degrees two, three, four

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1: 1.11 Zeros of Polynomials

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§1.11(iii) Polynomials of Degrees Two, Three, and Four

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2: Bille C. Carlson

… ►Carlson, in 1981, and is survived by his companion, Jody Stadler, two children, Marian Carlson and John Carlson, and four grandchildren. … ►After the war he returned to Harvard and completed Bachelor’s and Master’s degrees in physics and mathematics. He then went to Oxford as a Rhodes Scholar and completed a doctoral degree in physics. After four years in the Physics Department at Princeton, he went to the Ames Laboratory and Iowa State University in 1954, where he was a Professor in the Physics and Mathematics Departments. … ►This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

3: 14.15 Uniform Asymptotic Approximations

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14.15.3 $${\mathit{Q}}_{\nu }^{\mu }\left(x\right)=\frac{1}{{\mu }^{\nu +(1/2)}}{\left(\frac{\pi u}{2}\right)}^{1/2}{I}_{\nu +\frac{1}{2}}\left(\mu u\right)\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $x$: real variable, $\mu$: general order, $\nu$: general degree and $u$

Referenced by:

§14.15(i)

Permalink:

http://dlmf.nist.gov/14.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(i), §14.15 and Ch.14

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14.15.5

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Symbols:

$\mu$: general order, $\nu$: general degree and $\alpha$

Referenced by:

§14.15(ii)

Permalink:

http://dlmf.nist.gov/14.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(ii), §14.15 and Ch.14

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§14.15(iii) Large $\nu$, Fixed $\mu$

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14.15.19

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Symbols:

$\mu$: general order, $\nu$: general degree and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iv), §14.15 and Ch.14

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4: 22.19 Physical Applications

… ►The periodicity and symmetry of the pendulum imply that the motion in each four intervals $\theta \in (0,\pm \alpha )$ and $\theta \in (\pm \alpha ,0)$ have the same “quarter periods” $K=K\left(\sin \left(\frac{1}{2}\alpha \right)\right)$. … ►For $\beta$ real and positive, three of the four possible combinations of signs give rise to bounded oscillatory motions. …The subsequent position as a function of time, $x(t)$, for the three cases is given with results expressed in terms of $a$ and the dimensionless parameter $\eta =\frac{1}{2}\beta a^2$. … ►Two types of oscillatory motion are possible. … ►Hyperelliptic functions $u(z)$ are solutions of the equation $z={\int }_0^u{(f(x))}^{-1/2}dx$, where $f(x)$ is a polynomial of degree higher than 4. …