change of modulus

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11: Bibliography D

… ►

Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

…

12: 31.7 Relations to Other Functions

… ►With $z={\mathrm{sn}}^2(\zeta ,k)$ and …

13: 22.9 Cyclic Identities

… ►

22.9.7 $$\kappa =\mathrm{dn}(2K\left(k\right)/3,k),$$

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Defines:

$\kappa$: change of variable (locally)

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind and $k$: modulus

Referenced by:

§22.9(iii), §22.9(iv)

Permalink:

http://dlmf.nist.gov/22.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

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22.9.8 $$s_{1,3}^{(4)}{s}_{2,3}^{(4)}+s_{2,3}^{(4)}{s}_{3,3}^{(4)}+s_{3,3}^{(4)}{s}_{1,3}^{(4)}=\frac{{\kappa }^2-1}{k^2},$$

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Symbols:

$k$: modulus, $s_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Referenced by:

Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)

Permalink:

http://dlmf.nist.gov/22.9.E8

Encodings:

pMML, png, TeX

Correction (effective with 1.0.25):

Originally this equation was written incorrectly for all $s_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout.

Suggested 2019-11-07 by Juan Miguel Nieto

See also:

Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

… ►

22.9.15 $$d_{1,3}^{(2)}{d}_{2,3}^{(2)}{d}_{3,3}^{(2)}=\frac{{\kappa }^2+k^2-1}{1-{\kappa }^2}\left(d_{1,3}^{(2)}+d_{2,3}^{(2)}+d_{3,3}^{(2)}\right),$$

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Symbols:

$k$: modulus, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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22.9.22 $$s_{1,3}^{(2)}{c}_{1,3}^{(2)}{d}_{2,3}^{(2)}{d}_{3,3}^{(2)}+s_{2,3}^{(2)}{c}_{2,3}^{(2)}{d}_{3,3}^{(2)}{d}_{1,3}^{(2)}+s_{3,3}^{(2)}{c}_{3,3}^{(2)}{d}_{1,3}^{(2)}{d}_{2,3}^{(2)}=\frac{{\kappa }^2+k^2-1}{1-{\kappa }^2}\left(s_{1,3}^{(2)}{c}_{1,3}^{(2)}+s_{2,3}^{(2)}{c}_{2,3}^{(2)}+s_{3,3}^{(2)}{c}_{3,3}^{(2)}\right),$$

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Symbols:

$k$: modulus, $s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable

Permalink:

http://dlmf.nist.gov/22.9.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §22.9(iv), §22.9(iv), §22.9 and Ch.22

…

14: 15.8 Transformations of Variable

… ►With $\zeta ={\mathrm{e}}^{2\pi \mathrm{i}/3}(1-z)/\left(z-{\mathrm{e}}^{4\pi \mathrm{i}/3}\right)$ …

15: 9.6 Relations to Other Functions

… ►

9.6.1 $$\zeta =\frac{2}{3}{z}^{3/2},$$

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Defines:

$\zeta$: change of variable (locally)

Symbols:

$z$: complex variable

Referenced by:

§9.6(iii), (9.7.17), §9.7(iv)

Permalink:

http://dlmf.nist.gov/9.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(i), §9.6 and Ch.9

… ►

9.6.10 $$z={(\frac{3}{2}\zeta )}^{2/3},$$

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Defines:

$\zeta (z)$: change of variable (locally)

Symbols:

$z$: complex variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

Permalink:

http://dlmf.nist.gov/9.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

►

9.6.11 $$J_{\pm 1/3}\left(\zeta \right)=\frac{1}{2}\sqrt{3/z}\left(\sqrt{3}\mathrm{Ai}\left(-z\right)\mp \mathrm{Bi}\left(-z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $z$: complex variable and $\zeta (z)$: change of variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

A&S Ref:

10.4.22

Permalink:

http://dlmf.nist.gov/9.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

►

9.6.12 $$J_{\pm 2/3}\left(\zeta \right)=\frac{1}{2}(\sqrt{3}/z)\left(\pm \sqrt{3}{\mathrm{Ai}}^{\prime }\left(-z\right)+{\mathrm{Bi}}^{\prime }\left(-z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $z$: complex variable and $\zeta (z)$: change of variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

A&S Ref:

10.4.27

Permalink:

http://dlmf.nist.gov/9.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

►

9.6.13 $$I_{\pm 1/3}\left(\zeta \right)=\frac{1}{2}\sqrt{3/z}\left(\mp \sqrt{3}\mathrm{Ai}\left(z\right)+\mathrm{Bi}\left(z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\zeta (z)$: change of variable

Proof sketch:

Derivable from those given in Olver (1997b, pp. 392–393).

A&S Ref:

10.4.25

Permalink:

http://dlmf.nist.gov/9.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

…

16: 9.7 Asymptotic Expansions

… ►

9.7.1 $$\zeta =\frac{2}{3}{z}^{3/2}.$$

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Defines:

$\zeta$: change of variable (locally)

Symbols:

$z$: complex variable

Permalink:

http://dlmf.nist.gov/9.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(i), §9.7 and Ch.9

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9.7.6 $${\mathrm{Ai}}^{\prime }\left(z\right)\sim -\frac{z^{1/4}{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

§9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

… ►

9.7.8 $${\mathrm{Bi}}^{\prime }\left(z\right)\sim \frac{z^{1/4}{\mathrm{e}}^{\zeta }}{\sqrt{\pi }}\sum _{k=0}^{\mathrm{\infty }}\frac{v_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.16), pp. 393 and 414)

Referenced by:

§9.7(iii), §9.7(iii), Erratum (V1.0.17) for Subsection 9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

… ►

9.7.10 $${\mathrm{Ai}}^{\prime }\left(-z\right)\sim \frac{z^{1/4}}{\sqrt{\pi }}\left(\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k}}{{\zeta }^{2k}}-\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.09), pp. 392 and 413)

Permalink:

http://dlmf.nist.gov/9.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

… ►

9.7.12 $${\mathrm{Bi}}^{\prime }\left(-z\right)\sim \frac{z^{1/4}}{\sqrt{\pi }}\left(\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k}}{{\zeta }^{2k}}+\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$.

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.18), pp. 393 and 414)

Referenced by:

§9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

…

17: 20.11 Generalizations and Analogs

… ►

20.11.2 $$\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum _{k=0}^{n-1}{\mathrm{e}}^{-\pi \mathrm{i}{k}^{2}m/n}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}\sum _{j=0}^{m-1}{\mathrm{e}}^{\pi \mathrm{i}{j}^{2}n/m}=\frac{{\mathrm{e}}^{-\pi \mathrm{i}/4}}{\sqrt{m}}G(-n,m).$$

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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, $m$: integer, $n$: integer and $G(m,n)$: Gauss sum

Permalink:

http://dlmf.nist.gov/20.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(i), §20.11 and Ch.20

… ►

20.11.3 $$f(a,b)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}^{n(n+1)/2}{b}^{n(n-1)/2},$$

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Symbols:

$n$: integer

Permalink:

http://dlmf.nist.gov/20.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.11(ii), §20.11 and Ch.20

… ►

§20.11(iii) Ramanujan’s Change of Base

►As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). … ►These results are called Ramanujan’s changes of base. …

18: 1.13 Differential Equations

… ►The substitution $\xi =1/z$ in (1.13.1) gives … ►

1.13.21 $$\{z,\zeta \}={(d\xi /d\zeta )}^2\{z,\xi \}+\{\xi ,\zeta \}.$$

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Symbols:

$\{z,\zeta \}$: Schwarzian derivative, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: variable, $\xi$: change of variable and $\zeta (z)$: thrice-differentiable function

Permalink:

http://dlmf.nist.gov/1.13.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §1.13(iv), §1.13(iv), §1.13 and Ch.1

…

19: 27.12 Asymptotic Formulas: Primes

… ►

27.12.5 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-c{(\ln x)}^{1/2}\right)\right),$$ $x\to \mathrm{\infty }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

Erratum (V1.0.18) for Equation (27.12.5)

Permalink:

http://dlmf.nist.gov/27.12.E5

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.18):

Replaced $\sqrt{\ln x}$ with ${(\ln x)}^{1/2}$ to be consistent with other equations in this section.

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

… ►

27.12.6 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-d{(\ln x)}^{3/5}{(\ln \ln x)}^{-1/5}\right)\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number, $d$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.12.E6

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

… ►

27.12.8 $$\frac{\mathrm{li}\left(x\right)}{\varphi \left(m\right)}+O\left(x\exp \left(-\lambda (\alpha ){(\ln x)}^{1/2}\right)\right),$$ $m\le {(\ln x)}^{\alpha }$, $\alpha >0$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $m$: positive integer and $x$: real number

Referenced by:

Erratum (V1.0.17) for Equation (27.12.8)

Permalink:

http://dlmf.nist.gov/27.12.E8

Encodings:

pMML, png, TeX

Errata (effective with 1.0.17):

Originally the first term was given incorrectly by $\frac{x}{\varphi \left(m\right)}$.

Reported 2017-12-04 by Gergő Nemes

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

…

20: 1.17 Integral and Series Representations of the Dirac Delta

… ►

1.17.10 $$\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/(4n)}{\mathrm{e}}^{\mathrm{i}(x-a)t}dt=\sqrt{\frac{n}{\pi }}{\mathrm{e}}^{-n{(x-a)}^2}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.17(ii)

Permalink:

http://dlmf.nist.gov/1.17.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(ii), §1.17 and Ch.1

… ►

1.17.11 $${\delta }_n\left(x-a\right)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/(4n)}{\mathrm{e}}^{\mathrm{i}(x-a)t}dt,$$

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Symbols:

${\delta }_n\left(x\right)$: Dirac delta sequence, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.17.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(ii), §1.17 and Ch.1

… ►

1.17.23 $$\delta \left(x-a\right)={\mathrm{e}}^{-(x+a)/2}\sum _{k=0}^{\mathrm{\infty }}{L}_k\left(x\right)L_k\left(a\right).$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\mathrm{e}$: base of natural logarithm, $L_n\left(x\right)=L_n^{(0)}\left(x\right)$: Laguerre polynomial and $k$: integer

Permalink:

http://dlmf.nist.gov/1.17.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

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1.17.24 $$\delta \left(x-a\right)=\frac{{\mathrm{e}}^{-(x^2+a^2)/2}}{\sqrt{\pi }}\sum _{k=0}^{\mathrm{\infty }}\frac{H_k\left(x\right)H_k\left(a\right)}{2^{k}k!}.$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$) and $k$: integer

Referenced by:

§1.17(iii), §1.17(iv)

Permalink:

http://dlmf.nist.gov/1.17.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

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