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21: 20.11 Generalizations and Analogs

… ►

§20.11(iii) Ramanujan’s Change of Base

… ►These results are called Ramanujan’s changes of base. …

22: 9.6 Relations to Other Functions

… ►

9.6.1

\zeta=\tfrac{2}{3}z^{3/2},

ⓘ

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=(\tfrac{3}{2}\zeta)^{2/3},

ⓘ

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)=\tfrac{1}{2}\sqrt{3/z}\left(\sqrt{3}% \operatorname{Ai}\left(-z\right)\mp\operatorname{Bi}\left(-z\right)\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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)=\tfrac{1}{2}(\sqrt{3}/z)\left(\pm\sqrt{3}% \operatorname{Ai}'\left(-z\right)+\operatorname{Bi}'\left(-z\right)\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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)=\tfrac{1}{2}\sqrt{3/z}\left(\mp\sqrt{3}% \operatorname{Ai}\left(z\right)+\operatorname{Bi}\left(z\right)\right),

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : 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

…

23: 15.12 Asymptotic Approximations

… ►where ►

15.12.6

\zeta=\operatorname{arccosh}z.

ⓘ

Symbols:: $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $z$ : complex variable and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/15.12.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.10

\zeta=\operatorname{arccosh}\left(\tfrac{1}{4}z-1\right),

ⓘ

Symbols:: $\operatorname{arccosh}\NVar{z}$ : inverse hyperbolic cosine function, $z$ : complex variable and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/15.12.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

►

15.12.11

\beta=\left(-\frac{3}{2}\zeta+\frac{9}{4}\ln\left(\frac{2+{\mathrm{e}}^{\zeta}% }{2+{\mathrm{e}}^{-\zeta}}\right)\right)^{1/3},

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $\ln\NVar{z}$ : principal branch of logarithm function, $\zeta$ : change of variable and $\beta$
Permalink:: http://dlmf.nist.gov/15.12.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.13

G_{0}(\pm\beta)=\left(2+{\mathrm{e}}^{\pm\zeta}\right)^{c-b-(\ifrac{1}{2})}% \left(1+{\mathrm{e}}^{\pm\zeta}\right)^{a-c+(\ifrac{1}{2})}\left(z-1-{\mathrm{% e}}^{\pm\zeta}\right)^{-a+(\ifrac{1}{2})}\sqrt{\frac{\beta}{{\mathrm{e}}^{% \zeta}-{\mathrm{e}}^{-\zeta}}}.

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\zeta$ : change of variable, $\beta$ and $G_{0}(\pm\zeta)$ : function
Permalink:: http://dlmf.nist.gov/15.12.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

…

24: 2.5 Mellin Transform Methods

… ►

2.5.1

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\int_{0}^{\infty}t^{z-1}f(t% )\,\mathrm{d}t,

ⓘ

Symbols:: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $f(x)$ : locally integrable function
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/2.5.E1
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{M}(f;z)$ .
See also:: Annotations for §2.5(i), §2.5 and Ch.2

… ►

2.5.4

\mathscr{M}\mskip-3.0muI\mskip 3.0mu\left(z\right)=\mathscr{M}\mskip-3.0muf% \mskip 3.0mu\left(1-z\right)\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right).

ⓘ

Symbols:: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $f(x)$ : locally integrable function, $I(x)$ : convolution integral and $h(x)$ : function
Keywords:: Mellin transform
Permalink:: http://dlmf.nist.gov/2.5.E4
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{M}(f;z)$ .
See also:: Annotations for §2.5(i), §2.5 and Ch.2

… ►

2.5.26

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\mathscr{M}\mskip-3.0muf_{1% }\mskip 3.0mu\left(z\right)+\mathscr{M}\mskip-3.0muf_{2}\mskip 3.0mu\left(z\right)

ⓘ

Symbols:: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $f(x)$ : locally integrable function and $f_{j}(t)$ : truncated functions
Keywords:: Mellin transform
Referenced by:: §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E26
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{M}(f;z)$ .
See also:: Annotations for §2.5(ii), §2.5 and Ch.2

… ►

2.5.28

\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right)=\mathscr{M}\mskip-3.0muh_{1% }\mskip 3.0mu\left(z\right)+\mathscr{M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)

ⓘ

Symbols:: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $f(x)$ : locally integrable function and $h(x)$ : locally integrable function
Referenced by:: §2.5(iii), §2.5(ii)
Permalink:: http://dlmf.nist.gov/2.5.E28
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{M}(f;z)$ .
See also:: Annotations for §2.5(ii), §2.5 and Ch.2

… ►

2.5.38

\zeta\mathscr{L}\mskip-3.0muh\mskip 3.0mu\left(\zeta\right)=I_{1}(x)+I_{2}(x),

ⓘ

Symbols:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform, $h(x)$ : locally integrable function and $I_{j}(x)$ : integral
Keywords:: Laplace transform
Referenced by:: §2.5(iii)
Permalink:: http://dlmf.nist.gov/2.5.E38
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(z\right)$ from $\mathscr{L}(f;z)$ .
See also:: Annotations for §2.5(iii), §2.5 and Ch.2

…

25: 5.10 Continued Fractions

… ►

5.10.1

\operatorname{Ln}\Gamma\left(z\right)+z-\left(z-\tfrac{1}{2}\right)\ln z-% \tfrac{1}{2}\ln\left(2\pi\right)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}% {z+\cfrac{a_{3}}{z+\cfrac{a_{4}}{z+\cfrac{a_{5}}{z+}}}}}}\cdots,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable and $a_{k}$ : coefficient
Referenced by:: Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)
Permalink:: http://dlmf.nist.gov/5.10.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.0.10):: To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\operatorname{Ln}\Gamma\left(z\right)$ , where $\operatorname{Ln}$ is the general logarithm. Originally $\ln\Gamma\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.
Suggested 2015-02-13 by Philippe Spindel
See also:: Annotations for §5.10 and Ch.5

…

26: 9.1 Special Notation

… ►Other notations that have been used are as follows:

\operatorname{Ai}\left(-x\right)

and

\operatorname{Bi}\left(-x\right)

for

\operatorname{Ai}\left(x\right)

and

\operatorname{Bi}\left(x\right)

(Jeffreys (1928), later changed to

\operatorname{Ai}\left(x\right)

and

\operatorname{Bi}\left(x\right)

);

U(x)=\sqrt{\pi}\operatorname{Bi}\left(x\right)

V(x)=\sqrt{\pi}\operatorname{Ai}\left(x\right)

(Fock (1945));

A(x)=3^{-\ifrac{1}{3}}\pi\operatorname{Ai}\left(-3^{-\ifrac{1}{3}}x\right)

(Szegő (1967, §1.81));

e_{0}(x)=\pi\operatorname{Hi}(-x)

\widetilde{e}_{0}(x)=-\pi\operatorname{Gi}(-x)

(Tumarkin (1959)).

27: 9.5 Integral Representations

… ►

9.5.6

\operatorname{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi}\int_{0}^{\infty}\exp% \left(-\frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}\right)\,\mathrm{d}t,

|\operatorname{ph}z|<\tfrac{1}{6}\pi

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\operatorname{ph}$ : phase and $z$ : complex variable
Source:: Reid (1995, (5.4), p. 170, with change of variable and using analytic continuation to complex $z$ )
Referenced by:: Erratum (V1.0.15) for Equation (9.5.6)
Permalink:: http://dlmf.nist.gov/9.5.E6
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.15):: The validity constraint $|\operatorname{ph}z|<\tfrac{1}{6}\pi$ was added.
See also:: Annotations for §9.5(ii), §9.5 and Ch.9

►

9.5.7

\operatorname{Ai}\left(z\right)=\frac{e^{-\zeta}}{\pi}\int_{0}^{\infty}\exp% \left(-z^{\ifrac{1}{2}}t^{2}\right)\cos\left(\tfrac{1}{3}t^{3}\right)\,\mathrm% {d}t,

|\operatorname{ph}z|<\pi

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable and $\zeta$ : change of variable
Source:: Copson (1963, (2.1))
Referenced by:: §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §9.5(ii), §9.5 and Ch.9

►

9.5.8

\operatorname{Ai}\left(z\right)=\frac{e^{-\zeta}\zeta^{\ifrac{-1}{6}}}{\sqrt{% \pi}(48)^{\ifrac{1}{6}}\Gamma\left(\frac{5}{6}\right)}\int_{0}^{\infty}e^{-t}t% ^{-\ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-\ifrac{1}{6}}\,\mathrm{d}t,

|\operatorname{ph}z|<\frac{2}{3}\pi

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable and $\zeta$ : change of variable
Proof sketch:: Follows from (9.6.21) and (13.4.4).
Referenced by:: §9.17(iii), §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §9.5(ii), §9.5 and Ch.9

…

28: 19.15 Advantages of Symmetry

… ►(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …

29: 1.14 Integral Transforms

… ►

1.14.17

\mathscr{L}\left(f\right)\left(s\right)=\mathscr{L}\mskip-3.0muf\mskip 3.0mu% \left(s\right)=\int^{\infty}_{0}{\mathrm{e}}^{-st}f(t)\,\mathrm{d}t.

ⓘ

Defines:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm and $\int$ : integral
Keywords:: Laplace transform
A&S Ref:: 29.1.1
Permalink:: http://dlmf.nist.gov/1.14.E17
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Laplace transform was changed to $\mathscr{L}\left(f\right)\left(s\right)$ or $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{L}(f(t);s)$ .
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

… ►

1.14.19

\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)\to 0,

\Re s\to\infty

ⓘ

Symbols:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform and $\Re$ : real part
Keywords:: Laplace transform
Permalink:: http://dlmf.nist.gov/1.14.E19
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{L}(f(t);s)$ .
See also:: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

… ►

1.14.21

\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s-a\right)=\mathscr{L}\mskip-3.0muf_% {a}\mskip 3.0mu\left(s\right),

ⓘ

Symbols:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform and $\mathrm{e}$ : base of natural logarithm
Keywords:: Laplace transform
Permalink:: http://dlmf.nist.gov/1.14.E21
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{L}(f(t);s)$ . In addition, the auxiliary function $f_{a}(t)={\mathrm{e}}^{at}f(t)$ was introduced.
See also:: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

… ►

1.14.22

\mathscr{L}\mskip-3.0muf_{a}^{+}\mskip 3.0mu\left(s\right)={\mathrm{e}}^{-as}% \mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right),

ⓘ

Symbols:: $H\left(\NVar{x}\right)$ : Heaviside function, $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform and $\mathrm{e}$ : base of natural logarithm
Keywords:: Laplace transform
Permalink:: http://dlmf.nist.gov/1.14.E22
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Laplace transform was changed to $\mathscr{L}\left(f\right)\left(s\right)$ from $\mathscr{L}(f(t);s)$ . In addition, the auxiliary function $f^{+}_{a}(t)=H\left(t-a\right)f(t-a)$ was introduced.
See also:: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

… ►

1.14.46

\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathcal{H}\mskip-3.0muf\mskip 3.0% mu\left(u\right){\mathrm{e}}^{\mathrm{i}ux}\,\mathrm{d}u=-\mathrm{i}(% \operatorname{sign}x)\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right),

ⓘ

Symbols:: $\mathscr{F}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Fourier transform, $\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}\right)$ : Hilbert transform, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\operatorname{sign}\NVar{x}$ : sign of
Referenced by:: §1.14(v)
Permalink:: http://dlmf.nist.gov/1.14.E46
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.15):: The notation for the Hilbert transform was changed to $\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(t\right)$ from $\mathcal{H}(f;t)$ , and the notation for the Fourier transform was changed to $\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right)$ from $F(x)$ . In addition, the integration variable was changed from $t$ to $u$ .
See also:: Annotations for §1.14(v), §1.14(v), §1.14 and Ch.1

…

30: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

… ►Let ►

8.18.2

\xi=-\ln x.

ⓘ

Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : variable and $\xi$ : change of variable
Permalink:: http://dlmf.nist.gov/8.18.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

… ►

8.18.4

aF_{k+1}=(k+b-a\xi)F_{k}+k\xi F_{k-1},

ⓘ

Symbols:: $k$ : nonnegative integer, $a$ : parameter, $b$ : parameter, $\xi$ : change of variable and $F_{k}$ : functions
Permalink:: http://dlmf.nist.gov/8.18.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

… ►

8.18.6

\left(\frac{1-e^{-t}}{t}\right)^{b-1}=\sum_{k=0}^{\infty}d_{k}(t-\xi)^{k}.

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $k$ : nonnegative integer, $b$ : parameter, $\xi$ : change of variable and $d_{k}$ : coefficients
Permalink:: http://dlmf.nist.gov/8.18.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

…