compatibility%20condition

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1: 32.4 Isomonodromy Problems

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§32.4(i) Definition

… ►(32.4.3) is the compatibility condition of (32.4.1). … … ►The compatibility condition of (32.4.1) with … ►

2: Browsers

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3: 26.10 Integer Partitions: Other Restrictions

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26.10.1 $$p(𝒟,0)=p(𝒟k,0)=p(\in S,0)=1.$$

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Symbols:

$\in$: element of, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $k$: nonnegative integer and $S$: set

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http://dlmf.nist.gov/26.10.E1

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Annotations for §26.10(i), §26.10 and Ch.26

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Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from $A_{j,k}$.

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	$p(𝒟,n)$	$p(𝒟2,n)$	$p(𝒟2,\in T,n)$	$p({𝒟}^{\prime }3,n)$
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26.10.5 $$\sum _{n=0}^{\mathrm{\infty }}p(\in S,n)q^n=\prod _{j\in S}\frac{1}{1-q^j}.$$

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Symbols:

$\in$: element of, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer, $n$: nonnegative integer, $S$: set and $q$: parameter

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http://dlmf.nist.gov/26.10.E5

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Annotations for §26.10(ii), §26.10 and Ch.26

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26.10.6 $$p(𝒟,n)=\frac{1}{n}\sum _{t=1}^{n}p(𝒟,n-t)\sum _{\begin{array}{c}j|t\\ j\text{ odd}\end{array}}j,$$

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Symbols:

$p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer and $n$: nonnegative integer

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http://dlmf.nist.gov/26.10.E6

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Annotations for §26.10(iii), §26.10 and Ch.26

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26.10.12 $$p(𝒟,n)=p(𝒪,n),$$

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Symbols:

$p(\mathrm{condition},n)$: restricted number of partitions of $n$ and $n$: nonnegative integer

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http://dlmf.nist.gov/26.10.E12

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Annotations for §26.10(iv), §26.10 and Ch.26

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4: 20 Theta Functions

Chapter 20 Theta Functions

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5: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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Referenced by:

§33.22(iv).

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

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1st item.

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NJGRAF: An efficient program for calculation of general recoupling coefficients by graphical analysis, compatible with NJSYM. Comput. Phys. Comm. 50 (3), pp. 375–393.

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1st item.

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Double-precision Fortran code.

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1st item, 4th item.

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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Referenced by:

§10.73(iv).

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6: 3.8 Nonlinear Equations

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§3.8(vi) Conditioning of Zeros

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3.8.15 $$p(x)=(x-1)(x-2)\mathrm{\cdots }(x-20)$$

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http://dlmf.nist.gov/3.8.E15

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Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

►are well separated but extremely ill-conditioned. Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

7: 34.10 Zeros

… ►In a $\mathit{3}j$ symbol, if the three angular momenta $j_1,j_2,j_3$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3}j$ symbol is zero. Similarly the $\mathit{6}j$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3}j$ symbols in the summation. …However, the $\mathit{3}j$ and $\mathit{6}j$ symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …

8: 36.5 Stokes Sets

… ►The Stokes sets are defined by the exponential dominance condition: … ►

36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

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Symbols:

$x$: real parameter

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http://dlmf.nist.gov/36.5.E4

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Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

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Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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9: 10.73 Physical Applications

… ►and on separation of variables we obtain solutions of the form ${\mathrm{e}}^{\pm \mathrm{i}n\varphi }{\mathrm{e}}^{\pm \kappa z}{J}_n\left(\kappa r\right)$, from which a solution satisfying prescribed boundary conditions may be constructed. … ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►With the spherical harmonic $Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ defined as in §14.30(i), the solutions are of the form $f=g_{\mathrm{\ell }}(k\rho )Y_{\mathrm{\ell },m}\left(\theta ,\varphi \right)$ with $g_{\mathrm{\ell }}={𝗃}_{\mathrm{\ell }}$, ${𝗒}_{\mathrm{\ell }}$, ${𝗁}_{\mathrm{\ell }}^{(1)}$, or ${𝗁}_{\mathrm{\ell }}^{(2)}$, depending on the boundary conditions. …

10: 8 Incomplete Gamma and Related
Functions

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