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1: 1.13 Differential Equations

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§1.13(vii) Closed-Form Solutions

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§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms

►A standard form for second order ordinary differential equations with $x\in ℝ$, and with a real parameter $\lambda$, and real valued functions $p(x),q(x),$ and $\rho (x)$, with $p(x)$ and $\rho (x)$ positive, is …Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form … ►

Transformation to Liouville normal Form

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2: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

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§4.23(i) General Definitions

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§4.23(iv) Logarithmic Forms

… ►Care needs to be taken on the cuts, for example, if then $1/(x+\mathrm{i}0)=(1/x)-\mathrm{i}0$. … ►

§4.23(vii) Special Values and Interrelations

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3: 20 Theta Functions

Chapter 20 Theta Functions

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4: 26.3 Lattice Paths: Binomial Coefficients

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Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

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$m$	$n$
	…
6	1	6	15	20	15	6	1
…

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Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

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$m$	$n$
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3	1	4	10	20	35	56	84	120	165
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§26.3(ii) Generating Functions

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26.3.4 $$\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)x^m=\frac{1}{{(1-x)}^{n+1}},$$ .

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.1 (in slightly different form)

Permalink:

http://dlmf.nist.gov/26.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.3(ii), §26.3 and Ch.26

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§26.3(v) Limiting Form

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5: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

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§8.17(ii) Hypergeometric Representations

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§8.17(iii) Integral Representation

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§8.17(iv) Recurrence Relations

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§8.17(vi) Sums

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6: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

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§15.2(ii) Analytic Properties

… ►Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. … ►For comparison of $F(a,b;c;z)$ and $𝐅(a,b;c;z)$, with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7. …

7: 26.5 Lattice Paths: Catalan Numbers

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Table 26.5.1: Catalan numbers.

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$n$	$C\left(n\right)$	$n$	$C\left(n\right)$	$n$	$C\left(n\right)$
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6	132	13	7 42900	20	65641 20420

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§26.5(ii) Generating Function

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§26.5(iv) Limiting Forms

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26.5.7 $$\underset{n\to \mathrm{\infty }}{lim}\frac{C\left(n+1\right)}{C\left(n\right)}=4.$$

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Symbols:

$C\left(n\right)$: Catalan number and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.5(iv), §26.5 and Ch.26

8: 11.9 Lommel Functions

§11.9 Lommel Functions

… ►The inhomogeneous Bessel differential equation … ►the right-hand side being replaced by its limiting form when $\mu \pm \nu$ is an odd negative integer. … ► … ►

9: 5.2 Definitions

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§5.2(i) Gamma and Psi Functions

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Euler’s Integral

… ►It is a meromorphic function with no zeros, and with simple poles of residue ${(-1)}^n/n!$ at $z=-n$. … ►

5.2.2 $$\psi \left(z\right)={\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right),$$ $z\ne 0,-1,-2,\mathrm{\dots }$.

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Defines:

$\psi \left(z\right)$: psi (or digamma) function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $z$: complex variable

A&S Ref:

6.3.1

Permalink:

http://dlmf.nist.gov/5.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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5.2.3 $$\gamma =\underset{n\to \mathrm{\infty }}{lim}\left(1+\frac{1}{2}+\frac{1}{3}+\mathrm{\cdots }+\frac{1}{n}-\ln n\right)=\mathrm{0.57721\hspace{0.33em}56649\hspace{0.33em}01532\hspace{0.33em}86060}\mathrm{\dots }.$$

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Defines:

$\gamma$: Euler’s constant

Symbols:

$\ln z$: principal branch of logarithm function and $n$: nonnegative integer

A&S Ref:

6.1.3 (where the 10D value is given, and Table 1.1 where the 24D value is given.)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Referenced by:

§4.4(iii)

Permalink:

http://dlmf.nist.gov/5.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.2(ii), §5.2 and Ch.5

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10: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

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§25.11(i) Definition

… ►The Riemann zeta function is a special case: … ►

§25.11(ii) Graphics

… ►For an exponentially-improved form of (25.11.43) see Paris (2005b).