transcendental functions

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11: 19.20 Special Cases

… ►In this subsection, and also §§19.20(ii)–19.20(v), the variables of all $R$-functions satisfy the constraints specified in §19.16(i) unless other conditions are stated. … ► Schneider that this is a transcendental number. … ►

19.20.5 $$2R_G(x,y,y)=y{R}_C(x,y)+\sqrt{x}.$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_G(x,y,z)$: symmetric elliptic integral of second kind

Referenced by:

§19.20(ii)

Permalink:

http://dlmf.nist.gov/19.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(ii), §19.20 and Ch.19

… ►When the variables are real and distinct, the various cases of $R_J(x,y,z,p)$ are called circular (hyperbolic) cases if $(p-x)(p-y)(p-z)$ is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. … ► Schneider that this is a transcendental number. …

12: 30.3 Eigenvalues

§30.3 Eigenvalues

… ►With $\mu =m=0,1,2,\mathrm{\dots }$, the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are solutions of Equation (30.2.1) which are bounded on $(-1,1)$, or equivalently, which are of the form ${(1-x^2)}^{\frac{1}{2}m}g(x)$ where $g(z)$ is an entire function of $z$. … ►The eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are analytic functions of the real variable ${\gamma }^2$ and satisfy … ►

§30.3(iii) Transcendental Equation

… ►

§30.3(iv) Power-Series Expansion

…