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21: 22.7 Landen Transformations

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22.7.1 $$k_1=\frac{1-k^{\prime }}{1+k^{\prime }},$$

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Defines:

$k_1$: change of variable (locally)

Symbols:

$k^{\prime }$: complementary modulus

A&S Ref:

16.12.1

Referenced by:

§22.20(iii)

Permalink:

http://dlmf.nist.gov/22.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.7(i), §22.7 and Ch.22

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22: 28.11 Expansions in Series of Mathieu Functions

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28.11.1 $$f(z)={\alpha }_0{\mathrm{ce}}_0(z,q)+\sum _{n=1}^{\mathrm{\infty }}\left({\alpha }_n{\mathrm{ce}}_n(z,q)+{\beta }_n{\mathrm{se}}_n(z,q)\right),$$

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Defines:

$f(z)$: function (locally)

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{se}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, ${\alpha }_n$ and ${\beta }_n$

Referenced by:

§28.11

Permalink:

http://dlmf.nist.gov/28.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.11 and Ch.28

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23: 31.2 Differential Equations

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31.2.3 $$\frac{d^{2}W}{{dz}^2}=\left(\frac{A}{z}+\frac{B}{z-1}+\frac{C}{z-a}+\frac{D}{z^2}+\frac{E}{{(z-1)}^2}+\frac{F}{{(z-a)}^2}\right)W,$$ $A+B+C=0$,

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Defines:

$W(z)$: solution (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: complex parameter, $A$ , $B$ , $C$ , $D$ , $E$ and $F$

Permalink:

http://dlmf.nist.gov/31.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(ii), §31.2 and Ch.31

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31.2.11 $$d^{2}W/{d\xi }^2+\left(H+b_0\mathrm{\wp }\left(\xi \right)+b_1\mathrm{\wp }\left(\xi +{\omega }_1\right)+b_2\mathrm{\wp }\left(\xi +{\omega }_2\right)+b_3\mathrm{\wp }\left(\xi +{\omega }_3\right)\right)W=0,$$

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Defines:

$W(\xi )$: solution (locally)

Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${\omega }_j$: nonzero complex number, $𝕃$: lattice, $b_j$: coefficients and $H$

Permalink:

http://dlmf.nist.gov/31.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

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24: 31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►For an infinite set of discrete values $q_m$, $m=0,1,2,\mathrm{\dots }$, of the accessory parameter $q$, the function $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is analytic at $z=1$, and hence also throughout the disk . …

25: 33.8 Continued Fractions

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26: 19.11 Addition Theorems

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19.11.14 $$\sin \psi =(\sin 2\theta )\mathrm{\Delta }(\theta )/(1-k^2{\sin }^4\theta ),$$

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Defines:

$\psi$: angle (locally)

Symbols:

$\sin z$: sine function, $k$: real or complex modulus, $\theta$: amplitude and $\mathrm{\Delta }(\theta )$

Permalink:

http://dlmf.nist.gov/19.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

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27: 18.14 Inequalities

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§18.14(iii) Local Maxima and Minima

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Jacobi

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Laguerre

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28: 18.22 Hahn Class: Recurrence Relations and Differences

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29: 5.8 Infinite Products

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5.8.4 $$\sum _{k=1}^m{a}_k=\sum _{k=1}^m{b}_k,$$

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Defines:

$a_k$: coefficient (locally) and $b_k$: coefficient (locally)

Symbols:

$m$: nonnegative integer and $k$: nonnegative integer

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

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30: 12.4 Power-Series Expansions

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12.4.3 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(1+(a+\frac{1}{2})\frac{z^2}{2!}+(a+\frac{1}{2})(a+\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

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Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.1 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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12.4.4 $$u_2(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(z+(a+\frac{3}{2})\frac{z^3}{3!}+(a+\frac{3}{2})(a+\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

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Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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