special values

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1: 18.6 Symmetry, Special Values, and Limits to Monomials

§18.6 Symmetry, Special Values, and Limits to Monomials

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§18.6(i) Symmetry and Special Values

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Table 18.6.1: Classical OP’s: symmetry and special values.

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$p_n(x)$	$p_n(-x)$	$p_n(1)$	$p_{2n}(0)$	$p_{2n+1}^{\prime }(0)$
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2: 4.31 Special Values and Limits

§4.31 Special Values and Limits

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3: 4.4 Special Values and Limits

§4.4 Special Values and Limits

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§4.4(i) Logarithms

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§4.4(ii) Powers

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4: 4.17 Special Values and Limits

§4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

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$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
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5: 23.17 Elementary Properties

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§23.17(i) Special Values

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6: 22.5 Special Values

§22.5 Special Values

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§22.5(i) Special Values of $z$

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Table 22.5.1: Jacobian elliptic function values, together with derivatives or residues, for special values of the variable.

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	$z$
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Table 22.5.2: Other special values of Jacobian elliptic functions.

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	$z$
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7: 5.4 Special Values and Extrema

§5.4 Special Values and Extrema

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§5.4(ii) Psi Function

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5.4.19 $$\psi \left(\frac{p}{q}\right)=-\gamma -\ln q-\frac{\pi }{2}\cot \left(\frac{\pi p}{q}\right)+\frac{1}{2}\sum _{k=1}^{q-1}\cos \left(\frac{2\pi kp}{q}\right)\ln \left(2-2\cos \left(\frac{2\pi k}{q}\right)\right).$$

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Symbols:

$\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, $q$: real or complex variable and $k$: nonnegative integer

Referenced by:

§5.19(i), §5.4(ii)

Permalink:

http://dlmf.nist.gov/5.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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8: 8.4 Special Values

§8.4 Special Values

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9: 26.14 Permutations: Order Notation

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§26.14(iv) Special Values

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10: 15.17 Mathematical Applications

… ►For special values of $\alpha$ and $\beta$ there are many group-theoretic interpretations. …