transformation of variable

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11: 19.7 Connection Formulas

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Reciprocal-Modulus Transformation

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Imaginary-Modulus Transformation

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Imaginary-Argument Transformation

►With $\sinh \varphi =\tan \psi$, … ►For two further transformations of this type see Erdélyi et al. (1953b, p. 316). …

12: 31.2 Differential Equations

… ►By composing these three steps, there result $2^3=8$ possible transformations of the dependent variable (including the identity transformation) that preserve the form of (31.2.1). …

13: 35.2 Laplace Transform

… ►where the integration variable $𝐗$ ranges over the space $𝛀$. …

14: 12.16 Mathematical Applications

… ► ►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …Integral transforms and sampling expansions are considered in Jerri (1982).

15: 19.8 Quadratic Transformations

§19.8 Quadratic Transformations

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§19.8(ii) Landen Transformations

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Descending Landen Transformation

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Ascending Landen Transformation

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§19.8(iii) Gauss Transformation

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16: 31.10 Integral Equations and Representations

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Transformation of Independent Variable

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17: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►In this section we give asymptotic expansions of PCFs for large values of the parameter $a$ that are uniform with respect to the variable $z$, when both $a$ and $z$ $(=x)$ are real. These expansions follow from Olver (1959), where detailed information is also given for complex variables. ►With the transformations … ►The variable $\zeta$ is defined by … ►

12.10.40 $$\varphi (\zeta )={\left(\frac{\zeta }{t^2-1}\right)}^{\frac{1}{4}}.$$

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Defines:

$\varphi (\zeta )$: function (locally)

Symbols:

$\zeta$: change of variable

Referenced by:

§12.10(vii)

Permalink:

http://dlmf.nist.gov/12.10.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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18: 1.14 Integral Transforms

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1.14.1 $$ℱ\left(f\right)\left(x\right)=ℱf\left(x\right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t){\mathrm{e}}^{\mathrm{i}xt}dt.$$

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Defines:

$ℱ\left(f\right)\left(s\right)$: Fourier transform

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 and $\int$: integral

Referenced by:

§1.14(v), §1.14(i), item (i), §1.17(ii), §1.8(iv), item (i)

Permalink:

http://dlmf.nist.gov/1.14.E1

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

This equation defines $ℱ\left(f\right)\left(x\right)$ or $ℱf\left(x\right)$ as the Fourier transform of functions of a single variable. An analogous notation is defined for the Fourier transform of tempered distributions in (1.16.29) and the Fourier transform of special distributions in (1.16.38).

See also:

Annotations for §1.14(i), §1.14 and Ch.1

… ►The same notation $ℱ$ is used for Fourier transforms of functions of several variables and for Fourier transforms of distributions; see §1.16(vii). … ►

1.14.46 $$\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathscr{H}f\left(u\right){\mathrm{e}}^{\mathrm{i}ux}du=-\mathrm{i}(\mathrm{sign}x)ℱf\left(x\right),$$

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Symbols:

$ℱ\left(f\right)\left(s\right)$: Fourier transform, $\mathscr{H}\left(f\right)\left(x\right)$: Hilbert transform, $\pi$: 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 $\mathrm{sign}x$: 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 $\mathscr{H}f\left(t\right)$ from $\mathscr{H}(f;t)$, and the notation for the Fourier transform was changed to $ℱf\left(x\right)$ 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

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19: 1.10 Functions of a Complex Variable

… ►Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic. …

20: 22.7 Landen Transformations

§22.7 Landen Transformations

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§22.7(i) Descending Landen Transformation

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§22.7(ii) Ascending Landen Transformation

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$k_2={\displaystyle \frac{2\sqrt{k}}{1+k}},$

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§22.7(iii) Generalized Landen Transformations

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