# §1.14 Integral Transforms

## §1.14(i) Fourier Transform

The Fourier transform of a real- or complex-valued function $f(t)$ is defined by

 1.14.1 $\mathscr{F}\left(f\right)\left(x\right)=\mathscr{F}\mskip-3.0mu f\mskip 3.0mu % \left(x\right)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}f(t)e^{ixt}\mathrm{% d}t.$ ⓘ Defines: $\mathscr{F}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Fourier transform Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $\int$: integral Referenced by: §1.14(v), §1.14(i), item (i), §1.17(ii), item (i) Permalink: http://dlmf.nist.gov/1.14.E1 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): This equation defines $\mathscr{F}\left(f\right)\left(x\right)$ or $\mathscr{F}\mskip-3.0mu f\mskip 3.0mu \left(x\right)$ 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

(Some references replace $ixt$ by $-ixt$). The same notation $\mathscr{F}$ is used for Fourier transforms of functions of several variables and for Fourier transforms of distributions; see §1.16(vii).

In this subsection we let $F(x)=\mathscr{F}\left(f\right)\left(x\right)$.

If $f(t)$ is absolutely integrable on $(-\infty,\infty)$, then $F(x)$ is continuous, $F(x)\to 0$ as $x\to\pm\infty$, and

 1.14.2 $|F(x)|\leq\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}|f(t)|\mathrm{d}t.$

### Inversion

Suppose that $f(t)$ is absolutely integrable on $(-\infty,\infty)$ and of bounded variation in a neighborhood of $t=u$1.4(v)). Then

 1.14.3 $\tfrac{1}{2}(f(u+)+f(u-))=\frac{1}{\sqrt{2\pi}}\pvint^{\infty}_{-\infty}F(x)e^% {-ixu}\mathrm{d}x,$

where the last integral denotes the Cauchy principal value (1.4.25).

In many applications $f(t)$ is absolutely integrable and $f^{\prime}(t)$ is continuous on $(-\infty,\infty)$. Then

 1.14.4 $f(t)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}F(x)e^{-ixt}\mathrm{d}x.$

### Convolution

For Fourier transforms, the convolution $(f*g)(t)$ of two functions $f(t)$ and $g(t)$ defined on $(-\infty,\infty)$ is given by

 1.14.5 $(f*g)(t)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}f(t-s)g(s)\mathrm{d}s.$ ⓘ Defines: $*$: convolution (Fourier) (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.14.E5 Encodings: TeX, pMML, png See also: Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

If $f(t)$ and $g(t)$ are absolutely integrable on $(-\infty,\infty)$, then so is $(f*g)(t)$, and its Fourier transform is $F(x)G(x)$, where $G(x)$ is the Fourier transform of $g(t)$.

### Parseval’s Formula

Suppose $f(t)$ and $g(t)$ are absolutely integrable on $(-\infty,\infty)$, then

 1.14.6 $(f*g)(t)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}F(x)G(x)e^{-itx}\mathrm{d% }x,$
 1.14.7 $\int^{\infty}_{-\infty}F(x)G(x)\mathrm{d}x=\int^{\infty}_{-\infty}f(t)g(-t)% \mathrm{d}t,$
 1.14.8 $\int^{\infty}_{-\infty}|F(x)|^{2}\mathrm{d}x=\int^{\infty}_{-\infty}|f(t)|^{2}% \mathrm{d}t.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $F(x)$: Fourier transform of $f(t)$ Referenced by: §1.14(i) Permalink: http://dlmf.nist.gov/1.14.E8 Encodings: TeX, pMML, png See also: Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

(1.14.8) is Parseval’s formula.

### Uniqueness

If $f(t)$ and $g(t)$ are continuous and absolutely integrable on $(-\infty,\infty)$, and $F(x)=G(x)$ for all $x$, then $f(t)=g(t)$ for all $t$.

## §1.14(ii) Fourier Cosine and Sine Transforms

The Fourier cosine transform and Fourier sine transform are defined respectively by

 1.14.9 $\displaystyle\mathscr{F}_{\mkern-3.0mu c}\left(f\right)\left(x\right)$ $\displaystyle=\mathscr{F}_{\mkern-3.0mu c}\mskip-1.0mu f\mskip 3.0mu \left(x% \right)=\sqrt{\frac{2}{\pi}}\int^{\infty}_{0}f(t)\cos\left(xt\right)\mathrm{d}t,$ ⓘ Defines: $\mathscr{F}_{\mkern-3.0mu c}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Fourier cosine transform Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $F_{c}(x)$: cosine transformation of $f(t)$ Permalink: http://dlmf.nist.gov/1.14.E9 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): This equation defines the general notation $\mathscr{F}_{\mkern-3.0mu c}\left(f\right)\left(x\right)$ or $\mathscr{F}_{\mkern-3.0mu c}\mskip-1.0mu f\mskip 3.0mu \left(x\right)$. Previously it defined the local notation $F_{c}(x)$ used in this subsection. See also: Annotations for §1.14(ii), §1.14 and Ch.1 1.14.10 $\displaystyle\mathscr{F}_{\mkern-2.0mu s}\left(f\right)\left(x\right)$ $\displaystyle=\mathscr{F}_{\mkern-2.0mu s}\mskip-1.0mu f\mskip 3.0mu \left(x% \right)=\sqrt{\frac{2}{\pi}}\int^{\infty}_{0}f(t)\sin\left(xt\right)\mathrm{d}t.$ ⓘ Defines: $\mathscr{F}_{\mkern-2.0mu s}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Fourier sine transform Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\int$: integral, $\sin\NVar{z}$: sine function and $F_{s}(x)$: sine transformation of $f(t)$ Permalink: http://dlmf.nist.gov/1.14.E10 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): This equation defines the general notation $\mathscr{F}_{\mkern-2.0mu s}\left(f\right)\left(x\right)$ or $\mathscr{F}_{\mkern-2.0mu s}\mskip-1.0mu f\mskip 3.0mu \left(x\right)$. Previously it defined the local notation $F_{s}(x)$ used in this subsection. See also: Annotations for §1.14(ii), §1.14 and Ch.1

In this subsection we let $F_{c}(x)=\mathscr{F}_{\mkern-3.0mu c}\mskip-1.0mu f\mskip 3.0mu \left(x\right)$, $F_{s}(x)=\mathscr{F}_{\mkern-2.0mu s}\mskip-1.0mu f\mskip 3.0mu \left(x\right)$, $G_{c}(x)=\mathscr{F}_{\mkern-3.0mu c}\mskip-1.0mu g\mskip 3.0mu \left(x\right)$, and $G_{s}(x)=\mathscr{F}_{\mkern-2.0mu s}\mskip-1.0mu g\mskip 3.0mu \left(x\right)$.

### Inversion

If $f(t)$ is absolutely integrable on $[0,\infty)$ and of bounded variation (§1.4(v)) in a neighborhood of $t=u$, then

 1.14.11 $\displaystyle\tfrac{1}{2}(f(u+)+f(u-))$ $\displaystyle=\sqrt{\frac{2}{\pi}}\int^{\infty}_{0}F_{c}(x)\cos\left(ux\right)% \mathrm{d}x,$ 1.14.12 $\displaystyle\tfrac{1}{2}(f(u+)+f(u-))$ $\displaystyle=\sqrt{\frac{2}{\pi}}\int^{\infty}_{0}F_{s}(x)\sin\left(ux\right)% \mathrm{d}x.$

### Parseval’s Formula

If $\int^{\infty}_{0}|f(t)|\mathrm{d}t<\infty$, $g(t)$ is of bounded variation on $(0,\infty)$ and $g(t)\to 0$ as $t\to\infty$, then

 1.14.13 $\displaystyle\int^{\infty}_{0}F_{c}(x)G_{c}(x)\mathrm{d}x$ $\displaystyle=\int^{\infty}_{0}f(t)g(t)\mathrm{d}t,$ 1.14.14 $\displaystyle\int^{\infty}_{0}F_{s}(x)G_{s}(x)\mathrm{d}x$ $\displaystyle=\int^{\infty}_{0}f(t)g(t)\mathrm{d}t,$
 1.14.15 $\displaystyle\int^{\infty}_{0}(F_{c}(x))^{2}\mathrm{d}x$ $\displaystyle=\int^{\infty}_{0}(f(t))^{2}\mathrm{d}t,$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $F_{c}(x)$: cosine transformation of $f(t)$ Permalink: http://dlmf.nist.gov/1.14.E15 Encodings: TeX, pMML, png See also: Annotations for §1.14(ii), §1.14(ii), §1.14 and Ch.1 1.14.16 $\displaystyle\int^{\infty}_{0}(F_{s}(x))^{2}\mathrm{d}x$ $\displaystyle=\int^{\infty}_{0}(f(t))^{2}\mathrm{d}t.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $F_{s}(x)$: sine transformation of $f(t)$ Permalink: http://dlmf.nist.gov/1.14.E16 Encodings: TeX, pMML, png See also: Annotations for §1.14(ii), §1.14(ii), §1.14 and Ch.1

## §1.14(iii) Laplace Transform

Suppose $f(t)$ is a real- or complex-valued function and $s$ is a real or complex parameter. The Laplace transform of $f$ is defined by

 1.14.17 $\mathscr{L}\left(f\right)\left(s\right)=\mathscr{L}\mskip-3.0mu f\mskip 3.0mu % \left(s\right)=\int^{\infty}_{0}e^{-st}f(t)\mathrm{d}t.$ ⓘ Defines: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform Symbols: $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: 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: TeX, pMML, png 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 $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{L}(f(t);s)$. See also: Annotations for §1.14(iii), §1.14 and Ch.1

### Convergence and Analyticity

Assume that $f(t)$ is piecewise continuous on $[0,\infty)$ and of exponential growth, that is, constants $M$ and $\alpha$ exist such that

 1.14.18 $|f(t)|\leq Me^{\alpha t},$ $0\leq t<\infty$. ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $M$: constant and $\alpha$: constant Referenced by: §1.14(iii), §1.14(iii), §1.14(iii) Permalink: http://dlmf.nist.gov/1.14.E18 Encodings: TeX, pMML, png See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

Then $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ is an analytic function of $s$ for $\Re s>\alpha$. Moreover,

 1.14.19 $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)\to 0,$ $\Re s\to\infty$. ⓘ 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: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{L}(f(t);s)$. See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

Throughout the remainder of this subsection we assume (1.14.18) is satisfied and $\Re s>\alpha$.

### Inversion

If $f(t)$ is continuous and $f^{\prime}(t)$ is piecewise continuous on $[0,\infty)$, then

 1.14.20 $f(t)=\frac{1}{2\pi i}\lim_{T\to\infty}\int^{\sigma+iT}_{\sigma-iT}e^{ts}% \mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)\mathrm{d}s,$ $\sigma>\alpha$. ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sigma\in(a,b)$: parameter and $\alpha$: constant Keywords: Laplace transform A&S Ref: 29.2.2 Referenced by: §1.14(iii) Permalink: http://dlmf.nist.gov/1.14.E20 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{L}(f(t);s)$. See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

Moreover, if $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)=O\left(s^{-K}\right)$ in some half-plane $\Re s\geq\gamma$ and $K>1$, then (1.14.20) holds for $\sigma>\gamma$.

### Translation

If $\Re s>\max(\Re\left(a+\alpha\right),\alpha)$, then

 1.14.21 $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s-a\right)=\mathscr{L}\mskip-3.0mu% f_{a}\mskip 3.0mu \left(s\right),$ ⓘ 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: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{L}(f(t);s)$. In addition, the auxiliary function $f_{a}(t)=e^{at}f(t)$ was introduced. See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

where $f_{a}(t)=e^{at}f(t)$. Also, if $a\geq 0$ then

 1.14.22 $\mathscr{L}\mskip-3.0mu f_{a}^{+}\mskip 3.0mu \left(s\right)=e^{-as}\mathscr{L% }\mskip-3.0mu f\mskip 3.0mu \left(s\right),$ ⓘ 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: TeX, pMML, png 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

where $f^{+}_{a}(t)=H\left(t-a\right)f(t-a)$ and $H$ is the Heaviside function; see (1.16.13).

### Differentiation and Integration

If $f(t)$ is piecewise continuous, then

 1.14.23 ${(-1)}^{n}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}s}^{n}}\mathscr{L}\mskip-3.0mu f% \mskip 3.0mu \left(s\right)=\mathscr{L}\mskip-3.0mu f_{n}\mskip 3.0mu \left(s% \right),$ $n=1,2,3,\dots$, ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative and $n$: nonnegative integer Keywords: Laplace transform Permalink: http://dlmf.nist.gov/1.14.E23 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{L}(f(t);s)$. In addition, the auxiliary function $f_{n}(t)=t^{n}f(t)$ was introduced. See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

where $f_{n}(t)=t^{n}f(t)$. If also $\lim_{t\to 0+}f(t)/t$ exists, then

 1.14.24 $\int^{\infty}_{s}\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(u\right)\mathrm{d% }u=\mathscr{L}\mskip-3.0mu f_{-1}\mskip 3.0mu \left(s\right),$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\mathrm{d}\NVar{x}$: differential and $\int$: integral Keywords: Laplace transform Permalink: http://dlmf.nist.gov/1.14.E24 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{L}(f(t);s)$. In addition, the auxiliary function $f_{-1}(t)=\frac{f(t)}{t}$ was introduced. See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

where $f_{-1}(t)=\frac{f(t)}{t}$.

### Periodic Functions

If $a>0$ and $f(t+a)=f(t)$ for $t>0$, then

 1.14.25 $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)=\frac{1}{1-e^{-as}}\int^{% a}_{0}e^{-st}f(t)\mathrm{d}t.$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm and $\int$: integral Keywords: Laplace transform Permalink: http://dlmf.nist.gov/1.14.E25 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{L}(f(t);s)$. See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

Alternatively if $f(t+a)=-f(t)$ for $t>0$, then

 1.14.26 $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)=\frac{1}{1+e^{-as}}\int^{% a}_{0}e^{-st}f(t)\mathrm{d}t.$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm and $\int$: integral Keywords: Laplace transform Permalink: http://dlmf.nist.gov/1.14.E26 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{L}(f(t);s)$. See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

### Derivatives

If $f(t)$ is continuous on $[0,\infty)$ and $f^{\prime}(t)$ is piecewise continuous on $(0,\infty)$, then

 1.14.27 $\mathscr{L}\left(f^{\prime}\right)\left(s\right)=s\mathscr{L}\left(f\right)% \left(s\right)-f(0+).$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform Keywords: Laplace transform A&S Ref: 29.2.4 Permalink: http://dlmf.nist.gov/1.14.E27 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transform was changed to $\mathscr{L}\left(f^{\prime}\right)\left(s\right)$ and $\mathscr{L}\left(f\right)\left(s\right)$ from $\mathscr{L}(f^{\prime}(t);s)$ and $\mathscr{L}(f(t);s)$. See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

If $f(t)$ and $f^{\prime}(t)$ are piecewise continuous on $[0,\infty)$ with discontinuities at ($0=$) $t_{0}, then

 1.14.28 $\mathscr{L}\left(f^{\prime}\right)\left(s\right)=s\mathscr{L}\left(f\right)% \left(s\right)-f(0+)-\sum^{n}_{k=1}e^{-st_{k}}(f(t_{k}+)-f(t_{k}-)).$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform, $\mathrm{e}$: base of natural logarithm, $k$: integer and $n$: nonnegative integer Keywords: Laplace transform Permalink: http://dlmf.nist.gov/1.14.E28 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transforms was changed to $\mathscr{L}\left(f^{\prime}\right)\left(s\right)$ and $\mathscr{L}\left(f\right)\left(s\right)$ from $\mathscr{L}(f^{\prime}(t);s)$ and $\mathscr{L}(f(t);s)$. See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

Next, assume $f(t)$, $f^{\prime}(t)$, $\dots$, $f^{(n-1)}(t)$ are continuous and each satisfies (1.14.18). Also assume that $f^{(n)}(t)$ is piecewise continuous on $[0,\infty)$. Then

 1.14.29 $\mathscr{L}\left(f^{(n)}\right)\left(s\right)=s^{n}\mathscr{L}\left(f\right)% \left(s\right)-s^{n-1}f(0+)-s^{n-2}f^{\prime}(0+)-\dots-f^{(n-1)}(0+).$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform and $n$: nonnegative integer Keywords: Laplace transform A&S Ref: 29.2.5 Permalink: http://dlmf.nist.gov/1.14.E29 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Laplace transforms was changed to $\mathscr{L}\left(f^{(n)}\right)\left(s\right)$ and $\mathscr{L}\left(f\right)\left(s\right)$ from $\mathscr{L}(f^{(n)}(t);s)$ and $\mathscr{L}(f(t);s)$. See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

### Convolution

For Laplace transforms, the convolution of two functions $f(t)$ and $g(t)$, defined on $[0,\infty)$, is

 1.14.30 $(f*g)(t)=\int^{t}_{0}f(u)g(t-u)\mathrm{d}u.$ ⓘ Defines: $*$: convolution (Laplace) (locally) Symbols: $\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.14.E30 Encodings: TeX, pMML, png See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

If $f(t)$ and $g(t)$ are piecewise continuous, then

 1.14.31 $\mathscr{L}\left(f*g\right)=\mathscr{L}\left(f\right)\mathscr{L}\left(g\right).$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform and $*$: convolution (Laplace) Keywords: Laplace transform Permalink: http://dlmf.nist.gov/1.14.E31 Encodings: TeX, pMML, png See also: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

### Uniqueness

If $f(t)$ and $g(t)$ are continuous and $\mathscr{L}\mskip-3.0mu f\mskip 3.0mu \left(s\right)=\mathscr{L}\mskip-3.0mu g% \mskip 3.0mu \left(s\right)$, then $f(t)=g(t)$.

## §1.14(iv) Mellin Transform

The Mellin transform of a real- or complex-valued function $f(x)$ is defined by

 1.14.32 $\mathscr{M}\left(f\right)\left(s\right)=\mathscr{M}\mskip-3.0mu f\mskip 3.0mu % \left(s\right)=\int^{\infty}_{0}x^{s-1}f(x)\mathrm{d}x.$ ⓘ Defines: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform Symbols: $\mathrm{d}\NVar{x}$: differential and $\int$: integral Keywords: Mellin transform Referenced by: §1.14(iv), §1.14(iv) Permalink: http://dlmf.nist.gov/1.14.E32 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\left(f\right)\left(s\right)$ or $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §1.14(iv), §1.14 and Ch.1

If $x^{\sigma-1}f(x)$ is integrable on $(0,\infty)$ for all $\sigma$ in $a<\sigma, then the integral (1.14.32) converges and $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ is an analytic function of $s$ in the vertical strip $a<\Re s. Moreover, for $a<\sigma,

 1.14.33 $\lim_{t\to\pm\infty}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(\sigma+it% \right)=0.$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\mathrm{i}$: imaginary unit and $\sigma\in(a,b)$: parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/1.14.E33 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §1.14(iv), §1.14 and Ch.1

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 $x\to\infty$, then $x^{\sigma-1}f(x)$ is integrable on $(0,\infty)$ for all $\sigma\in(-\alpha,-\beta)$.

### Inversion

Suppose the integral (1.14.32) is absolutely convergent on the line $\Re s=\sigma$ and $f(x)$ is of bounded variation in a neighborhood of $x=u$. Then

 1.14.34 $\tfrac{1}{2}(f(u+)+f(u-))=\frac{1}{2\pi i}\lim_{T\to\infty}\int^{\sigma+iT}_{% \sigma-iT}u^{-s}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)\mathrm{d}s.$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral and $\sigma\in(a,b)$: parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/1.14.E34 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

If $f(x)$ is continuous on $(0,\infty)$ and $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(\sigma+it\right)$ is integrable on $(-\infty,\infty)$, then

 1.14.35 $f(x)=\frac{1}{2\pi i}\int^{\sigma+i\infty}_{\sigma-i\infty}x^{-s}\mathscr{M}% \mskip-3.0mu f\mskip 3.0mu \left(s\right)\mathrm{d}s.$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral and $\sigma\in(a,b)$: parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/1.14.E35 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

### Parseval-type Formulas

Suppose $x^{-\sigma}f(x)$ and $x^{\sigma-1}g(x)$ are absolutely integrable on $(0,\infty)$ and either $\mathscr{M}\mskip-3.0mu g\mskip 3.0mu \left(\sigma+it\right)$ or $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-\sigma-it\right)$ is absolutely integrable on $(-\infty,\infty)$. Then for $y>0$,

 1.14.36 $\int^{\infty}_{0}f(x)g(yx)\mathrm{d}x=\frac{1}{2\pi i}\*\int^{\sigma+i\infty}_% {\sigma-i\infty}y^{-s}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-s\right)% \mathscr{M}\mskip-3.0mu g\mskip 3.0mu \left(s\right)\mathrm{d}s,$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral and $\sigma\in(a,b)$: parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/1.14.E36 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1
 1.14.37 $\int^{\infty}_{0}f(x)g(x)\mathrm{d}x=\frac{1}{2\pi i}\*\int^{\sigma+i\infty}_{% \sigma-i\infty}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-s\right)\mathscr{% M}\mskip-3.0mu g\mskip 3.0mu \left(s\right)\mathrm{d}s.$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral and $\sigma\in(a,b)$: parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/1.14.E37 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

When $f$ is real and $\sigma=\tfrac{1}{2}$,

 1.14.38 $\int^{\infty}_{0}(f(x))^{2}\mathrm{d}x=\frac{1}{2\pi}\int^{\infty}_{-\infty}{% \left|\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(\tfrac{1}{2}+it\right)\right% |}^{2}\mathrm{d}t.$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit and $\int$: integral Keywords: Mellin transform Permalink: http://dlmf.nist.gov/1.14.E38 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

### Convolution

Let

 1.14.39 $(f*g)(x)=\int^{\infty}_{0}f(y)g\left(\frac{x}{y}\right)\frac{\mathrm{d}y}{y}.$ ⓘ Defines: $*$: convolution (Mellin) (locally) Symbols: $\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.14.E39 Encodings: TeX, pMML, png See also: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

If $x^{\sigma-1}f(x)$ and $x^{\sigma-1}g(x)$ are absolutely integrable on $(0,\infty)$, then for $s=\sigma+it$,

 1.14.40 $\int^{\infty}_{0}x^{s-1}(f*g)(x)\mathrm{d}x=\mathscr{M}\mskip-3.0mu f\mskip 3.% 0mu \left(s\right)\mathscr{M}\mskip-3.0mu g\mskip 3.0mu \left(s\right).$ ⓘ Symbols: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Mellin transform, $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $*$: convolution (Mellin) Keywords: Mellin transform Permalink: http://dlmf.nist.gov/1.14.E40 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Mellin transform was changed to $\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathscr{M}(f;s)$. See also: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

## §1.14(v) Hilbert Transform

The Hilbert transform of a real-valued function $f(t)$ is defined in the following equivalent ways:

 1.14.41 $\displaystyle\mathcal{H}\left(f\right)\left(x\right)$ $\displaystyle=\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)=\frac{1}{% \pi}\pvint^{\infty}_{-\infty}\frac{f(t)}{t-x}\mathrm{d}t,$ ⓘ Symbols: $\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}\right)$: Hilbert transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential and $\pvint_{\NVar{a}}^{\NVar{b}}$: Cauchy principal value Permalink: http://dlmf.nist.gov/1.14.E41 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Hilbert transform was changed to $\mathcal{H}\left(f\right)\left(x\right)$ or $\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)$ from $\mathcal{H}(f;x)$. See also: Annotations for §1.14(v), §1.14 and Ch.1 1.14.42 $\displaystyle\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)$ $\displaystyle=\lim_{y\to 0+}\frac{1}{\pi}\int^{\infty}_{-\infty}\frac{t-x}{(t-% x)^{2}+y^{2}}f(t)\mathrm{d}t,$ ⓘ Symbols: $\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}\right)$: Hilbert transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.14.E42 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Hilbert transform was changed to $\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)$ from $\mathcal{H}(f;x)$. See also: Annotations for §1.14(v), §1.14 and Ch.1 1.14.43 $\displaystyle\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)$ $\displaystyle=\lim_{\epsilon\to 0+}\frac{1}{\pi}\int^{\infty}_{\epsilon}\frac{% f(x+t)-f(x-t)}{t}\mathrm{d}t.$ ⓘ Symbols: $\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}\right)$: Hilbert transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential and $\int$: integral Permalink: http://dlmf.nist.gov/1.14.E43 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Hilbert transform was changed to $\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)$ from $\mathcal{H}(f;x)$. See also: Annotations for §1.14(v), §1.14 and Ch.1

### Inversion

Suppose $f(t)$ is continuously differentiable on $(-\infty,\infty)$ and vanishes outside a bounded interval. Then

 1.14.44 $f(x)=-\frac{1}{\pi}\pvint^{\infty}_{-\infty}\frac{\mathcal{H}\mskip-3.0mu f% \mskip 3.0mu \left(u\right)}{u-x}\mathrm{d}u.$ ⓘ Symbols: $\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}\right)$: Hilbert transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential and $\pvint_{\NVar{a}}^{\NVar{b}}$: Cauchy principal value Referenced by: §1.14(v) Permalink: http://dlmf.nist.gov/1.14.E44 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Hilbert transform was changed to $\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(u\right)$ from $\mathcal{H}(f;u)$. See also: Annotations for §1.14(v), §1.14(v), §1.14 and Ch.1

### Inequalities

If $|f(t)|^{p}$, $p>1$, is integrable on $(-\infty,\infty)$, then so is $|\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)|^{p}$ and

 1.14.45 $\int^{\infty}_{-\infty}|\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)|^% {p}\mathrm{d}x\leq A_{p}\int^{\infty}_{-\infty}|f(t)|^{p}\mathrm{d}t,$ ⓘ Symbols: $\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}\right)$: Hilbert transform, $\mathrm{d}\NVar{x}$: differential, $\int$: integral and $A_{p}$: coefficient Permalink: http://dlmf.nist.gov/1.14.E45 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Hilbert transform was changed to $\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(x\right)$ from $\mathcal{H}(f;x)$. See also: Annotations for §1.14(v), §1.14(v), §1.14 and Ch.1

where $A_{p}=\tan\left(\tfrac{1}{2}\pi/p\right)$ when $1, or $\cot\left(\tfrac{1}{2}\pi/p\right)$ when $p\geq 2$. These bounds are sharp, and equality holds when $p=2$.

### Fourier Transform

When $f(t)$ satisfies the same conditions as those for (1.14.44),

 1.14.46 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathcal{H}\mskip-3.0mu f\mskip 3.% 0mu \left(u\right)e^{iux}\mathrm{d}u=-\mathrm{i}(\operatorname{sign}x)\mathscr% {F}\mskip-3.0mu f\mskip 3.0mu \left(x\right),$ ⓘ 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, $\mathrm{e}$: 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: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Hilbert transform was changed to $\mathcal{H}\mskip-3.0mu f\mskip 3.0mu \left(t\right)$ from $\mathcal{H}(f;t)$, and the notation for the Fourier transform was changed to $\mathscr{F}\mskip-3.0mu f\mskip 3.0mu \left(x\right)$ 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

where $\mathscr{F}\mskip-3.0mu f\mskip 3.0mu \left(x\right)$ is given by (1.14.1).

## §1.14(vi) Stieltjes Transform

The Stieltjes transform of a real-valued function $f(t)$ is defined by

 1.14.47 $\mathcal{S}\left(f\right)\left(s\right)=\mathcal{S}\mskip-3.0mu f\mskip 3.0mu % \left(s\right)=\int^{\infty}_{0}\frac{f(t)}{s+t}\mathrm{d}t.$ ⓘ Defines: $\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Stieltjes transform Symbols: $\mathrm{d}\NVar{x}$: differential and $\int$: integral Referenced by: §1.14(vi), §1.14(vi) Permalink: http://dlmf.nist.gov/1.14.E47 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Stieltjes transform was changed to $\mathcal{S}\left(f\right)\left(s\right)$ or $\mathcal{S}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathcal{S}(f;s)$. See also: Annotations for §1.14(vi), §1.14 and Ch.1

Sufficient conditions for the integral to converge are that $s$ is a positive real number, and $f(t)=O\left(t^{-\delta}\right)$ as $t\to\infty$, where $\delta>0$.

If the integral converges, then it converges uniformly in any compact domain in the complex $s$-plane not containing any point of the interval $(-\infty,0]$. In this case, $\mathcal{S}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ represents an analytic function in the $s$-plane cut along the negative real axis, and

 1.14.48 $\frac{{\mathrm{d}}^{m}}{{\mathrm{d}s}^{m}}\mathcal{S}\mskip-3.0mu f\mskip 3.0% mu \left(s\right)=(-1)^{m}m!\int^{\infty}_{0}\frac{f(t)\mathrm{d}t}{(s+t)^{m+1% }},$ $m=0,1,2,\dots$. ⓘ Symbols: $\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Stieltjes transform, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\mathrm{d}\NVar{x}$: differential, $!$: factorial (as in $n!$), $\int$: integral and $m$: nonnegative integer Permalink: http://dlmf.nist.gov/1.14.E48 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Stieltjes transform was changed to $\mathcal{S}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathcal{S}(f;s)$. See also: Annotations for §1.14(vi), §1.14 and Ch.1

### Inversion

If $f(t)$ is absolutely integrable on $[0,R]$ for every finite $R$, and the integral (1.14.47) converges, then

 1.14.49 $\lim_{t\to 0+}\frac{\mathcal{S}\mskip-3.0mu f\mskip 3.0mu \left(-\sigma-it% \right)-\mathcal{S}\mskip-3.0mu f\mskip 3.0mu \left(-\sigma+it\right)}{2\pi i}% =\tfrac{1}{2}(f(\sigma+)+f(\sigma-)),$ ⓘ Symbols: $\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Stieltjes transform, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit and $\sigma\in(a,b)$: parameter Permalink: http://dlmf.nist.gov/1.14.E49 Encodings: TeX, pMML, png Notational Change (effective with 1.0.15): The notation for the Stieltjes transform was changed to $\mathcal{S}\mskip-3.0mu f\mskip 3.0mu \left(s\right)$ from $\mathcal{S}(f;s)$. See also: Annotations for §1.14(vi), §1.14(vi), §1.14 and Ch.1

for all values of the positive constant $\sigma$ for which the right-hand side exists.

### Laplace Transform

If $f(t)$ is piecewise continuous on $[0,\infty)$ and the integral (1.14.47) converges, then

 1.14.50 $\mathcal{S}\left(f\right)=\mathscr{L}\left(\mathscr{L}\left(f\right)\right).$ ⓘ Symbols: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Laplace transform and $\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Stieltjes transform Permalink: http://dlmf.nist.gov/1.14.E50 Encodings: TeX, pMML, png See also: Annotations for §1.14(vi), §1.14(vi), §1.14 and Ch.1

## §1.14(viii) Compendia

For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2000), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).