瓦贝希学院学位证〖办证V信ATV1819〗el3

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1: 19.39 Software

… ►For research software see Bulirsch (1965b, function $\mathrm{el2}$), Bulirsch (1969b, function $\mathrm{el3}$), Jefferson (1961), and Neuman (1969a, functions $E(\varphi ,k)$ and $\mathrm{\Pi }(\varphi ,k^2,k)$). …

2: 19.1 Special Notation

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$\mathrm{el3}(x,k_c,p),$

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3: 19.2 Definitions

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19.2.16 $$\mathrm{el3}(x,k_c,p)={\int }_0^{\arctan x}\frac{d\theta }{({\cos }^2\theta +p{\sin }^2\theta )\sqrt{{\cos }^2\theta +k_c^2{\sin }^2\theta }}=\mathrm{\Pi }(\arctan x,1-p,k),$$ $x^2\ne -1/p$.

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Defines:

$\mathrm{el3}(x,k_c,p)$: Bulirsch’s incomplete elliptic integral of the third kind

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\int$: integral, $\arctan z$: arctangent function, $\sin z$: sine function, $k$: real or complex modulus, $p$: real parameter not equal to zero, $k_c$: real or complex variable not equal to zero and $x$: real or complex variable

Permalink:

http://dlmf.nist.gov/19.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iii), §19.2 and Ch.19

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4: 19.36 Methods of Computation

… ►Bulirsch (1969a, b) extend Bartky’s transformation to $\mathrm{el3}(x,k_c,p)$ by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. …

5: 19.25 Relations to Other Functions

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19.25.22 $$\mathrm{el3}(x,k_c,p)=\mathrm{el1}(x,k_c)+\frac{1}{3}(1-p)R_J(r,r+k_c^2,r+1,r+p).$$

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Symbols:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind, $\mathrm{el3}(x,k_c,p)$: Bulirsch’s incomplete elliptic integral of the third kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $k$: real or complex modulus and $r$: inverse

Permalink:

http://dlmf.nist.gov/19.25.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(ii), §19.25 and Ch.19

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