pages

31: 18.15 Asymptotic Approximations

… ►The case

M=1

of (18.15.1) goes back to Darboux. … ►Then as

n\rightarrow\infty

, …For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term in a related expansion see Wong and Zhao (2003). …These expansions are in terms of Whittaker functions (§13.14). … ►With

\mu=\sqrt{2n+1}

the expansions in Chapter 12 are for the parabolic cylinder function

U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)

, which is related to the Hermite polynomials via …

32: 22.15 Inverse Functions

… ►The inverse Jacobian elliptic functions can be defined in an analogous manner to the inverse trigonometric functions (§4.23). With real variables, the solutions of the equations … ►Equations (22.15.1) and (22.15.4), for

\operatorname{arcsn}\left(x,k\right)

, are equivalent to (22.15.12) and also to … ►

§22.15(ii) Representations as Elliptic Integrals

…