§1.14 Integral Transforms

Keywords:: integral transforms
Permalink:: http://dlmf.nist.gov/1.14

§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

Notes:: See Titchmarsh (1986a, pp. 3–15, 42, 50–59).
Keywords:: Fourier transform, convergence
Referenced by:: ¶ ‣ §1.15(v)
Permalink:: http://dlmf.nist.gov/1.14.i

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

1.14.1 $F(x)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{{-\infty}}f(t)e^{{ixt}}dt.$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $F(x)$ : Fourier transform of $f(t)$
Referenced by:: ¶ ‣ §1.14(v), §1.17(ii)
Permalink:: http://dlmf.nist.gov/1.14.E1
Encodings:: TeX, pMML, png

(Some references replace $ixt$ by $-ixt$ .)

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $F(x)$ : Fourier transform of $f(t)$
Permalink:: http://dlmf.nist.gov/1.14.E2
Encodings:: TeX, pMML, png

¶ Inversion

Keywords:: Fourier transform

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

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\pvint _{a}^{b}$ : Cauchy principal value and $F(x)$ : Fourier transform of $f(t)$
Permalink:: http://dlmf.nist.gov/1.14.E3
Encodings:: TeX, pMML, png

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

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $F(x)$ : Fourier transform of $f(t)$
Referenced by:: §1.17(ii)
Permalink:: http://dlmf.nist.gov/1.14.E4
Encodings:: TeX, pMML, png

¶ Convolution

Keywords:: Fourier transform

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

Defines:: $*$ : $f*g$ : convolution for Fourier transforms
Symbols:: $dx$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.14.E5
Encodings:: TeX, pMML, png

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

Keywords:: Fourier transform, Parseval’s formula

Suppose $f(t)$ and $g(t)$ are absolutely integrable on $(-\infty,\infty)$ , and $F(x)$ and $G(x)$ are their respective Fourier transforms. Then

1.14.6 $(f*g)(t)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{{-\infty}}F(x)G(x)e^{{-itx}}dx,$

Symbols:: $*$ : $f*g$ : convolution for Fourier transforms, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $F(x)$ : Fourier transform of $f(t)$ and $G(x)$ : Fourier transform of $g(t)$
Permalink:: http://dlmf.nist.gov/1.14.E6
Encodings:: TeX, pMML, png

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $F(x)$ : Fourier transform of $f(t)$ and $G(x)$ : Fourier transform of $g(t)$
Permalink:: http://dlmf.nist.gov/1.14.E7
Encodings:: TeX, pMML, png

1.14.8 $\int^{\infty}_{{-\infty}}|F(x)|^{2}dx=\int^{\infty}_{{-\infty}}|f(t)|^{2}dt.$

Symbols:: $dx$ : differential of $x$ , $\int$ : 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

(1.14.8) is Parseval’s formula.

¶ Uniqueness

Keywords:: Fourier transform

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

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

These are defined respectively by

1.14.9 $F_{c}(x)=\sqrt{\frac{2}{\pi}}\int^{\infty}_{0}f(t)\mathop{\cos\/}\nolimits\!\left(xt\right)dt,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $F_{c}(x)$ : cosine transformation of $f(t)$
Permalink:: http://dlmf.nist.gov/1.14.E9
Encodings:: TeX, pMML, png

1.14.10 $F_{s}(x)=\sqrt{\frac{2}{\pi}}\int^{\infty}_{0}f(t)\mathop{\sin\/}\nolimits\!\left(xt\right)dt.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $F_{s}(x)$ : sine transformation of $f(t)$
Permalink:: http://dlmf.nist.gov/1.14.E10
Encodings:: TeX, pMML, png

¶ 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 $\tfrac{1}{2}(f(u+)+f(u-))=\sqrt{\frac{2}{\pi}}\int^{\infty}_{0}F_{c}(x)\mathop{\cos\/}\nolimits\!\left(ux\right)dx,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $F_{c}(x)$ : cosine transformation of $f(t)$
Permalink:: http://dlmf.nist.gov/1.14.E11
Encodings:: TeX, pMML, png

1.14.12 $\tfrac{1}{2}(f(u+)+f(u-))=\sqrt{\frac{2}{\pi}}\int^{\infty}_{0}F_{s}(x)\mathop{\sin\/}\nolimits\!\left(ux\right)dx.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $F_{s}(x)$ : sine transformation of $f(t)$
Permalink:: http://dlmf.nist.gov/1.14.E12
Encodings:: TeX, pMML, png

¶ Parseval’s Formula

Keywords:: Fourier cosine and sine transforms, Fourier transform, Parseval’s formula

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

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $F_{c}(x)$ : cosine transformation of $f(t)$ and $G_{c}(x)$ : cosine transformation of $g(t)$
Permalink:: http://dlmf.nist.gov/1.14.E13
Encodings:: TeX, pMML, png

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $F_{s}(x)$ : sine transformation of $f(t)$ and $G_{s}(x)$ : sine transformation of $g(t)$
Permalink:: http://dlmf.nist.gov/1.14.E14
Encodings:: TeX, pMML, png

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $F_{c}(x)$ : cosine transformation of $f(t)$
Permalink:: http://dlmf.nist.gov/1.14.E15
Encodings:: TeX, pMML, png

1.14.16 $\int^{\infty}_{0}(F_{s}(x))^{2}dx=\int^{\infty}_{0}(f(t))^{2}dt,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $F_{s}(x)$ : sine transformation of $f(t)$
Permalink:: http://dlmf.nist.gov/1.14.E16
Encodings:: TeX, pMML, png

where $G_{c}(x)$ and $G_{s}(x)$ are respectively the cosine and sine transforms of $g(t)$ .

§1.14(iii) Laplace Transform

Notes:: See Schiff (1999, pp. 12–57, 91–93, 151–157).
Keywords:: Laplace transform
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

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 $\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right)=\int^{\infty}_{0}e^{{-st}}f(t)dt.$

Defines:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform
Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function and $\int$ : integral
A&S Ref:: 29.1.1
Permalink:: http://dlmf.nist.gov/1.14.E17
Encodings:: TeX, pMML, png

Alternative notations are $\mathop{\mathscr{L}\/}\nolimits(f(t))$ , $\mathop{\mathscr{L}\/}\nolimits(f;s)$ , or even $\mathop{\mathscr{L}\/}\nolimits(f)$ , when it is not important to display all the variables.

¶ Convergence and Analyticity

Keywords:: Laplace transform, convergence, exponential growth

Assume that on $[0,\infty)$ $f(t)$ is piecewise continuous 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:: $e$ : base of exponential function, $M$ : constant and $\alpha$ : constant
Referenced by:: ¶ ‣ §1.14(iii), ¶ ‣ §1.14(iii)
Permalink:: http://dlmf.nist.gov/1.14.E18
Encodings:: TeX, pMML, png

Then $\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right)$ is an analytic function of $s$ for $\realpart{s}>\alpha$ . Moreover,

1.14.19 $\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right)\to 0,$ $\realpart{s}\to\infty$ .

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform and $\realpart{}$ : real part
Permalink:: http://dlmf.nist.gov/1.14.E19
Encodings:: TeX, pMML, png

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

¶ Inversion

Keywords:: Fourier cosine and sine transforms, Laplace transform

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

1.14.20 $f(t)=\frac{1}{2\pi i}\lim _{{T\to\infty}}\int^{{\sigma+iT}}_{{\sigma-iT}}e^{{ts}}\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right)ds,$ $\sigma>\alpha$ .

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\sigma\in(a,b)$ : parameter
A&S Ref:: 29.2.2
Referenced by:: ¶ ‣ §1.14(iii)
Permalink:: http://dlmf.nist.gov/1.14.E20
Encodings:: TeX, pMML, png

Moreover, if $\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right)=\mathop{O\/}\nolimits\!\left(s^{{-K}}\right)$ in some half-plane $\realpart{s}\geq\gamma$ and $K>1$ , then (1.14.20) holds for $\sigma>\gamma$ .

¶ Translation

Keywords:: Laplace transform

If $\realpart{s}>\max(\realpart{(a+\alpha)},\alpha)$ , then

1.14.21 $\mathop{\mathscr{L}\/}\nolimits\left(f(t);s-a\right)=\mathop{\mathscr{L}\/}\nolimits\left(e^{{at}}f(t);s\right).$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform and $e$ : base of exponential function
Permalink:: http://dlmf.nist.gov/1.14.E21
Encodings:: TeX, pMML, png

Also, if $a\geq 0$ then

1.14.22 $\mathop{\mathscr{L}\/}\nolimits\left(\mathop{H\/}\nolimits\!\left(t-a\right)f(t-a);s\right)=e^{{-as}}\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right),$

Symbols:: $\mathop{H\/}\nolimits\!\left(x\right)$ : Heaviside function, $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform and $e$ : base of exponential function
Permalink:: http://dlmf.nist.gov/1.14.E22
Encodings:: TeX, pMML, png

where $\mathop{H\/}\nolimits$ is the Heaviside function; see (1.16.13).

¶ Differentiation and Integration

Keywords:: Laplace transform, differentiation, integration

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

1.14.23 $\frac{{d}^{n}}{{ds}^{n}}\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right)=\mathop{\mathscr{L}\/}\nolimits\left((-t)^{n}f(t);s\right),$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.14.E23
Encodings:: TeX, pMML, png

If also $\lim _{{t\to 0+}}f(t)/t$ exists, then

1.14.24 $\int^{\infty}_{s}\mathop{\mathscr{L}\/}\nolimits\left(f(t);u\right)du=\mathop{\mathscr{L}\/}\nolimits\left(\frac{f(t)}{t};s\right).$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $dx$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.14.E24
Encodings:: TeX, pMML, png

¶ Periodic Functions

Keywords:: Laplace transform

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

1.14.25 $\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right)=\frac{1}{1-e^{{-as}}}\int^{a}_{0}e^{{-st}}f(t)dt.$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $dx$ : differential of $x$ , $e$ : base of exponential function and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.14.E25
Encodings:: TeX, pMML, png

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

1.14.26 $\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right)=\frac{1}{1+e^{{-as}}}\int^{a}_{0}e^{{-st}}f(t)dt.$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $dx$ : differential of $x$ , $e$ : base of exponential function and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.14.E26
Encodings:: TeX, pMML, png

¶ Derivatives

Keywords:: Laplace transform, derivatives

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

1.14.27 $\mathop{\mathscr{L}\/}\nolimits\left(f^{{\prime}}(t);s\right)=s\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right)-f(0+).$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform
A&S Ref:: 29.2.4
Permalink:: http://dlmf.nist.gov/1.14.E27
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform, $e$ : base of exponential function, $k$ : integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.14.E28
Encodings:: TeX, pMML, png

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 $\mathop{\mathscr{L}\/}\nolimits\left(f^{{(n)}}(t);s\right)=s^{n}\mathop{\mathscr{L}\/}\nolimits\left(f(t);s\right)-s^{{n-1}}f(0+)-s^{{n-2}}f^{{\prime}}(0+)-\dots-f^{{(n-1)}}(0+).$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform and $n$ : nonnegative integer
A&S Ref:: 29.2.5
Permalink:: http://dlmf.nist.gov/1.14.E29
Encodings:: TeX, pMML, png

¶ Convolution

Keywords:: Laplace transform

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

1.14.30 $(f*g)(t)=\int^{t}_{0}f(u)g(t-u)du.$

Symbols:: $dx$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.14.E30
Encodings:: TeX, pMML, png

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

1.14.31 $\mathop{\mathscr{L}\/}\nolimits(f*g)=\mathop{\mathscr{L}\/}\nolimits(f)\mathop{\mathscr{L}\/}\nolimits(g).$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform
Permalink:: http://dlmf.nist.gov/1.14.E31
Encodings:: TeX, pMML, png

¶ Uniqueness

Keywords:: Laplace transform

If $f(t)$ and $g(t)$ are continuous and $\mathop{\mathscr{L}\/}\nolimits(f)=\mathop{\mathscr{L}\/}\nolimits(g)$ , then $f(t)=g(t)$ .

§1.14(iv) Mellin Transform

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

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

1.14.32 $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)=\int^{\infty}_{0}x^{{s-1}}f(x)dx.$

Defines:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform
Symbols:: $dx$ : differential of $x$ and $\int$ : integral
Referenced by:: ¶ ‣ §1.14(iv), §1.14(iv)
Permalink:: http://dlmf.nist.gov/1.14.E32
Encodings:: TeX, pMML, png

Alternative notations for $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ are $\mathop{\mathscr{M}\/}\nolimits\left(f(x);s\right)$ and $\mathop{\mathscr{M}\/}\nolimits(f)$ .

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 $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ is an analytic function of $s$ in the vertical strip $a<\realpart{s}<b$ . Moreover, for $a<\sigma<b$ ,

1.14.33 $\lim _{{t\to\pm\infty}}\mathop{\mathscr{M}\/}\nolimits\left(f;\sigma+it\right)=0.$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform and $\sigma\in(a,b)$ : parameter
Permalink:: http://dlmf.nist.gov/1.14.E33
Encodings:: TeX, pMML, png

Note: If $f(x)$ is continuous and $\alpha$ and $\beta$ are real numbers such that $f(x)=\mathop{O\/}\nolimits\!\left(x^{\alpha}\right)$ as $x\to 0+$ and $f(x)=\mathop{O\/}\nolimits\!\left(x^{\beta}\right)$ 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

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

1.14.34 $\tfrac{1}{2}(f(u+)+f(u-))=\frac{1}{2\pi i}\lim _{{T\to\infty}}\int^{{\sigma+iT}}_{{\sigma-iT}}u^{{-s}}\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)ds.$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral and $\sigma\in(a,b)$ : parameter
Permalink:: http://dlmf.nist.gov/1.14.E34
Encodings:: TeX, pMML, png

If $f(x)$ is continuous on $(0,\infty)$ and $\mathop{\mathscr{M}\/}\nolimits\left(f;\sigma+it\right)$ is integrable on $(-\infty,\infty)$ , then

1.14.35 $f(x)=\frac{1}{2\pi i}\int^{{\sigma+i\infty}}_{{\sigma-i\infty}}x^{{-s}}\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)ds.$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral and $\sigma\in(a,b)$ : parameter
Permalink:: http://dlmf.nist.gov/1.14.E35
Encodings:: TeX, pMML, png

¶ Parseval-type Formulas

Keywords:: Mellin transform, Parseval-type formulas

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

1.14.36 $\int^{\infty}_{0}f(x)g(yx)dx=\frac{1}{2\pi i}\*\int^{{\sigma+i\infty}}_{{\sigma-i\infty}}y^{{-s}}\mathop{\mathscr{M}\/}\nolimits\left(f;1-s\right)\mathop{\mathscr{M}\/}\nolimits\left(g;s\right)ds,$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral and $\sigma\in(a,b)$ : parameter
Permalink:: http://dlmf.nist.gov/1.14.E36
Encodings:: TeX, pMML, png

1.14.37 $\int^{\infty}_{0}f(x)g(x)dx=\frac{1}{2\pi i}\*\int^{{\sigma+i\infty}}_{{\sigma-i\infty}}\mathop{\mathscr{M}\/}\nolimits\left(f;1-s\right)\mathop{\mathscr{M}\/}\nolimits\left(g;s\right)ds.$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ , $\int$ : integral and $\sigma\in(a,b)$ : parameter
Permalink:: http://dlmf.nist.gov/1.14.E37
Encodings:: TeX, pMML, png

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

1.14.38 $\int^{\infty}_{0}(f(x))^{2}dx=\frac{1}{2\pi}\int^{\infty}_{{-\infty}}\left|\mathop{\mathscr{M}\/}\nolimits\left(f;\tfrac{1}{2}+it\right)\right|^{2}dt.$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.14.E38
Encodings:: TeX, pMML, png

¶ Convolution

Keywords:: Mellin transform

Let

1.14.39 $(f*g)(x)=\int^{\infty}_{0}f(y)g\left(\frac{x}{y}\right)\frac{dy}{y}.$

Symbols:: $dx$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.14.E39
Encodings:: TeX, pMML, png

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 $\int^{\infty}_{0}x^{{s-1}}(f*g)(x)dx=\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)\mathop{\mathscr{M}\/}\nolimits\left(g;s\right).$

Symbols:: $\mathop{\mathscr{M}\/}\nolimits\left(f;s\right)$ : Mellin transform, $dx$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.14.E40
Encodings:: TeX, pMML, png

§1.14(v) Hilbert Transform

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

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

1.14.41 $\mathop{\mathcal{H}\/}\nolimits\left(f;x\right)=\mathop{\mathcal{H}\/}\nolimits\left(f(t);x\right)=\mathop{\mathcal{H}\/}\nolimits(f)=\frac{1}{\pi}\pvint^{\infty}_{{-\infty}}\frac{f(t)}{t-x}dt,$

Symbols:: $\mathop{\mathcal{H}\/}\nolimits\left(f;x\right)$ : Hilbert transform, $dx$ : differential of $x$ and $\pvint _{a}^{b}$ : Cauchy principal value
Permalink:: http://dlmf.nist.gov/1.14.E41
Encodings:: TeX, pMML, png

1.14.42 $\mathop{\mathcal{H}\/}\nolimits\left(f;x\right)=\lim _{{y\to 0+}}\frac{1}{\pi}\int^{\infty}_{{-\infty}}\frac{t-x}{(t-x)^{2}+y^{2}}f(t)dt,$

Symbols:: $\mathop{\mathcal{H}\/}\nolimits\left(f;x\right)$ : Hilbert transform, $dx$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.14.E42
Encodings:: TeX, pMML, png

1.14.43 $\mathop{\mathcal{H}\/}\nolimits\left(f;x\right)=\lim _{{\epsilon\to 0+}}\frac{1}{\pi}\int^{\infty}_{\epsilon}\frac{f(x+t)-f(x-t)}{t}dt.$

Symbols:: $\mathop{\mathcal{H}\/}\nolimits\left(f;x\right)$ : Hilbert transform, $dx$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.14.E43
Encodings:: TeX, pMML, png

¶ Inversion

Keywords:: Hilbert transform

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

1.14.44 $f(x)=-\frac{1}{\pi}\pvint^{\infty}_{{-\infty}}\frac{\mathop{\mathcal{H}\/}\nolimits\left(f;u\right)}{u-x}du.$

Symbols:: $\mathop{\mathcal{H}\/}\nolimits\left(f;x\right)$ : Hilbert transform, $dx$ : differential of $x$ and $\pvint _{a}^{b}$ : Cauchy principal value
Referenced by:: ¶ ‣ §1.14(v)
Permalink:: http://dlmf.nist.gov/1.14.E44
Encodings:: TeX, pMML, png

¶ Inequalities

Keywords:: Hilbert transform, inequalities

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

1.14.45 $\int^{\infty}_{{-\infty}}|\mathop{\mathcal{H}\/}\nolimits\left(f;x\right)|^{p}dx\leq A_{p}\int^{\infty}_{{-\infty}}|f(t)|^{p}dt,$

Symbols:: $\mathop{\mathcal{H}\/}\nolimits\left(f;x\right)$ : Hilbert transform, $dx$ : differential of $x$ , $\int$ : integral and $A_{p}$ : coefficient
Permalink:: http://dlmf.nist.gov/1.14.E45
Encodings:: TeX, pMML, png

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

¶ Fourier Transform

Keywords:: Hilbert transform

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

1.14.46 $\frac{1}{\sqrt{2\pi}}\int _{{-\infty}}^{\infty}\mathop{\mathcal{H}\/}\nolimits\left(f;t\right)e^{{ixt}}dt=-i(\mathop{\mathrm{sign}\/}\nolimits x)F(x),$

Symbols:: $\mathop{\mathcal{H}\/}\nolimits\left(f;x\right)$ : Hilbert transform, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\mathop{\mathrm{sign}\/}\nolimits x$ : sign of $x$
Referenced by:: §1.14(v)
Permalink:: http://dlmf.nist.gov/1.14.E46
Encodings:: TeX, pMML, png

where $F(x)$ is given by (1.14.1).

§1.14(vi) Stieltjes Transform

Notes:: See Widder (1941, pp. 325–328, 340–341).
Keywords:: Stieltjes transform, convergence, derivatives
Permalink:: http://dlmf.nist.gov/1.14.vi

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

1.14.47 $\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)=\mathop{\mathcal{S}\/}\nolimits\left(f(t);s\right)=\mathop{\mathcal{S}\/}\nolimits(f)=\int^{\infty}_{0}\frac{f(t)}{s+t}dt.$

Defines:: $\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)$ : Stieltjes transform
Symbols:: $dx$ : differential of $x$ and $\int$ : integral
Referenced by:: ¶ ‣ §1.14(vi), ¶ ‣ §1.14(vi)
Permalink:: http://dlmf.nist.gov/1.14.E47
Encodings:: TeX, pMML, png

Sufficient conditions for the integral to converge are that $s$ is a positive real number, and $f(t)=\mathop{O\/}\nolimits\!\left(t^{{-\delta}}\right)$ 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, $\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)$ represents an analytic function in the $s$ -plane cut along the negative real axis, and

1.14.48 $\frac{{d}^{m}}{{ds}^{m}}\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)=(-1)^{m}m!\int^{\infty}_{0}\frac{f(t)dt}{(s+t)^{{m+1}}},$ $m=0,1,2,\dots$ .

Symbols:: $\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)$ : Stieltjes transform, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral and $m$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.14.E48
Encodings:: TeX, pMML, png

¶ Inversion

Keywords:: Stieltjes transform

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

1.14.49 $\lim _{{t\to 0+}}\frac{\mathop{\mathcal{S}\/}\nolimits\left(f;-\sigma-it\right)-\mathop{\mathcal{S}\/}\nolimits\left(f;-\sigma+it\right)}{2\pi i}=\tfrac{1}{2}(f(\sigma+)+f(\sigma-)),$

Symbols:: $\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)$ : Stieltjes transform and $\sigma\in(a,b)$ : parameter
Permalink:: http://dlmf.nist.gov/1.14.E49
Encodings:: TeX, pMML, png

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

¶ Laplace Transform

Keywords:: Stieltjes transform

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

1.14.50 $\mathop{\mathcal{S}\/}\nolimits(f)=\mathop{\mathscr{L}\/}\nolimits(\mathop{\mathscr{L}\/}\nolimits(f)).$

Symbols:: $\mathop{\mathscr{L}\/}\nolimits\left(f;s\right)$ : Laplace transform and $\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)$ : Stieltjes transform
Permalink:: http://dlmf.nist.gov/1.14.E50
Encodings:: TeX, pMML, png

§1.14(vii) Tables

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

Table 1.14.1: Fourier transforms.

$f(t)$	$\dfrac{1}{\sqrt{2\pi}}\displaystyle\int^{\infty}_{{-\infty}}f(t)e^{{ixt}}dt$


$\begin{cases}1,&\|t\|<a,\\ 0,&\mbox{otherwise}\end{cases}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{\mathop{\sin\/}\nolimits\!\left(ax\right)}{x}}$
$e^{{-a\|t\|}}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{a}{a^{2}+x^{2}}}$ ,	$a>0$
$te^{{-a\|t\|}}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{2iax}{(a^{2}+x^{2})^{2}}}$ ,	$a>0$
$\|t\|e^{{-a\|t\|}}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{a^{2}-x^{2}}{(a^{2}+x^{2})^{2}}}$ ,	$a>0$
$\dfrac{e^{{-a\|t\|}}}{\|t\|^{{1/2}}}$	$\dfrac{(a+(a^{2}+x^{2})^{{1/2}})^{{1/2}}}{(a^{2}+x^{2})^{{1/2}}}$ ,	$a>0$
$\dfrac{\mathop{\sinh\/}\nolimits\!\left(at\right)}{\mathop{\sinh\/}\nolimits\!\left(\pi t\right)}$	$\dfrac{1}{\sqrt{2\pi}}\dfrac{\mathop{\sin\/}\nolimits a}{\mathop{\cosh\/}\nolimits x+\mathop{\cos\/}\nolimits a}$ ,	$-\pi<a<\pi$
$\dfrac{\mathop{\cosh\/}\nolimits\!\left(at\right)}{\mathop{\cosh\/}\nolimits\!\left(\pi t\right)}$	$\sqrt{\dfrac{2}{\pi}}\dfrac{\mathop{\cos\/}\nolimits\!\left(\frac{1}{2}a\right)\mathop{\cosh\/}\nolimits\!\left(\frac{1}{2}x\right)}{\mathop{\cosh\/}\nolimits x+\mathop{\cos\/}\nolimits a}$ ,	$-\pi<a<\pi$
$e^{{-at^{2}}}$	$\dfrac{1}{\sqrt{2a}}e^{{-x^{2}/(4a)}}$ ,	$a>0$
$\mathop{\sin\/}\nolimits\!\left(at^{2}\right)$	$-\dfrac{1}{\sqrt{2a}}\mathop{\sin\/}\nolimits\!\left(\dfrac{x^{2}}{4a}-\dfrac{\pi}{4}\right)$ ,	$a>0$
$\mathop{\cos\/}\nolimits\!\left(at^{2}\right)$	$\dfrac{1}{\sqrt{2a}}\mathop{\cos\/}\nolimits\!\left(\dfrac{x^{2}}{4a}-\dfrac{\pi}{4}\right)$ ,	$a>0$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral and $\mathop{\sin\/}\nolimits z$ : sine function
Keywords:: Fourier transform
Permalink:: http://dlmf.nist.gov/1.14.T1

Table 1.14.2: Fourier cosine transforms.

$f(t)$	$\sqrt{\dfrac{2}{\pi}}\displaystyle\int^{\infty}_{0}\!\!\! f(t)\mathop{\cos\/}\nolimits\!\left(xt\right)dt$ ,	$x>0$


$\begin{cases}1,&0<t\leq a,\\ 0,&\mbox{otherwise}\end{cases}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{\mathop{\sin\/}\nolimits\!\left(ax\right)}{x}}$
$\dfrac{1}{a^{2}+t^{2}}$	$\displaystyle{\sqrt{\frac{\pi}{2}}\frac{e^{{-ax}}}{a}}$ ,	$\realpart{a}>0$
$\dfrac{1}{(a^{2}+t^{2})^{2}}$	$\displaystyle{\sqrt{\frac{\pi}{2}}\frac{(1+ax)e^{{-ax}}}{2a^{3}}}$ ,	$\realpart{a}>0$
$\dfrac{4a^{3}}{4a^{4}+t^{4}}$	$\sqrt{\pi}e^{{-ax}}\mathop{\sin\/}\nolimits\!\left(ax+\frac{1}{4}\pi\right)$ ,	$\realpart{a}>0$
$e^{{-at}}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{a}{a^{2}+x^{2}}}$ ,	$\realpart{a}>0$
$e^{{-at^{2}}}$	$\dfrac{1}{\sqrt{2a}}e^{{-x^{2}/(4a)}}$ ,	$\realpart{a}>0$
$\mathop{\sin\/}\nolimits\!\left(at^{2}\right)$	$-\dfrac{1}{\sqrt{2a}}\mathop{\sin\/}\nolimits\!\left(\dfrac{x^{2}}{4a}-\dfrac{\pi}{4}\right)$ ,	$a>0$
$\mathop{\cos\/}\nolimits\!\left(at^{2}\right)$	$\dfrac{1}{\sqrt{2a}}\mathop{\cos\/}\nolimits\!\left(\dfrac{x^{2}}{4a}-\dfrac{\pi}{4}\right)$ ,	$a>0$
$\mathop{\ln\/}\nolimits\!\left(1+\dfrac{a^{2}}{t^{2}}\right)$	$\displaystyle{\sqrt{2\pi}\frac{1-e^{{-ax}}}{x}}$ ,	$\realpart{a}>0$
$\mathop{\ln\/}\nolimits\!\left(\dfrac{a^{2}+t^{2}}{b^{2}+t^{2}}\right)$	$\displaystyle{\sqrt{2\pi}\frac{e^{{-bx}}-e^{{-ax}}}{x}}$ ,	$\realpart{a}>0$ , $\realpart{b}>0$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : real part and $\mathop{\sin\/}\nolimits z$ : sine function
Keywords:: Fourier cosine and sine transforms
Permalink:: http://dlmf.nist.gov/1.14.T2

Table 1.14.3: Fourier sine transforms.

$f(t)$	$\sqrt{\dfrac{2}{\pi}}\displaystyle\int^{\infty}_{0}\!\!\! f(t)\mathop{\sin\/}\nolimits\!\left(xt\right)dt$ ,	$x>0$


$t^{{-1}}$	$\displaystyle{\sqrt{\frac{\pi}{2}}}$
$t^{{-1/2}}$	$x^{{-1/2}}$
$t^{{-3/2}}$	$2x^{{1/2}}$
$\dfrac{t}{a^{2}+t^{2}}$	$\displaystyle{\sqrt{\frac{\pi}{2}}e^{{-ax}}}$ ,	$\realpart{a}>0$
$\dfrac{t}{(a^{2}+t^{2})^{2}}$	$\displaystyle{\sqrt{\frac{\pi}{8}}\frac{x}{a}e^{{-ax}}}$ ,	$\realpart{a}>0$
$\dfrac{1}{t(a^{2}+t^{2})}$	$\displaystyle{\sqrt{\frac{\pi}{2}}\frac{1-e^{{-ax}}}{a^{2}}}$ ,	$\realpart{a}>0$
$\dfrac{e^{{-at}}}{t}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\mathop{\mathrm{arctan}\/}\nolimits\!\left(\frac{x}{a}\right)}$ ,	$\realpart{a}>0$
$e^{{-at}}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{x}{a^{2}+x^{2}}}$ ,	$\realpart{a}>0$
$te^{{-at}}$	$\displaystyle{\sqrt{\frac{2}{\pi}}\frac{2ax}{(a^{2}+x^{2})^{2}}}$ ,	$\realpart{a}>0$
$te^{{-at^{2}}}$	$(2a)^{{-3/2}}xe^{{-x^{2}/(4a)}}$ ,	$\|\mathop{\mathrm{ph}\/}\nolimits a\|<\tfrac{1}{2}\pi$
$\dfrac{\mathop{\sin\/}\nolimits\!\left(at\right)}{t}$	$\dfrac{1}{\sqrt{2\pi}}\mathop{\ln\/}\nolimits\left\|\dfrac{x+a}{x-a}\right\|$ ,	$a>0$
$\mathop{\mathrm{arctan}\/}\nolimits\!\left(\dfrac{t}{a}\right)$	$\displaystyle{\sqrt{\frac{\pi}{2}}\frac{e^{{-ax}}}{x}}$ ,	$a>0$
$\mathop{\ln\/}\nolimits\left\|\dfrac{t+a}{t-a}\right\|$	$\displaystyle{\sqrt{2\pi}\frac{\mathop{\sin\/}\nolimits\!\left(ax\right)}{x}}$ ,	$a>0$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part and $\mathop{\sin\/}\nolimits z$ : sine function
Keywords:: Fourier cosine and sine transforms
Permalink:: http://dlmf.nist.gov/1.14.T3

Table 1.14.4: Laplace transforms.

$f(t)$	$\displaystyle\int^{\infty}_{0}e^{{-st}}f(t)dt$

1	$\dfrac{1}{s}$ ,	$\realpart{s}>0$
$\dfrac{t^{n}}{n!}$	$\dfrac{1}{s^{{n+1}}}$ ,	$\realpart{s}>0$
$\dfrac{1}{\sqrt{\pi t}}$	$\dfrac{1}{\sqrt{s}}$ ,	$\realpart{s}>0$
$e^{{-at}}$	$\dfrac{1}{s+a}$ ,	$\realpart{(s+a)}>0$
$\dfrac{t^{n}e^{{-at}}}{n!}$	$\dfrac{1}{(s+a)^{{n+1}}}$ ,	$\realpart{(s+a)}>0$
$\dfrac{e^{{-at}}-e^{{-bt}}}{b-a}$	$\dfrac{1}{(s+a)(s+b)}$ ,	$a\not=b$ , $\realpart{s}>-\realpart{a}$ , $\realpart{s}>-\realpart{b}$
$\mathop{\sin\/}\nolimits\!\left(at\right)$	$\dfrac{a}{s^{2}+a^{2}}$ ,	$\realpart{s}>\|\imagpart{a}\|$
$\mathop{\cos\/}\nolimits\!\left(at\right)$	$\dfrac{s}{s^{2}+a^{2}}$ ,	$\realpart{s}>\|\imagpart{a}\|$
$\mathop{\sinh\/}\nolimits\!\left(at\right)$	$\dfrac{a}{s^{2}-a^{2}}$ ,	$\realpart{s}>\|\realpart{a}\|$
$\mathop{\cosh\/}\nolimits\!\left(at\right)$	$\dfrac{s}{s^{2}-a^{2}}$ ,	$\realpart{s}>\|\realpart{a}\|$
$t\mathop{\sin\/}\nolimits\!\left(at\right)$	$\dfrac{2as}{(s^{2}+a^{2})^{2}}$ ,	$\realpart{s}>\|\imagpart{a}\|$
$t\mathop{\cos\/}\nolimits\!\left(at\right)$	$\dfrac{s^{2}-a^{2}}{(s^{2}+a^{2})^{2}}$ ,	$\realpart{s}>\|\imagpart{a}\|$
$\dfrac{e^{{-bt}}-e^{{-at}}}{t}$	$\displaystyle{\mathop{\ln\/}\nolimits\!\left(\frac{s+a}{s+b}\right)}$ ,	$\realpart{s}>-\realpart{a}$ , $\realpart{s}>-\realpart{b}$
$\dfrac{2(1-\mathop{\cosh\/}\nolimits\!\left(at\right))}{t}$	$\displaystyle{\mathop{\ln\/}\nolimits\!\left(1-\frac{a^{2}}{s^{2}}\right)}$ ,	$\realpart{(s+a)}>0$
$\dfrac{2(1-\mathop{\cos\/}\nolimits\!\left(at\right))}{t}$	$\displaystyle{\mathop{\ln\/}\nolimits\!\left(1+\frac{a^{2}}{s^{2}}\right)}$ ,	$\realpart{s}>0$
$\dfrac{\mathop{\sin\/}\nolimits\!\left(at\right)}{t}$	$\mathop{\mathrm{arctan}\/}\nolimits\!\left(\dfrac{a}{s}\right)$ ,	$\realpart{s}>0$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\imagpart{}$ : imaginary part, $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function and $n$ : nonnegative integer
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
Keywords:: Laplace transform
Permalink:: http://dlmf.nist.gov/1.14.T4

Table 1.14.5: Mellin transforms.

$f(x)$	$\displaystyle\int^{\infty}_{0}x^{{s-1}}f(x)dx$


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

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{arccot}\/}\nolimits z$ : arccotangent function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $\mathop{\sec\/}\nolimits z$ : secant function, $\mathop{\sin\/}\nolimits z$ : sine function and $\mathop{\tan\/}\nolimits z$ : tangent function
Keywords:: Mellin transform
Referenced by:: ¶ ‣ §2.5(i)
Permalink:: http://dlmf.nist.gov/1.14.T5

§1.14(viii) Compendia

Keywords:: Fourier cosine and sine transforms, Fourier transform, Laplace transform, Mellin transform, integral transforms
Permalink:: http://dlmf.nist.gov/1.14.viii

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).

§1.14 Integral Transforms

Contents

§1.14(i) Fourier Transform

¶ Inversion

¶ Convolution

¶ Parseval’s 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