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21: 1.14 Integral Transforms

… ►Note: If $f(x)$ is continuous and $\alpha$ and $\beta$ are real numbers such that $f(x)=O\left(x^{\alpha }\right)$ as $x\to 0+$ and $f(x)=O\left(x^{\beta }\right)$ as $x\to \mathrm{\infty }$, then $x^{\sigma -1}f(x)$ is integrable on $(0,\mathrm{\infty })$ for all $\sigma \in (-\alpha ,-\beta )$. … ►where $A_p=\tan \left(\frac{1}{2}\pi /p\right)$ when , or $\cot \left(\frac{1}{2}\pi /p\right)$ when $p\ge 2$. … ►Sufficient conditions for the integral to converge are that $s$ is a positive real number, and $f(t)=O\left(t^{-\delta }\right)$ as $t\to \mathrm{\infty }$, where $\delta >0$. … ►

Table 1.14.3: Fourier sine transforms.

► ►►►

$f(t)$	$\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle {\int }_0^{\mathrm{\infty }}}f(t)\sin \left(xt\right)dt$,	$x>0$
…
${\displaystyle \frac{{\mathrm{e}}^{-at}}{t}}$	$\sqrt{{\displaystyle \frac{2}{\pi }}}\arctan \left({\displaystyle \frac{x}{a}}\right)$,	$\mathrm{\Re }a>0$
…

►

►

Table 1.14.4: Laplace transforms.

► ►►►

$f(t)$	${\displaystyle {\int }_0^{\mathrm{\infty }}}{\mathrm{e}}^{-st}f(t)dt$
…
${\displaystyle \frac{\sin \left(at\right)}{t}}$	$\arctan \left({\displaystyle \frac{a}{s}}\right)$,	$\mathrm{\Re }s>0$

►

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22: 25.14 Lerch’s Transcendent

… ►If $s$ is not an integer then ; if $s$ is a positive integer then $a\ne 0,-1,-2,\mathrm{\dots }$; if $s$ is a non-positive integer then $a$ can be any complex number. … ►

25.14.6 $$\mathrm{\Phi }(z,s,a)=\frac{1}{2}{a}^{-s}+{\int }_0^{\mathrm{\infty }}\frac{z^x}{{(a+x)}^s}dx-2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(x\ln z-s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$ $\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$.

ⓘ

Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.4), p. 28)

Notes:

For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\mathrm{\Re }s>1$.

Referenced by:

Erratum (V1.1.4) for Equation (25.14.6)

Permalink:

http://dlmf.nist.gov/25.14.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint which originally read “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $|z|=1,\mathrm{\Re }a>0$” has been improved to be “$\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$”.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

…

23: 1.4 Calculus of One Variable

… ►If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is piecewise (or sectionally) continuous on $I$. … ►Lastly, whether or not the real numbers $a$ and $b$ satisfy , and whether or not they are finite, we define ${𝒱}_{a,b}\left(f\right)$ by (1.4.34) whenever this integral exists. … ►A continuously differentiable function is convex iff the curve does not lie below its tangent at any point. …

24: 19.20 Special Cases

… ► Schneider that this is a transcendental number. … ► Schneider that this is a transcendental number. …