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1: 26.14 Permutations: Order Notation

§26.14 Permutations: Order Notation

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2: 11.1 Special Notation

§11.1 Special Notation

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3: 14.1 Special Notation

§14.1 Special Notation

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4: 10.1 Special Notation

… ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

5: 10.3 Graphics

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§10.3(i) Real Order and Variable

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§10.3(ii) Real Order, Complex Variable

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§10.3(iii) Imaginary Order, Real Variable

►For the notation see §10.24. … ►

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Figure 10.3.19: ${\tilde{J}}_5\left(x\right)$, ${\tilde{Y}}_5\left(x\right)$, $0.01\le x\le 10$. Magnify

6: 10.75 Tables

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MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function $J_0\left(z\right)-\mathrm{i}{J}_1\left(z\right)$, 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).

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§10.75(viii) Modified Bessel Functions of Imaginary or Complex Order

►For the notation see §10.45. ►

Žurina and Karmazina (1967) tabulates ${\tilde{K}}_{\nu }\left(x\right)$ for $\nu =0.01(.01)10$, $x=0.1(.1)10.2$, 7S.

… ►For the notation see §10.58. …

7: 10.26 Graphics

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§10.26(i) Real Order and Variable

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§10.26(ii) Real Order, Complex Variable

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§10.26(iii) Imaginary Order, Real Variable

►For the notation, see §10.45. ►

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Figure 10.26.7: ${\tilde{I}}_{1/2}\left(x\right)$, ${\tilde{K}}_{1/2}\left(x\right)$, $0.01\le x\le 3$. Magnify

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8: 35.1 Special Notation

§35.1 Special Notation

►(For other notation see Notation for the Special Functions.) ►All matrices are of order $m\times m$, unless specified otherwise. … ►An alternative notation for the multivariate gamma function is ${\mathrm{\Pi }}_m(a)={\mathrm{\Gamma }}_m\left(a+\frac{1}{2}(m+1)\right)$ (Herz (1955, p. 480)). Related notations for the Bessel functions are ${𝒥}_{\nu +\frac{1}{2}(m+1)}(𝐓)=A_{\nu }\left(𝐓\right)/A_{\nu }\left(𝟎\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_m(0,\mathrm{\dots },0,\nu |𝐒,𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Terras (1988, pp. 49–64)), and ${𝒦}_{\nu }(𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Faraut and Korányi (1994, pp. 357–358)).

9: 26.13 Permutations: Cycle Notation

… ►In cycle notation, the elements in each cycle are put inside parentheses, ordered so that $\sigma (j)$ immediately follows $j$ or, if $j$ is the last listed element of the cycle, then $\sigma (j)$ is the first element of the cycle. …

10: 28.12 Definitions and Basic Properties

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§28.12(i) Eigenvalues ${\lambda }_{\nu +2n}\left(q\right)$

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§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►For $q=0$, … ► … ►