mixed base Heine-type transformations

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11: 6.14 Integrals

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§6.14(i) Laplace Transforms

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6.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_1\left(t\right)dt=\frac{1}{a}\ln \left(1+a\right),$$ $\mathrm{\Re }a>-1$,

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.1.34 (special case of)

Permalink:

http://dlmf.nist.gov/6.14.E1

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Annotations for §6.14(i), §6.14 and Ch.6

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6.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{Ci}\left(t\right)dt=-\frac{1}{2a}\ln \left(1+a^2\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.2.28

Referenced by:

§6.14(i)

Permalink:

http://dlmf.nist.gov/6.14.E2

Encodings:

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See also:

Annotations for §6.14(i), §6.14 and Ch.6

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6.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{si}\left(t\right)dt=-\frac{1}{a}\arctan a,$$ $\mathrm{\Re }a>0$.

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.29

Permalink:

http://dlmf.nist.gov/6.14.E3

Encodings:

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See also:

Annotations for §6.14(i), §6.14 and Ch.6

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12: 13.10 Integrals

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§13.10(ii) Laplace Transforms

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§13.10(iii) Mellin Transforms

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§13.10(iv) Fourier Transforms

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§13.10(v) Hankel Transforms

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13: 13.23 Integrals

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§13.23(i) Laplace and Mellin Transforms

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§13.23(ii) Fourier Transforms

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§13.23(iii) Hankel Transforms

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§13.23(iv) Integral Transforms in terms of Whittaker Functions

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14: 2.4 Contour Integrals

… ►Then … ►For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985). ►

§2.4(ii) Inverse Laplace Transforms

… ►Then the Laplace transform … ►For examples see Olver (1997b, pp. 315–320). …

15: 21.5 Modular Transformations

§21.5 Modular Transformations

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§21.5(i) Riemann Theta Functions

… ►Equation (21.5.4) is the modular transformation property for Riemann theta functions. … ►

§21.5(ii) Riemann Theta Functions with Characteristics

… ►For explicit results in the case $g=1$, see §20.7(viii).

16: 10.74 Methods of Computation

… ►The integral representation used is based on (10.32.8). … ►

Hankel Transform

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Spherical Bessel Transform

►The spherical Bessel transform is the Hankel transform (10.22.76) in the case when $\nu$ is half an odd positive integer. … ►

Kontorovich–Lebedev Transform

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17: 7.7 Integral Representations

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7.7.1 $$\mathrm{erfc}z=\frac{2}{\pi }{\mathrm{e}}^{-z^2}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-z^2{t}^2}}{t^2+1}dt,$$ $|\mathrm{ph}z|\le \frac{1}{4}\pi$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

7.4.11 (in different form)

Referenced by:

§7.11, §7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E1

Encodings:

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See also:

Annotations for §7.7(i), §7.7 and Ch.7

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7.7.4 $${\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-at}}{\sqrt{t+z^2}}dt=\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{a{z}^2}\mathrm{erfc}\left(\sqrt{a}z\right),$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }z>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform

A&S Ref:

7.4.8

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E4

Encodings:

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Annotations for §7.7(i), §7.7 and Ch.7

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7.7.9 $${\int }_0^x\mathrm{erf}tdt=x\mathrm{erf}x+\frac{1}{\sqrt{\pi }}\left({\mathrm{e}}^{-x^2}-1\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

A&S Ref:

7.4.35 (in different form)

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E9

Encodings:

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Annotations for §7.7(i), §7.7 and Ch.7

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7.7.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\cos \left(t^2\right)dt=\sqrt{\frac{\pi }{2}}\mathrm{f}\left(\frac{a}{\sqrt{2\pi }}\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.22 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.7.E15

Encodings:

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See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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7.7.16 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\sin \left(t^2\right)dt=\sqrt{\frac{\pi }{2}}\mathrm{g}\left(\frac{a}{\sqrt{2\pi }}\right),$$ $\mathrm{\Re }a>0$.

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Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function

Keywords:

Laplace transform

A&S Ref:

7.4.23 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.7.E16

Encodings:

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See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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18: 1.16 Distributions

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§1.16(vii) Fourier Transforms of Tempered Distributions

… ►Then its Fourier transform is … ►If $u\in {𝒯}_n^{\ast }$ is a tempered distribution, then its Fourier transform $ℱ\left(u\right)$ is defined by …The Fourier transform $ℱ\left(u\right)$ of a tempered distribution is again a tempered distribution, and … ►

§1.16(viii) Fourier Transforms of Special Distributions

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19: 9.10 Integrals

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§9.10(v) Laplace Transforms

… ►For Laplace transforms of products of Airy functions see Shawagfeh (1992). ►

§9.10(vi) Mellin Transform

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§9.10(vii) Stieltjes Transforms

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§9.10(ix) Compendia

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20: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(vi) Direct Numerical Transformations

… ► … ►The following example, based on Weniger (1996), illustrates their power. … ►However, direct numerical transformations need to be used with care. Their extrapolation is based on assumed forms of remainder terms that may not always be appropriate for asymptotic expansions. …