# essential

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## 11—15 of 15 matching pages

##### 11: Bibliography K

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Essential singularity of the Painlevé function of the second kind and the nonlinear Stokes phenomenon.
Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.
(LOMI) 187, pp. 139–170 (Russian).
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##### 12: 2.4 Contour Integrals

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►However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential.
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##### 13: 19.36 Methods of Computation

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►All cases of ${R}_{F}$, ${R}_{C}$, ${R}_{J}$, and ${R}_{D}$ are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)).
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##### 14: 32.10 Special Function Solutions

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►The solution (32.10.34) is an essentially transcendental function of both constants of integration since ${\text{P}}_{\text{VI}}$ with $\alpha =\beta =\gamma =0$ and $\delta =\frac{1}{2}$ does not admit an algebraic first integral of the form $P(z,w,{w}^{\prime},C)=0$, with $C$ a constant.
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##### 15: 9.13 Generalized Airy Functions

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►The function on the right-hand side is recessive in the sector $-(2j-1)\pi /m\le \mathrm{ph}z\le (2j+1)\pi /m$, and is therefore an essential member of any numerically satisfactory pair of solutions in this region.
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