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11: 2.11 Remainder Terms; Stokes Phenomenon

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2.11.20 $$R_n^{(1)}(z)={(-1)}^{n-1}\mathrm{i}{\mathrm{e}}^{({\mu }_2-{\mu }_1)\pi \mathrm{i}}{\mathrm{e}}^{{\lambda }_{2}z}{z}^{{\mu }_2}\left(C_1\sum _{s=0}^{m-1}{(-1)}^s{a}_{s,2}\frac{F_{n+{\mu }_2-{\mu }_1-s}\left(z\right)}{z^s}+R_{m,n}^{(1)}(z)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F_p\left(z\right)$: terminant function, $R_n^{(j)}(z)$: remainder, $a_{s,j}$: coefficients, ${\lambda }_j$: roots, ${\mu }_j$: quantities and $C_1$, $C_2$: Stokes multipliers

Referenced by:

§2.11(v)

Permalink:

http://dlmf.nist.gov/2.11.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(v), §2.11 and Ch.2

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2.11.21 $$R_n^{(2)}(z)={(-1)}^n\mathrm{i}{\mathrm{e}}^{({\mu }_2-{\mu }_1)\pi \mathrm{i}}{\mathrm{e}}^{{\lambda }_{1}z}{z}^{{\mu }_1}\left(C_2\sum _{s=0}^{m-1}{(-1)}^s{a}_{s,1}\frac{F_{n+{\mu }_1-{\mu }_2-s}\left(z{\mathrm{e}}^{-\pi \mathrm{i}}\right)}{z^s}+R_{m,n}^{(2)}(z)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F_p\left(z\right)$: terminant function, $R_n^{(j)}(z)$: remainder, $a_{s,j}$: coefficients, ${\lambda }_j$: roots, ${\mu }_j$: quantities and $C_1$, $C_2$: Stokes multipliers

Referenced by:

§2.11(v)

Permalink:

http://dlmf.nist.gov/2.11.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(v), §2.11 and Ch.2

… ►Multiplying these differences by ${(-1)}^j{2}^{-j-1}$ and summing, we obtain …

12: 1.13 Differential Equations

… ►(More generally in (1.13.5) for $n$th-order differential equations, $f(z)$ is the coefficient multiplying the $(n-1)$th-order derivative of the solution divided by the coefficient multiplying the $n$th-order derivative of the solution, see Ince (1926, §5.2).) …

13: 7.7 Integral Representations

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14: 14.19 Toroidal (or Ring) Functions

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14.19.2 $$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;1-{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and $\xi >0$: variable

Referenced by:

§14.19(ii), Erratum (V1.0.1) for Equation (14.19.2)

Permalink:

http://dlmf.nist.gov/14.19.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the argument to $𝐅$ was incorrect and the condition on $\mu$ too weak:

$$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(1-2\mu \right)2^{2\mu }}{\mathrm{\Gamma }\left(1-\mu \right){\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2}$.

Also, the factor multiplying $𝐅$ was rewritten to clarify the poles, which determine the correct condition on $\mu$.

Reported 2010-11-02 by Alvaro Valenzuela

See also:

Annotations for §14.19(ii), §14.19 and Ch.14

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15: 21.2 Definitions

… ►It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor: …

16: 21.6 Products

… ►that is, $𝒟$ is the number of elements in the set containing all $h$-dimensional vectors obtained by multiplying ${𝐓}^{\mathrm{T}}$ on the right by a vector with integer elements. …

17: 1.2 Elementary Algebra

… ►To find the polynomials $f_j(x)$, $j=1,2,\mathrm{\dots },n$, multiply both sides by the denominator of the left-hand side and equate coefficients. … ►If $det(𝐀)=0$ then $𝐀𝐁=𝐀𝐂$ does not imply that $𝐁=𝐂$; if $det(𝐀)\ne 0$, then $𝐁=𝐂$, as both sides may be multiplied by ${𝐀}^{-1}$. …

18: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►If all the elements of a row (column) of a determinant are multiplied by an arbitrary factor $\mu$, then the result is a determinant which is $\mu$ times the original. …

19: 3.8 Nonlinear Equations

… ►For multiple zeros the convergence is linear, but if the multiplicity $m$ is known then quadratic convergence can be restored by multiplying the ratio $f(z_n)/f^{\prime }(z_n)$ in (3.8.4) by $m$. …

20: 27.14 Unrestricted Partitions

… ►Multiplying the power series for $f\left(x\right)$ with that for $1/f\left(x\right)$ and equating coefficients, we obtain the recursion formula …