# other forms

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## 1—10 of 129 matching pages

##### 3: Guide to Searching the DLMF
Note that the first form may match other functions $K$ than the Bessel $K$ function, so if you are sure you want Bessel $K$, you might as well enter one of the other 3 forms. …
##### 4: Mathematical Introduction
Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions (§14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). … For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed. …
##### 5: 18.2 General Orthogonal Polynomials
If polynomials $p_{n}(x)$ are generated by recurrence relation (18.2.8) under assumption of inequality (18.2.9_5) (or similarly for the other three forms) then the $p_{n}(x)$ are orthogonal by Favard’s theorem, see §18.2(viii), in that the existence of a bounded non-decreasing function $\alpha(x)$ on $(a,b)$ yielding the orthogonality realtion (18.2.4_5) is guaranteed. …
##### 6: 2.11 Remainder Terms; Stokes Phenomenon
In the transition through $\theta=\pi$, $\operatorname{erfc}\left(\sqrt{\frac{1}{2}\rho}\;c(\theta)\right)$ changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case $\rho=100$. …
##### 7: 3.6 Linear Difference Equations
For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6). …
##### 9: 18.34 Bessel Polynomials
18.34.8 $\lim_{\alpha\to\infty}\frac{P^{(\alpha,a-\alpha-2)}_{n}\left(1+\alpha x\right)% }{P^{(\alpha,a-\alpha-2)}_{n}\left(1\right)}=y_{n}\left(x;a\right).$
##### 10: 4.37 Inverse Hyperbolic Functions
$\operatorname{Arcsinh}z$ and $\operatorname{Arccsch}z$ have branch points at $z=\pm i$; the other four functions have branch points at $z=\pm 1$. …
###### Other Inverse Functions
For the corresponding results for $\operatorname{arccsch}z$, $\operatorname{arcsech}z$, and $\operatorname{arccoth}z$, use (4.37.7)–(4.37.9); compare §4.23(iv). …