1 Algebraic and Analytic Methods Topics of Discussion 1.13 Differential Equations 1.15 Summability Methods

§1.14 Integral Transforms

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Keywords:

integral transforms

Referenced by:

Erratum (V1.0.15) for Section 1.14, Erratum (V1.0.15) for Section 1.14

Permalink:

http://dlmf.nist.gov/1.14

Addition (effective with 1.0.15):

The following table summarizes the changes that were made to the notation for integral transforms throughout the DLMF.

Notation for Integral Transforms

Transform	New Notation	Abbreviated Notation	Old Notation
Fourier	$\mathscr{F}\left(f\right)\left(x\right)$	$\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right)$
Fourier Cosine	$\mathscr{F}_{\mkern-3.0muc}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Fourier Sine	$\mathscr{F}_{\mkern-2.0mus}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Laplace	$\mathscr{L}\left(f\right)\left(s\right)$	$\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{L}(f(t);s)$
Mellin	$\mathscr{M}\left(f\right)\left(s\right)$	$\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{M}(f;s)$
Hilbert	$\mathcal{H}\left(f\right)\left(s\right)$	$\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{H}(f;s)$
Stieltjes	$\mathcal{S}\left(f\right)\left(s\right)$	$\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{S}(f;s)$

See also:

Annotations for Ch.1

§1.14(i) Fourier Transform
§1.14(ii) Fourier Cosine and Sine Transforms
§1.14(iii) Laplace Transform
§1.14(iv) Mellin Transform
§1.14(v) Hilbert Transform
§1.14(vi) Stieltjes Transform
§1.14(vii) Tables
§1.14(viii) Compendia

§1.14(i) Fourier Transform

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Keywords:: Fourier transform, convergence, definitions
Notes:: See Titchmarsh (1986a, pp. 3–15, 42, 50–59).
Referenced by:: §1.15(v), (1.16.29), Erratum (V1.2.0) §1.14(i)
Permalink:: http://dlmf.nist.gov/1.14.i
Addition (effective with 1.2.0):: Equation (1.14.7_5) was added.
Notational Addition (effective with 1.0.15):: A general notation for the Fourier transform of functions of a single variable was not defined in the DLMF. In this release we provide such a definition in (1.14.1).
See also:: Annotations for §1.14 and Ch.1

The Fourier transform of a real- or complex-valued function is defined by

1.14.1 $\mathscr{F}\left(f\right)\left(x\right)=\mathscr{F}\mskip-3.0muf\mskip 3.0mu% \left(x\right)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}f(t){\mathrm{e}}^{% \mathrm{i}xt}\,\mathrm{d}t.$

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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 of , $\mathrm{e}$ : 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:: TeX, pMML, png
Notational Change (effective with 1.0.15):: This equation defines $\mathscr{F}\left(f\right)\left(x\right)$ or $\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right)$ 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 $\mathrm{i}xt$ by $-\mathrm{i}xt$ ). 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 is absolutely integrable on $(-\infty,\infty)$ , then is continuous, $F(x)\to 0$ as $x\to\pm\infty$ , and

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : Fourier transform of and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.14.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §1.14(i), §1.14 and Ch.1

Inversion

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Keywords:: Fourier transform, inversion
See also:: Annotations for §1.14(i), §1.14 and Ch.1

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value and : Fourier transform of
Permalink:: http://dlmf.nist.gov/1.14.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

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

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and : Fourier transform of
Referenced by:: §1.17(ii)
Permalink:: http://dlmf.nist.gov/1.14.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

Convolution

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Keywords:: Fourier transform, convolution
See also:: Annotations for §1.14(i), §1.14 and Ch.1

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

1.14.5 $(f*g)(t)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}f(t-s)g(s)\,\mathrm{d}s.$

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Defines:: : convolution (Fourier) (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of 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 and are absolutely integrable on $(-\infty,\infty)$ , then so is , and its Fourier transform is , where is the Fourier transform of .

Parseval’s Formula

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Keywords:: Fourier transform, Parseval’s formula
Modification (effective with 1.2.0):: In the first sentence of this paragraph “absolutely integrable” was replaced with “absolutely and square integrable”.
See also:: Annotations for §1.14(i), §1.14 and Ch.1

Suppose and are absolutely and square integrable on $(-\infty,\infty)$ , then

1.14.6 $(f*g)(t)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}F(x)G(x){\mathrm{e}}^{-% \mathrm{i}tx}\,\mathrm{d}x,$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, : Fourier transform of , : convolution (Fourier) and : Fourier transform of
Permalink:: http://dlmf.nist.gov/1.14.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

1.14.7 $\int^{\infty}_{-\infty}F(x)G(x)\,\mathrm{d}x=\int^{\infty}_{-\infty}f(t)g(-t)% \,\mathrm{d}t,$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : Fourier transform of and : Fourier transform of
Permalink:: http://dlmf.nist.gov/1.14.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

1.14.7_5 $\int^{\infty}_{-\infty}F(x)\overline{G(x)}\,\mathrm{d}x=\int^{\infty}_{-\infty% }f(t)\overline{g(t)}\,\mathrm{d}t,$

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : Fourier transform of and : Fourier transform of
Referenced by:: §1.14(i), §1.14(i), Erratum (V1.2.0) §1.14(i)
Permalink:: http://dlmf.nist.gov/1.14.E7_5
Encodings:: TeX, pMML, png
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

1.14.8 $\int^{\infty}_{-\infty}{\left|F(x)\right|}^{2}\,\mathrm{d}x=\int^{\infty}_{-% \infty}{\left|f(t)\right|}^{2}\,\mathrm{d}t.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : Fourier transform of and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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

(1.14.7_5) and (1.14.8) are Parseval’s formulas.

Poisson’s Summation Formula

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Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §1.14(i), §1.14 and Ch.1

See §1.8(iv) and especially (1.8.14).

Uniqueness

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Keywords:: Fourier transform, uniqueness
See also:: Annotations for §1.14(i), §1.14 and Ch.1

If and are continuous and absolutely integrable on $(-\infty,\infty)$ , and for all , then for all .

§1.14(ii) Fourier Cosine and Sine Transforms

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Keywords:: Fourier cosine and sine transforms, definition
Notes:: See Titchmarsh (1986a, pp. 3–15, 50–59).
Referenced by:: §15.17(iii)
Permalink:: http://dlmf.nist.gov/1.14.ii
See also:: Annotations for §1.14 and Ch.1

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

1.14.9 $\mathscr{F}_{\mkern-3.0muc}\left(f\right)\left(x\right)=\mathscr{F}_{\mkern-3.% 0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)=\sqrt{\frac{2}{\pi}}\int^{\infty}% _{0}f(t)\cos\left(xt\right)\,\mathrm{d}t,$

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Defines:: $\mathscr{F}_{\mkern-3.0muc}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : 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 of , $\int$ : integral and $F_{c}(x)$ : cosine transformation of
Referenced by:: §1.17(ii), §1.18(vi), §1.18(vi)
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 $\mathscr{F}_{\mkern-3.0muc}\left(f\right)\left(x\right)$ or $\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)$ . 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 $\mathscr{F}_{\mkern-2.0mus}\left(f\right)\left(x\right)=\mathscr{F}_{\mkern-2.% 0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)=\sqrt{\frac{2}{\pi}}\int^{\infty}% _{0}f(t)\sin\left(xt\right)\,\mathrm{d}t.$

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Defines:: $\mathscr{F}_{\mkern-2.0mus}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Fourier sine transform
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function and $F_{s}(x)$ : sine transformation of
Referenced by:: §1.17(ii), §1.18(vi), §1.18(vi)
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 $\mathscr{F}_{\mkern-2.0mus}\left(f\right)\left(x\right)$ or $\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)$ . 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 $F_{c}(x)=\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)$ , $F_{s}(x)=\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)$ , $G_{c}(x)=\mathscr{F}_{\mkern-3.0muc}\mskip-1.0mug\mskip 3.0mu\left(x\right)$ , and $G_{s}(x)=\mathscr{F}_{\mkern-2.0mus}\mskip-1.0mug\mskip 3.0mu\left(x\right)$ .

Inversion

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Keywords:: Fourier cosine and sine transforms, inversion
See also:: Annotations for §1.14(ii), §1.14 and Ch.1

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

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $F_{c}(x)$ : cosine transformation of
Referenced by:: §1.17(ii), §1.18(vi), §1.18(vi)
Permalink:: http://dlmf.nist.gov/1.14.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §1.14(ii), §1.14(ii), §1.14 and Ch.1

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\sin\NVar{z}$ : sine function and $F_{s}(x)$ : sine transformation of
Referenced by:: §1.17(ii), §1.18(vi), §1.18(vi)
Permalink:: http://dlmf.nist.gov/1.14.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §1.14(ii), §1.14(ii), §1.14 and Ch.1

Parseval’s Formula

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Keywords:: Fourier cosine and sine transforms, Fourier transform, Parseval’s formula
Modification (effective with 1.2.0):: The sentence immediately before (1.14.13) was replaced.
See also:: Annotations for §1.14(ii), §1.14 and Ch.1

Suppose and are absolutely and square integrable on $[0,\infty)$ , then

1.14.13 $\int^{\infty}_{0}F_{c}(x)G_{c}(x)\,\mathrm{d}x=\int^{\infty}_{0}f(t)g(t)\,% \mathrm{d}t,$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $F_{c}(x)$ : cosine transformation of and $G_{c}(x)$ : cosine transformation of
Referenced by:: §1.14(ii)
Permalink:: http://dlmf.nist.gov/1.14.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §1.14(ii), §1.14(ii), §1.14 and Ch.1

1.14.14 $\int^{\infty}_{0}F_{s}(x)G_{s}(x)\,\mathrm{d}x=\int^{\infty}_{0}f(t)g(t)\,% \mathrm{d}t,$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $F_{s}(x)$ : sine transformation of and $G_{s}(x)$ : sine transformation of
Permalink:: http://dlmf.nist.gov/1.14.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §1.14(ii), §1.14(ii), §1.14 and Ch.1

1.14.15 $\int^{\infty}_{0}(F_{c}(x))^{2}\,\mathrm{d}x=\int^{\infty}_{0}(f(t))^{2}\,% \mathrm{d}t,$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $F_{c}(x)$ : cosine transformation of
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 $\int^{\infty}_{0}(F_{s}(x))^{2}\,\mathrm{d}x=\int^{\infty}_{0}(f(t))^{2}\,% \mathrm{d}t.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and $F_{s}(x)$ : sine transformation of
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

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Keywords:: Laplace transform, definition, notation
Notes:: See Schiff (1999, pp. 12–57, 91–93, 151–157).
Referenced by:: §18.17(i), §2.4(ii), §3.11(iv), §3.5(viii), §3.5(ix)
Permalink:: http://dlmf.nist.gov/1.14.iii
See also:: Annotations for §1.14 and Ch.1

Suppose is a real- or complex-valued function and is a real or complex parameter. The Laplace transform of is defined by

1.14.17 $\mathscr{L}\left(f\right)\left(s\right)=\mathscr{L}\mskip-3.0muf\mskip 3.0mu% \left(s\right)=\int^{\infty}_{0}{\mathrm{e}}^{-st}f(t)\,\mathrm{d}t.$

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Defines:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : 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 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{L}(f(t);s)$ .
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

Convergence and Analyticity

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Keywords:: Laplace transform, analyticity, convergence, exponential growth
Modifications (effective with 1.0.15):: The wording of the sentence above (1.14.18) was modified for clarity.
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

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

1.14.18 $\left|f(t)\right|\leq M{\mathrm{e}}^{\alpha t},$ $0\leq t<\infty$ .

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Symbols:: $\mathrm{e}$ : base of natural logarithm, : constant, $\alpha$ : constant and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Referenced by:: §1.14(iii), §1.14(iii), §1.14(iii), §1.4(v)
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 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ is an analytic function of for $\Re s>\alpha$ . Moreover,

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:: TeX, pMML, png
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

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

Inversion

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Keywords:: Laplace transform, inversion
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

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

1.14.20 $f(t)=\frac{1}{2\pi\mathrm{i}}\lim_{T\to\infty}\int^{\sigma+\mathrm{i}T}_{% \sigma-\mathrm{i}T}{\mathrm{e}}^{ts}\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(% s\right)\,\mathrm{d}s,$ $\sigma>\alpha$ .

ⓘ

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 of , $\mathrm{e}$ : 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 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{L}(f(t);s)$ .
See also:: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

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

Translation

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Keywords:: Laplace transform, translation
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

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

1.14.21 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s-a\right)=\mathscr{L}\mskip-3.0muf_% {a}\mskip 3.0mu\left(s\right),$

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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 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ 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

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

1.14.22 $\mathscr{L}\mskip-3.0muf_{a}^{+}\mskip 3.0mu\left(s\right)={\mathrm{e}}^{-as}% \mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right),$

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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 is the Heaviside function; see (1.16.13).

Differentiation and Integration

ⓘ

Keywords:: Laplace transform, differentiation, integration
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

If is piecewise continuous, then

1.14.23 ${(-1)}^{n}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}s}^{n}}\mathscr{L}\mskip-3.0muf% \mskip 3.0mu\left(s\right)=\mathscr{L}\mskip-3.0muf_{n}\mskip 3.0mu\left(s% \right),$ $n=1,2,3,\dots$ ,

ⓘ

Symbols:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to and : 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 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ 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 $\int^{\infty}_{s}\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(u\right)\,\mathrm{d% }u=\mathscr{L}\mskip-3.0muf_{-1}\mskip 3.0mu\left(s\right),$

ⓘ

Symbols:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform, $\,\mathrm{d}\NVar{x}$ : differential of 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 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ 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

ⓘ

Keywords:: Laplace transform, of periodic functions
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

If and for , then

1.14.25 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)=\frac{1}{1-{\mathrm{e}}^{-% as}}\int^{a}_{0}{\mathrm{e}}^{-st}f(t)\,\mathrm{d}t.$

ⓘ

Symbols:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : 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 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{L}(f(t);s)$ .
See also:: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

Alternatively if for , then

1.14.26 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)=\frac{1}{1+{\mathrm{e}}^{-% as}}\int^{a}_{0}{\mathrm{e}}^{-st}f(t)\,\mathrm{d}t.$

ⓘ

Symbols:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : 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 $\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{L}(f(t);s)$ .
See also:: Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

Derivatives

ⓘ

Keywords:: Laplace transform, derivatives
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

If 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 and $f^{\prime}(t)$ are piecewise continuous on $[0,\infty)$ with discontinuities at () $t_{0}<t_{1}<\cdots<t_{n}$ , then

1.14.28 $\mathscr{L}\left(f^{\prime}\right)\left(s\right)=s\mathscr{L}\left(f\right)% \left(s\right)-f(0+)-\sum^{n}_{k=1}{\mathrm{e}}^{-st_{k}}(f(t_{k}+)-f(t_{k}-)).$

ⓘ

Symbols:: $\mathscr{L}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Laplace transform, $\mathrm{e}$ : base of natural logarithm, : integer and : 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 , $f^{\prime}(t)$ , $\dots$ , $f^{(n-1)}(t)$ 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 : 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

ⓘ

Keywords:: Laplace transform, convolution
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

For Laplace transforms, the convolution of two functions and , 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 of 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 and 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

ⓘ

Keywords:: Laplace transform, uniqueness
See also:: Annotations for §1.14(iii), §1.14 and Ch.1

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

§1.14(iv) Mellin Transform

ⓘ

Keywords:: Mellin transform, analyticity, convergence, definition, notation
Notes:: See Paris and Kaminski (2001, pp. 79–89), Titchmarsh (1986a, pp. 51–53, 60), and Wong (1989, pp. 147–152).
Referenced by:: §2.5(i), §3.5(ix), §7.7(ii)
Permalink:: http://dlmf.nist.gov/1.14.iv
See also:: Annotations for §1.14 and Ch.1

The Mellin transform of a real- or complex-valued function is defined by

1.14.32 $\mathscr{M}\left(f\right)\left(s\right)=\mathscr{M}\mskip-3.0muf\mskip 3.0mu% \left(s\right)=\int^{\infty}_{0}x^{s-1}f(x)\,\mathrm{d}x.$

ⓘ

Defines:: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of 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 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ 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 $a<\sigma<b$ , then the integral (1.14.32) converges and $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ is an analytic function of in the vertical strip $a<\Re s<b$ . Moreover, for $a<\sigma<b$ ,

1.14.33 $\lim_{t\to\pm\infty}\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(\sigma+\mathrm{i% }t\right)=0.$

ⓘ

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 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$ .
See also:: Annotations for §1.14(iv), §1.14 and Ch.1

Note: If 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

ⓘ

Keywords:: Mellin transform, inversion
See also:: Annotations for §1.14(iv), §1.14 and Ch.1

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

1.14.34 $\tfrac{1}{2}(f(u+)+f(u-))=\frac{1}{2\pi\mathrm{i}}\lim_{T\to\infty}\int^{% \sigma+\mathrm{i}T}_{\sigma-\mathrm{i}T}u^{-s}\mathscr{M}\mskip-3.0muf\mskip 3% .0mu\left(s\right)\,\mathrm{d}s.$

ⓘ

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 of , $\mathrm{i}$ : 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 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$ .
See also:: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

If is continuous on $(0,\infty)$ and $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(\sigma+\mathrm{i}t\right)$ is integrable on $(-\infty,\infty)$ , then

1.14.35 $f(x)=\frac{1}{2\pi\mathrm{i}}\int^{\sigma+\mathrm{i}\infty}_{\sigma-\mathrm{i}% \infty}x^{-s}\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)\,\mathrm{d}s.$

ⓘ

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 of , $\mathrm{i}$ : 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 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$ .
See also:: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

Parseval-type Formulas

ⓘ

Keywords:: Mellin transform, Parseval-type formulas
See also:: Annotations for §1.14(iv), §1.14 and Ch.1

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

1.14.36 $\int^{\infty}_{0}f(x)g(yx)\,\mathrm{d}x=\frac{1}{2\pi\mathrm{i}}\*\int^{\sigma% +\mathrm{i}\infty}_{\sigma-\mathrm{i}\infty}y^{-s}\mathscr{M}\mskip-3.0muf% \mskip 3.0mu\left(1-s\right)\mathscr{M}\mskip-3.0mug\mskip 3.0mu\left(s\right)% \,\mathrm{d}s,$

ⓘ

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 of , $\mathrm{i}$ : 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 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$ .
See also:: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

1.14.37 $\int^{\infty}_{0}f(x)g(x)\,\mathrm{d}x=\frac{1}{2\pi\mathrm{i}}\*\int^{\sigma+% \mathrm{i}\infty}_{\sigma-\mathrm{i}\infty}\mathscr{M}\mskip-3.0muf\mskip 3.0% mu\left(1-s\right)\mathscr{M}\mskip-3.0mug\mskip 3.0mu\left(s\right)\,\mathrm{% d}s.$

ⓘ

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 of , $\mathrm{i}$ : 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 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$ .
See also:: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

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

1.14.38 $\int^{\infty}_{0}(f(x))^{2}\,\mathrm{d}x=\frac{1}{2\pi}\int^{\infty}_{-\infty}% {\left|\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(\tfrac{1}{2}+\mathrm{i}t% \right)\right|}^{2}\,\mathrm{d}t.$

ⓘ

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 of , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathscr{M}(f;s)$ .
See also:: Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

Convolution

ⓘ

Keywords:: Mellin transform, convolution
See also:: Annotations for §1.14(iv), §1.14 and Ch.1

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 of 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 $s=\sigma+\mathrm{i}t$ ,

1.14.40 $\int^{\infty}_{0}x^{s-1}(f*g)(x)\,\mathrm{d}x=\mathscr{M}\mskip-3.0muf\mskip 3% .0mu\left(s\right)\mathscr{M}\mskip-3.0mug\mskip 3.0mu\left(s\right).$

ⓘ

Symbols:: $\mathscr{M}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Mellin transform, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : 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 $\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ 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

ⓘ

Defines:: $\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}\right)$ : Hilbert transform
Keywords:: Hilbert transform, definition
Notes:: See Titchmarsh (1986a, pp. 119–132) and Henrici (1986, vol. 3, pp. 197–202). For (1.14.46) see Sneddon (1972, p. 234).
Referenced by:: §1.15(v), §3.5(ix)
Permalink:: http://dlmf.nist.gov/1.14.v
See also:: Annotations for §1.14 and Ch.1

The Hilbert transform of a real-valued function is defined in the following equivalent ways:

1.14.41 $\mathcal{H}\left(f\right)\left(x\right)=\mathcal{H}\mskip-3.0muf\mskip 3.0mu% \left(x\right)=\frac{1}{\pi}\pvint^{\infty}_{-\infty}\frac{f(t)}{t-x}\,\mathrm% {d}t,$

ⓘ

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 of 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 $\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(x\right)$ from $\mathcal{H}(f;x)$ .
See also:: Annotations for §1.14(v), §1.14 and Ch.1

1.14.42 $\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(x\right)=\lim_{y\to 0+}\frac{1}{\pi}% \int^{\infty}_{-\infty}\frac{t-x}{(t-x)^{2}+y^{2}}f(t)\,\mathrm{d}t,$

ⓘ

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 of 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 $\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(x\right)$ from $\mathcal{H}(f;x)$ .
See also:: Annotations for §1.14(v), §1.14 and Ch.1

1.14.43 $\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(x\right)=\lim_{\epsilon\to 0+}\frac{% 1}{\pi}\int^{\infty}_{\epsilon}\frac{f(x+t)-f(x-t)}{t}\,\mathrm{d}t.$

ⓘ

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 of 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 $\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(x\right)$ from $\mathcal{H}(f;x)$ .
See also:: Annotations for §1.14(v), §1.14 and Ch.1

Inversion

ⓘ

Keywords:: Hilbert transform, inversion
See also:: Annotations for §1.14(v), §1.14 and Ch.1

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

1.14.44 $f(x)=-\frac{1}{\pi}\pvint^{\infty}_{-\infty}\frac{\mathcal{H}\mskip-3.0muf% \mskip 3.0mu\left(u\right)}{u-x}\,\mathrm{d}u.$

ⓘ

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 of 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 $\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(u\right)$ from $\mathcal{H}(f;u)$ .
See also:: Annotations for §1.14(v), §1.14(v), §1.14 and Ch.1

Inequalities

ⓘ

Keywords:: Hilbert transform, inequalities
See also:: Annotations for §1.14(v), §1.14 and Ch.1

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

1.14.45 $\int^{\infty}_{-\infty}{\left|\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(x% \right)\right|}^{p}\,\mathrm{d}x\leq A_{p}\int^{\infty}_{-\infty}{\left|f(t)% \right|}^{p}\,\mathrm{d}t,$

ⓘ

Symbols:: $\mathcal{H}\left(\NVar{f}\right)\left(\NVar{x}\right)$ : Hilbert transform, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $A_{p}$ : coefficient and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
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 $\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(x\right)$ 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 $1<p\leq 2$ , or $\cot\left(\tfrac{1}{2}\pi/p\right)$ when $p\geq 2$ . These bounds are sharp, and equality holds when .

Fourier Transform

ⓘ

Keywords:: Fourier transform of, Hilbert transform
See also:: Annotations for §1.14(v), §1.14 and Ch.1

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

1.14.46 $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathcal{H}\mskip-3.0muf\mskip 3.0% mu\left(u\right){\mathrm{e}}^{\mathrm{i}ux}\,\mathrm{d}u=-\mathrm{i}(% \operatorname{sign}x)\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right),$

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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 , $\mathrm{e}$ : 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 $\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(t\right)$ 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 . In addition, the integration variable was changed from to .
See also:: Annotations for §1.14(v), §1.14(v), §1.14 and Ch.1

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

§1.14(vi) Stieltjes Transform

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Keywords:: Stieltjes transform, analyticity, convergence, definition, derivatives
Notes:: See Widder (1941, pp. 325–328, 340–341).
Permalink:: http://dlmf.nist.gov/1.14.vi
See also:: Annotations for §1.14 and Ch.1

The Stieltjes transform of a real-valued function is defined by

1.14.47 $\mathcal{S}\left(f\right)\left(s\right)=\mathcal{S}\mskip-3.0muf\mskip 3.0mu% \left(s\right)=\int^{\infty}_{0}\frac{f(t)}{s+t}\,\mathrm{d}t.$

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Defines:: $\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Stieltjes transform
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of 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 $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ 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 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 -plane not containing any point of the interval $(-\infty,0]$ . In this case, $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ represents an analytic function in the -plane cut along the negative real axis, and

1.14.48 $\frac{{\mathrm{d}}^{m}}{{\mathrm{d}s}^{m}}\mathcal{S}\mskip-3.0muf\mskip 3.0mu% \left(s\right)=(-1)^{m}m!\int^{\infty}_{0}\frac{f(t)\,\mathrm{d}t}{(s+t)^{m+1}},$ $m=0,1,2,\dots$ .

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Symbols:: $\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$ : Stieltjes transform, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral and : 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 $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ from $\mathcal{S}(f;s)$ .
See also:: Annotations for §1.14(vi), §1.14 and Ch.1

Inversion

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Keywords:: Stieltjes transform, inversion
See also:: Annotations for §1.14(vi), §1.14 and Ch.1

If is absolutely integrable on for every finite , and the integral (1.14.47) converges, then

1.14.49 $\lim_{t\to 0+}\frac{\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(-\sigma-\mathrm{% i}t\right)-\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(-\sigma+\mathrm{i}t\right% )}{2\pi\mathrm{i}}=\tfrac{1}{2}(f(\sigma+)+f(\sigma-)),$

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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 $\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$ 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

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Keywords:: Stieltjes transform, representation as double Laplace transform
See also:: Annotations for §1.14(vi), §1.14 and Ch.1

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

1.14.50 $\mathcal{S}\left(f\right)=\mathscr{L}\left(\mathscr{L}\left(f\right)\right).$

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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(vii) Tables

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Notes:: See Davies (1984, pp. 11–13, 103–108, 152–153, 209–211), Pinkus and Zafrany (1997, pp. 147–149), Schiff (1999, pp. 209–218), Titchmarsh (1986a, pp. 176–210), and Wong (1989, pp. 192–194).
Permalink:: http://dlmf.nist.gov/1.14.vii
See also:: Annotations for §1.14 and Ch.1

Table 1.14.1: Fourier transforms.

	$\dfrac{1}{\sqrt{2\pi}}\displaystyle\int^{\infty}_{-\infty}f(t){\mathrm{e}}^{% \mathrm{i}xt}\,\mathrm{d}t$
$\begin{cases}1,&\left\|t\right\|<a,\\ 0,&\mbox{otherwise}\end{cases}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{\sin\left(ax\right)}{x}}$
${\mathrm{e}}^{-a\left\|t\right\|}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{a}{a^{2}+x^{2}}}$ ,
$t{\mathrm{e}}^{-a\left\|t\right\|}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{2\mathrm{i}ax}{(a^{2}+x^{2})^{2}}}$ ,
$\left\|t\right\|{\mathrm{e}}^{-a\left\|t\right\|}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{a^{2}-x^{2}}{(a^{2}+x^{2})^{2}}}$ ,
$\dfrac{{\mathrm{e}}^{-a\left\|t\right\|}}{{\left\|t\right\|}^{1/2}}$	$\dfrac{(a+(a^{2}+x^{2})^{1/2})^{1/2}}{(a^{2}+x^{2})^{1/2}}$ ,
$\dfrac{\sinh\left(at\right)}{\sinh\left(\pi t\right)}$	$\dfrac{1}{\sqrt{2\pi}}\dfrac{\sin a}{\cosh x+\cos a}$ ,	$-\pi<a<\pi$
$\dfrac{\cosh\left(at\right)}{\cosh\left(\pi t\right)}$	$\sqrt{\dfrac{2}{\pi}}\dfrac{\cos\left(\frac{1}{2}a\right)\cosh\left(\frac{1}{2% }x\right)}{\cosh x+\cos a}$ ,	$-\pi<a<\pi$
${\mathrm{e}}^{-at^{2}}$	$\dfrac{1}{\sqrt{2a}}{\mathrm{e}}^{-x^{2}/(4a)}$ ,
$\sin\left(at^{2}\right)$	$-\dfrac{1}{\sqrt{2a}}\sin\left(\dfrac{x^{2}}{4a}-\dfrac{\pi}{4}\right)$ ,
$\cos\left(at^{2}\right)$	$\dfrac{1}{\sqrt{2a}}\cos\left(\dfrac{x^{2}}{4a}-\dfrac{\pi}{4}\right)$ ,

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Keywords:: Fourier transform, tables
Permalink:: http://dlmf.nist.gov/1.14.T1
See also:: Annotations for §1.14(vii), §1.14 and Ch.1

Table 1.14.2: Fourier cosine transforms.

	$\sqrt{\dfrac{2}{\pi}}\displaystyle\int^{\infty}_{0}\!\!\!f(t)\cos\left(xt% \right)\,\mathrm{d}t$ ,
$\begin{cases}1,&0<t\leq a,\\ 0,&\mbox{otherwise}\end{cases}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{\sin\left(ax\right)}{x}}$
$\dfrac{1}{a^{2}+t^{2}}$	$\displaystyle{\sqrt{\frac{\pi}{2}}\frac{{\mathrm{e}}^{-ax}}{a}}$ ,	$\Re a>0$
$\dfrac{1}{(a^{2}+t^{2})^{2}}$	$\displaystyle{\sqrt{\frac{\pi}{2}}\frac{(1+ax){\mathrm{e}}^{-ax}}{2a^{3}}}$ ,	$\Re a>0$
$\dfrac{4a^{3}}{4a^{4}+t^{4}}$	$\sqrt{\pi}{\mathrm{e}}^{-ax}\sin\left(ax+\frac{1}{4}\pi\right)$ ,	$\Re a>0$
${\mathrm{e}}^{-at}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{a}{a^{2}+x^{2}}}$ ,	$\Re a>0$
${\mathrm{e}}^{-at^{2}}$	$\dfrac{1}{\sqrt{2a}}{\mathrm{e}}^{-x^{2}/(4a)}$ ,	$\Re a>0$
$\sin\left(at^{2}\right)$	$-\dfrac{1}{\sqrt{2a}}\sin\left(\dfrac{x^{2}}{4a}-\dfrac{\pi}{4}\right)$ ,
$\cos\left(at^{2}\right)$	$\dfrac{1}{\sqrt{2a}}\cos\left(\dfrac{x^{2}}{4a}-\dfrac{\pi}{4}\right)$ ,
$\ln\left(1+\dfrac{a^{2}}{t^{2}}\right)$	$\displaystyle{\sqrt{2\pi}\frac{1-{\mathrm{e}}^{-ax}}{x}}$ ,	$\Re a>0$
$\ln\left(\dfrac{a^{2}+t^{2}}{b^{2}+t^{2}}\right)$	$\displaystyle{\sqrt{2\pi}\frac{{\mathrm{e}}^{-bx}-{\mathrm{e}}^{-ax}}{x}}$ ,	$\Re a>0$ , $\Re b>0$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part and $\sin\NVar{z}$ : sine function
Keywords:: Fourier cosine and sine transforms, tables
Permalink:: http://dlmf.nist.gov/1.14.T2
See also:: Annotations for §1.14(vii), §1.14 and Ch.1

Table 1.14.3: Fourier sine transforms.

	$\sqrt{\dfrac{2}{\pi}}\displaystyle\int^{\infty}_{0}\!\!\!f(t)\sin\left(xt% \right)\,\mathrm{d}t$ ,
$t^{-1}$	$\displaystyle{\sqrt{\frac{\pi}{2}}}$
$t^{-1/2}$	$x^{-1/2}$
$t^{-3/2}$	$2x^{1/2}$
$\dfrac{t}{a^{2}+t^{2}}$	$\displaystyle{\sqrt{\frac{\pi}{2}}{\mathrm{e}}^{-ax}}$ ,	$\Re a>0$
$\dfrac{t}{(a^{2}+t^{2})^{2}}$	$\displaystyle{\sqrt{\frac{\pi}{8}}\frac{x}{a}{\mathrm{e}}^{-ax}}$ ,	$\Re a>0$
$\dfrac{1}{t(a^{2}+t^{2})}$	$\displaystyle{\sqrt{\frac{\pi}{2}}\frac{1-{\mathrm{e}}^{-ax}}{a^{2}}}$ ,	$\Re a>0$
$\dfrac{{\mathrm{e}}^{-at}}{t}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\operatorname{arctan}\left(\frac{x}{a}\right)}$ ,	$\Re a>0$
${\mathrm{e}}^{-at}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{x}{a^{2}+x^{2}}}$ ,	$\Re a>0$
$t{\mathrm{e}}^{-at}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{2ax}{(a^{2}+x^{2})^{2}}}$ ,	$\Re a>0$
$t{\mathrm{e}}^{-at^{2}}$	$(2a)^{-3/2}x{\mathrm{e}}^{-x^{2}/(4a)}$ ,	$\left\|\operatorname{ph}a\right\|<\tfrac{1}{2}\pi$
$\dfrac{\sin\left(at\right)}{t}$	$\dfrac{1}{\sqrt{2\pi}}\ln\left\|\dfrac{x+a}{x-a}\right\|$ ,
$\operatorname{arctan}\left(\dfrac{t}{a}\right)$	$\displaystyle{\sqrt{\frac{\pi}{2}}\frac{{\mathrm{e}}^{-ax}}{x}}$ ,
$\ln\left\|\dfrac{t+a}{t-a}\right\|$	$\displaystyle{\sqrt{2\pi}\frac{\sin\left(ax\right)}{x}}$ ,

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\Re$ : real part, $\sin\NVar{z}$ : sine function and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Keywords:: Fourier cosine and sine transforms, tables
Permalink:: http://dlmf.nist.gov/1.14.T3
See also:: Annotations for §1.14(vii), §1.14 and Ch.1

Table 1.14.4: Laplace transforms.

	$\displaystyle\int^{\infty}_{0}{\mathrm{e}}^{-st}f(t)\,\mathrm{d}t$
1	$\dfrac{1}{s}$ ,	$\Re s>0$
$\dfrac{t^{n}}{n!}$	$\dfrac{1}{s^{n+1}}$ ,	$\Re s>0$
$\dfrac{1}{\sqrt{\pi t}}$	$\dfrac{1}{\sqrt{s}}$ ,	$\Re s>0$
${\mathrm{e}}^{-at}$	$\dfrac{1}{s+a}$ ,	$\Re(s+a)>0$
$\dfrac{t^{n}{\mathrm{e}}^{-at}}{n!}$	$\dfrac{1}{(s+a)^{n+1}}$ ,	$\Re(s+a)>0$
$\dfrac{{\mathrm{e}}^{-at}-{\mathrm{e}}^{-bt}}{b-a}$	$\dfrac{1}{(s+a)(s+b)}$ ,	$a\not=b$ , $\Re s>-\Re a$ , $\Re s>-\Re b$
$\sin\left(at\right)$	$\dfrac{a}{s^{2}+a^{2}}$ ,	$\Re s>\left\|\Im a\right\|$
$\cos\left(at\right)$	$\dfrac{s}{s^{2}+a^{2}}$ ,	$\Re s>\left\|\Im a\right\|$
$\sinh\left(at\right)$	$\dfrac{a}{s^{2}-a^{2}}$ ,	$\Re s>\left\|\Re a\right\|$
$\cosh\left(at\right)$	$\dfrac{s}{s^{2}-a^{2}}$ ,	$\Re s>\left\|\Re a\right\|$
$t\sin\left(at\right)$	$\dfrac{2as}{(s^{2}+a^{2})^{2}}$ ,	$\Re s>\left\|\Im a\right\|$
$t\cos\left(at\right)$	$\dfrac{s^{2}-a^{2}}{(s^{2}+a^{2})^{2}}$ ,	$\Re s>\left\|\Im a\right\|$
$\dfrac{{\mathrm{e}}^{-bt}-{\mathrm{e}}^{-at}}{t}$	$\displaystyle{\ln\left(\frac{s+a}{s+b}\right)}$ ,	$\Re s>-\Re a$ , $\Re s>-\Re b$
$\dfrac{2(1-\cosh\left(at\right))}{t}$	$\displaystyle{\ln\left(1-\frac{a^{2}}{s^{2}}\right)}$ ,	$\Re(s+a)>0$
$\dfrac{2(1-\cos\left(at\right))}{t}$	$\displaystyle{\ln\left(1+\frac{a^{2}}{s^{2}}\right)}$ ,	$\Re s>0$
$\dfrac{\sin\left(at\right)}{t}$	$\operatorname{arctan}\left(\dfrac{a}{s}\right)$ ,	$\Re s>0$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\Im$ : imaginary part, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Keywords:: Laplace transform, tables
A&S Ref:: 29.3.1 29.3.3 29.3.4 29.3.8 29.3.10 29.3.12 29.3.15 29.3.16 29.3.17 29.3.18 29.3.22 29.3.22 29.3.104 29.3.109 29.3.108 29.3.110
Permalink:: http://dlmf.nist.gov/1.14.T4
See also:: Annotations for §1.14(vii), §1.14 and Ch.1

Table 1.14.5: Mellin transforms.

	$\displaystyle\int^{\infty}_{0}x^{s-1}f(x)\,\mathrm{d}x$
$\begin{cases}1,&x<a,\\ 0,&x\geq a\end{cases}$	$\dfrac{a^{s}}{s}$ ,	$a\geq 0$ , $\Re s>0$
$\begin{cases}\ln\left(a/x\right),&x<a,\\ 0,&x\geq a\end{cases}$	$\dfrac{a^{s}}{s^{2}}$ ,	$a\geq 0$ , $\Re s>1$
$\dfrac{1}{1-x}$	$\pi\cot\left(s\pi\right)$ ,	$0<\Re s<1$ , (Cauchy p. v.)
$\dfrac{1}{1+x}$	$\pi\csc\left(s\pi\right)$ ,	$0<\Re s<1$
$\ln\left(1+ax\right)$	$\dfrac{\pi\csc\left(s\pi\right)}{sa^{s}}$ ,	$\left\|\operatorname{ph}a\right\|<\pi$ , $-1<\Re s<0$
$\displaystyle{\ln\left\|\frac{1+x}{1-x}\right\|}$	$\dfrac{\pi\tan\left(\frac{1}{2}s\pi\right)}{s}$ ,	$-1<\Re s<1$
$\dfrac{\ln\left(1+x\right)}{x}$	$\dfrac{\pi\csc\left(s\pi\right)}{1-s}$ ,	$0<\Re s<1$
$\operatorname{arctan}x$	$-\dfrac{\pi\sec\left(\frac{1}{2}s\pi\right)}{2s}$ ,	$-1<\Re s<0$
$\operatorname{arccot}x$	$\dfrac{\pi\sec\left(\frac{1}{2}s\pi\right)}{2s}$ ,	$0<\Re s<1$
$\dfrac{1+x\cos\theta}{1+2x\cos\theta+x^{2}}$	$\dfrac{\pi\cos\left(s\theta\right)}{\sin\left(s\pi\right)}$ ,	$-\pi<\theta<\pi$ , $0<\Re s<1$
$\dfrac{x\sin\theta}{1+2x\cos\theta+x^{2}}$	$\dfrac{\pi\sin\left(s\theta\right)}{\sin\left(s\pi\right)}$ ,	$-\pi<\theta<\pi$ , $0<\Re s<1$

§1.14(viii) Compendia

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Keywords:: Fourier cosine and sine transforms, Fourier transform, Laplace transform, Mellin transform, compendia, integral transforms, tables
Permalink:: http://dlmf.nist.gov/1.14.viii
See also:: Annotations for §1.14 and Ch.1

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 (2015), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

1.13 Differential Equations 1.15 Summability Methods

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§1.14 Integral Transforms

Contents

§1.14(i) Fourier Transform

Inversion

Convolution

Parseval’s Formula

Poisson’s Summation Formula

Uniqueness

§1.14(ii) Fourier Cosine and Sine Transforms

Inversion

Parseval’s Formula

§1.14(iii) Laplace Transform

Convergence and Analyticity

Inversion

Translation

Differentiation and Integration

Periodic Functions

Derivatives

Convolution

Uniqueness

§1.14(iv) Mellin Transform

Inversion

Parseval-type Formulas

Convolution

§1.14(v) Hilbert Transform

Inversion

Inequalities

Fourier Transform

§1.14(vi) Stieltjes Transform

Inversion

Laplace Transform

§1.14(vii) Tables

§1.14(viii) Compendia