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11: 10.37 Inequalities; Monotonicity

… ►If $\nu$ $(\ge 0)$ is fixed, then throughout the interval , $I_{\nu }\left(x\right)$ is positive and increasing, and $K_{\nu }\left(x\right)$ is positive and decreasing. ►If $x$ $(>0)$ is fixed, then throughout the interval , $I_{\nu }\left(x\right)$ is decreasing, and $K_{\nu }\left(x\right)$ is increasing. … ►

10.37.1

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Symbols:

$K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.37, §10.37, Erratum (V1.0.17) for Section 10.37

Permalink:

http://dlmf.nist.gov/10.37.E1

Encodings:

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See also:

Annotations for §10.37 and Ch.10

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12: 26.1 Special Notation

… ► ►►

$x$	real variable.
…

… ► ►►►

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$	binomial coefficient.
…
$S(n,k)$	Stirling numbers of the second kind.

… ►Other notations for $S(n,k)$, the Stirling numbers of the second kind, include ${𝒮}_n^{(k)}$ (Fort (1948)), ${𝔖}_n^k$ (Jordan (1939)), ${\sigma }_n^k$ (Moser and Wyman (1958b)), $\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B_{n-k}^{(-k)}$ (Milne-Thomson (1933)), $S_2(k,n-k)$ (Carlitz (1960), Gould (1960)), $\left\{\genfrac{}{}{0pt}{}{n}{k}\right\}$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

13: 19.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. … ►of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals …of the first, second, and third kinds, respectively. … ►The second set of main functions treated in this chapter is … ►The first three functions are incomplete integrals of the first, second, and third kinds, and the $\mathrm{cel}$ function includes complete integrals of all three kinds.

14: 32.12 Asymptotic Approximations for Complex Variables

… ►

§32.12(ii) Second Painlevé Equation

…

15: David M. Bressoud

… ► 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. …

16: 10.28 Wronskians and Cross-Products

… ►

10.28.2 $$𝒲\left\{K_{\nu }\left(z\right),I_{\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)K_{\nu +1}\left(z\right)+I_{\nu +1}\left(z\right)K_{\nu }\left(z\right)=1/z.$$

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Symbols:

$𝒲$: Wronskian, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.15

Permalink:

http://dlmf.nist.gov/10.28.E2

Encodings:

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See also:

Annotations for §10.28 and Ch.10

17: 10.42 Zeros

… ►Properties of the zeros of $I_{\nu }\left(z\right)$ and $K_{\nu }\left(z\right)$ may be deduced from those of $J_{\nu }\left(z\right)$ and $H_{\nu }^{(1)}\left(z\right)$, respectively, by application of the transformations (10.27.6) and (10.27.8). … ►The distribution of the zeros of $K_n\left(nz\right)$ in the sector $-\frac{3}{2}\pi \le \mathrm{ph}z\le \frac{1}{2}\pi$ in the cases $n=1,5,10$ is obtained on rotating Figures 10.21.2, 10.21.4, 10.21.6, respectively, through an angle $-\frac{1}{2}\pi$ so that in each case the cut lies along the positive imaginary axis. … ► $K_n\left(z\right)$ has no zeros in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi$; this result remains true when $n$ is replaced by any real number $\nu$. For the number of zeros of $K_{\nu }\left(z\right)$ in the sector $|\mathrm{ph}z|\le \pi$, when $\nu$ is real, see Watson (1944, pp. 511–513). ►For $z$-zeros of $K_{\nu }\left(z\right)$, with complex $\nu$, see Ferreira and Sesma (2008). …

18: 29.17 Other Solutions

… ►

§29.17(i) Second Solution

►If (29.2.1) admits a Lamé polynomial solution $E$, then a second linearly independent solution $F$ is given by …

19: 10.27 Connection Formulas

… ►Other solutions of (10.25.1) are $I_{-\nu }\left(z\right)$ and $K_{-\nu }\left(z\right)$. … ►

10.27.2 $$I_{-\nu }\left(z\right)=I_{\nu }\left(z\right)+(2/\pi )\sin \left(\nu \pi \right)K_{\nu }\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.2

Permalink:

http://dlmf.nist.gov/10.27.E2

Encodings:

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See also:

Annotations for §10.27 and Ch.10

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10.27.3 $$K_{-\nu }\left(z\right)=K_{\nu }\left(z\right).$$

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Symbols:

$K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.6

Referenced by:

§10.25(iii), §10.30(i), §10.31, §10.47(ii)

Permalink:

http://dlmf.nist.gov/10.27.E3

Encodings:

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See also:

Annotations for §10.27 and Ch.10

… ►

10.27.9 $$\pi \mathrm{i}{J}_{\nu }\left(z\right)={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{-\pi \mathrm{i}/2}\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\pi \mathrm{i}/2}\right),$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.67(i)

Permalink:

http://dlmf.nist.gov/10.27.E9

Encodings:

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See also:

Annotations for §10.27 and Ch.10

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10.27.10 $$-\pi Y_{\nu }\left(z\right)={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\pi \mathrm{i}/2}\right),$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi$.

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.27.E10

Encodings:

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See also:

Annotations for §10.27 and Ch.10

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20: 19.6 Special Cases

… ►

$E\left(1\right)=E^{\prime }\left(0\right)=1.$

… ►Exact values of $K\left(k\right)$ and $E\left(k\right)$ for various special values of $k$ are given in Byrd and Friedman (1971, 111.10 and 111.11) and Cooper et al. (2006). … ►

§19.6(iii) $E(\varphi ,k)$

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$E(0,k)=0,$

►

$E(\varphi ,0)=\varphi ,$

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