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21: 10.65 Power Series

§10.65 Power Series

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§10.65(iii) Cross-Products and Sums of Squares

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§10.65(iv) Compendia

►For further power series summable in terms of Kelvin functions and their derivatives see Hansen (1975).

22: 27.2 Functions

… ►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer $n>1$ can be represented uniquely as a product of prime powers, … ►

27.2.6 $${\varphi }_k\left(n\right)=\sum _{\left(m,n\right)=1}{m}^k,$$

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Defines:

${\varphi }_k\left(n\right)$: sum of powers of integers relatively prime to a number

Symbols:

$\left(m,n\right)$: greatest common divisor (gcd), $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.2.E6

Encodings:

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See also:

Annotations for §27.2(i), §27.2 and Ch.27

►the sum of the $k$th powers of the positive integers $m\le n$ that are relatively prime to $n$. … ►is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. … ►where $p^a$ is a prime power with $a\ge 1$; otherwise $\mathrm{\Lambda }\left(n\right)=0$. …

23: 28.6 Expansions for Small $q$

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§28.6(i) Eigenvalues

►Leading terms of the power series for $a_m\left(q\right)$ and $b_m\left(q\right)$ for $m\le 6$ are: … ►Leading terms of the of the power series for $m=7,8,9,\mathrm{\dots }$ are: … ►

§28.6(ii) Functions ${\mathrm{ce}}_n$ and ${\mathrm{se}}_n$

►Leading terms of the power series for the normalized functions are: …

24: 4.4 Special Values and Limits

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§4.4(ii) Powers

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§4.4(iii) Limits

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25: 7.6 Series Expansions

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§7.6(i) Power Series

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26: 27.3 Multiplicative Properties

… ►If $f$ is multiplicative, then the values $f(n)$ for $n>1$ are determined by the values at the prime powers. … ►

27.3.6 $${\sigma }_{\alpha }\left(n\right)=\prod _{r=1}^{\nu \left(n\right)}\frac{p_r^{\alpha (1+a_r)}-1}{p_r^{\alpha }-1},$$ $\alpha \ne 0$.

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a_r$: positive exponent

A&S Ref:

24.3.3 I.C

Permalink:

http://dlmf.nist.gov/27.3.E6

Encodings:

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Annotations for §27.3 and Ch.27

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27.3.7 $${\sigma }_{\alpha }\left(m\right){\sigma }_{\alpha }\left(n\right)=\sum _{d|\left(m,n\right)}{d}^{\alpha }{\sigma }_{\alpha }\left(\frac{mn}{d^2}\right),$$

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Symbols:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number, $\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

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27: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … ►Noble (2004) obtains double-precision accuracy for $W_{-\eta ,\mu }\left(2\rho \right)$ for a wide range of parameters using a combination of recurrence techniques, power-series expansions, and numerical quadrature; compare (33.2.7). …

28: 8.24 Physical Applications

… ►The function $\gamma (a,x)$ appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). …

29: 12.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. …

30: 23.9 Laurent and Other Power Series

§23.9 Laurent and Other Power Series

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