Digital Library of Mathematical Functions
About the Project
NIST
1 Algebraic and Analytic MethodsAreas

§1.14 Integral Transforms

Contents

§1.14(i) Fourier Transform

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

1.14.1 F(x)=12π-f(t)xtt.

(Some references replace xt by -xt.)

If f(t) is absolutely integrable on (-,), then F(x) is continuous, F(x)0 as x±, and

1.14.2 |F(x)|12π-|f(t)|t.

Inversion

Suppose that f(t) is absolutely integrable on (-,) and of bounded variation in a neighborhood of t=u1.4(v)). Then

1.14.3 12(f(u+)+f(u-))=12π-F(x)-xux,

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

In many applications f(t) is absolutely integrable and f(t) is continuous on (-,). Then

1.14.4 f(t)=12π-F(x)-xtx.

Convolution

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

1.14.5 (f*g)(t)=12π-f(t-s)g(s)s.

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

Parseval’s Formula

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

1.14.6 (f*g)(t)=12π-F(x)G(x)-txx,
1.14.7 -F(x)G(x)x=-f(t)g(-t)t,
1.14.8 -|F(x)|2x=-|f(t)|2t.

(1.14.8) is Parseval’s formula.

Uniqueness

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

§1.14(ii) Fourier Cosine and Sine Transforms

These are defined respectively by

1.14.9 Fc(x) =2π0f(t)cos(xt)t,
1.14.10 Fs(x) =2π0f(t)sin(xt)t.

Inversion

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

1.14.11 12(f(u+)+f(u-)) =2π0Fc(x)cos(ux)x,
1.14.12 12(f(u+)+f(u-)) =2π0Fs(x)sin(ux)x.

Parseval’s Formula

If 0|f(t)|t<, g(t) is of bounded variation on (0,) and g(t)0 as t, then

1.14.13 0Fc(x)Gc(x)x =0f(t)g(t)t,
1.14.14 0Fs(x)Gs(x)x =0f(t)g(t)t,
1.14.15 0(Fc(x))2x =0(f(t))2t,
1.14.16 0(Fs(x))2x =0(f(t))2t,

where Gc(x) and Gs(x) are respectively the cosine and sine transforms of g(t).

§1.14(iii) Laplace Transform

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

1.14.17 (f(t);s)=0-stf(t)t.

Alternative notations are (f(t)), (f;s), or even (f), when it is not important to display all the variables.

Convergence and Analyticity

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

1.14.18 |f(t)|Mαt,
0t<.

Then (f(t);s) is an analytic function of s for s>α. Moreover,

1.14.19 (f(t);s)0,
s.

Throughout the remainder of this subsection we assume (1.14.18) is satisfied and s>α.

Inversion

If f(t) is continuous and f(t) is piecewise continuous on [0,), then

1.14.20 f(t)=12πlimTσ-Tσ+Tts(f(t);s)s,
σ>α.

Moreover, if (f(t);s)=O(s-K) in some half-plane sγ and K>1, then (1.14.20) holds for σ>γ.

Translation

If s>max((a+α),α), then

1.14.21 (f(t);s-a)=(atf(t);s).

Also, if a0 then

1.14.22 (H(t-a)f(t-a);s)=-as(f(t);s),

where H is the Heaviside function; see (1.16.13).

Differentiation and Integration

If f(t) is piecewise continuous, then

1.14.23 nsn(f(t);s)=((-t)nf(t);s),
n=1,2,3,.

If also limt0+f(t)/t exists, then

1.14.24 s(f(t);u)u=(f(t)t;s).

Periodic Functions

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

1.14.25 (f(t);s)=11--as0a-stf(t)t.

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

1.14.26 (f(t);s)=11+-as0a-stf(t)t.

Derivatives

If f(t) is continuous on [0,) and f(t) is piecewise continuous on (0,), then

1.14.27 (f(t);s)=s(f(t);s)-f(0+).

If f(t) and f(t) are piecewise continuous on [0,) with discontinuities at (0=) t0<t1<<tn, then

1.14.28 (f(t);s)=s(f(t);s)-f(0+)-k=1n-stk(f(tk+)-f(tk-)).

Next, assume f(t), f(t), , f(n-1)(t) are continuous and each satisfies (1.14.18). Also assume that f(n)(t) is piecewise continuous on [0,). Then

1.14.29 (f(n)(t);s)=sn(f(t);s)-sn-1f(0+)-sn-2f(0+)--f(n-1)(0+).

Convolution

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

1.14.30 (f*g)(t)=0tf(u)g(t-u)u.

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

1.14.31 (f*g)=(f)(g).

Uniqueness

If f(t) and g(t) are continuous and (f)=(g), then f(t)=g(t).

§1.14(iv) Mellin Transform

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

1.14.32 (f;s)=0xs-1f(x)x.

Alternative notations for (f;s) are (f(x);s) and (f).

If xσ-1f(x) is integrable on (0,) for all σ in a<σ<b, then the integral (1.14.32) converges and (f;s) is an analytic function of s in the vertical strip a<s<b. Moreover, for a<σ<b,

1.14.33 limt±(f;σ+t)=0.

Note: If f(x) is continuous and α and β are real numbers such that f(x)=O(xα) as x0+ and f(x)=O(xβ) as x, then xσ-1f(x) is integrable on (0,) for all σ(-α,-β).

Inversion

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

1.14.34 12(f(u+)+f(u-))=12πlimTσ-Tσ+Tu-s(f;s)s.

If f(x) is continuous on (0,) and (f;σ+t) is integrable on (-,), then

1.14.35 f(x)=12πσ-σ+x-s(f;s)s.

Parseval-type Formulas

Suppose x-σf(x) and xσ-1g(x) are absolutely integrable on (0,) and either (g;σ+t) or (f;1-σ-t) is absolutely integrable on (-,). Then for y>0,

1.14.36 0f(x)g(yx)x=12πσ-σ+y-s(f;1-s)(g;s)s,
1.14.37 0f(x)g(x)x=12πσ-σ+(f;1-s)(g;s)s.

When f is real and σ=12,

1.14.38 0(f(x))2x=12π-|(f;12+t)|2t.

Convolution

Let

1.14.39 (f*g)(x)=0f(y)g(xy)yy.

If xσ-1f(x) and xσ-1g(x) are absolutely integrable on (0,), then for s=σ+t,

1.14.40 0xs-1(f*g)(x)x=(f;s)(g;s).

§1.14(v) Hilbert Transform

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

1.14.41 (f;x) =(f(t);x)
=(f)
=1π-f(t)t-xt,
1.14.42 (f;x) =limy0+1π-t-x(t-x)2+y2f(t)t,
1.14.43 (f;x) =limϵ0+1πϵf(x+t)-f(x-t)tt.

Inversion

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

1.14.44 f(x)=-1π-(f;u)u-xu.

Inequalities

If |f(t)|p, p>1, is integrable on (-,), then so is |(f;x)|p and

1.14.45 -|(f;x)|pxAp-|f(t)|pt,

where Ap=tan(12π/p) when 1<p2, or cot(12π/p) when p2. These bounds are sharp, and equality holds when p=2.

Fourier Transform

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

1.14.46 12π-(f;t)xtt=-(signx)F(x),

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

§1.14(vi) Stieltjes Transform

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

1.14.47 𝒮(f;s)=𝒮(f(t);s)=𝒮(f)=0f(t)s+tt.

Sufficient conditions for the integral to converge are that s is a positive real number, and f(t)=O(t-δ) as t, where δ>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 (-,0]. In this case, 𝒮(f;s) represents an analytic function in the s-plane cut along the negative real axis, and

1.14.48 msm𝒮(f;s)=(-1)mm!0f(t)t(s+t)m+1,
m=0,1,2,.

Inversion

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

1.14.49 limt0+𝒮(f;-σ-t)-𝒮(f;-σ+t)2π=12(f(σ+)+f(σ-)),

for all values of the positive constant σ for which the right-hand side exists.

Laplace Transform

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

1.14.50 𝒮(f)=((f)).

§1.14(vii) Tables

Table 1.14.1: Fourier transforms.
f(t) 12π-f(t)xtt
{1,|t|<a,0,otherwise 2πsin(ax)x
-a|t| 2πaa2+x2, a>0
t-a|t| 2π2ax(a2+x2)2, a>0
|t|-a|t| 2πa2-x2(a2+x2)2, a>0
-a|t||t|1/2 (a+(a2+x2)1/2)1/2(a2+x2)1/2, a>0
sinh(at)sinh(πt) 12πsinacoshx+cosa, -π<a<π
cosh(at)cosh(πt) 2πcos(12a)cosh(12x)coshx+cosa, -π<a<π
-at2 12a-x2/(4a), a>0
sin(at2) -12asin(x24a-π4), a>0
cos(at2) 12acos(x24a-π4), a>0
Table 1.14.2: Fourier cosine transforms.
f(t) 2π0f(t)cos(xt)t, x>0
{1,0<ta,0,otherwise 2πsin(ax)x
1a2+t2 π2-axa, a>0
1(a2+t2)2 π2(1+ax)-ax2a3, a>0
4a34a4+t4 π-axsin(ax+14π), a>0
-at 2πaa2+x2, a>0
-at2 12a-x2/(4a), a>0
sin(at2) -12asin(x24a-π4), a>0
cos(at2) 12acos(x24a-π4), a>0
ln(1+a2t2) 2π1--axx, a>0
ln(a2+t2b2+t2) 2π-bx--axx, a>0, b>0
Table 1.14.3: Fourier sine transforms.
f(t) 2π0f(t)sin(xt)t, x>0
t-1 π2
t-1/2 x-1/2
t-3/2 2x1/2
ta2+t2 π2-ax, a>0
t(a2+t2)2 π8xa-ax, a>0
1t(a2+t2) π21--axa2, a>0
-att 2πarctan(xa), a>0
-at 2πxa2+x2, a>0
t-at 2π2ax(a2+x2)2, a>0
t-at2 (2a)-3/2x-x2/(4a), |pha|<12π
sin(at)t 12πln|x+ax-a|, a>0
arctan(ta) π2-axx, a>0
ln|t+at-a| 2πsin(ax)x, a>0
Table 1.14.4: Laplace transforms.
f(t) 0-stf(t)t
1 1s, s>0
tnn! 1sn+1, s>0
1πt 1s, s>0
-at 1s+a, (s+a)>0
tn-atn! 1(s+a)n+1, (s+a)>0
-at--btb-a 1(s+a)(s+b), ab, s>-a, s>-b
sin(at) as2+a2, s>|a|
cos(at) ss2+a2, s>|a|
sinh(at) as2-a2, s>|a|
cosh(at) ss2-a2, s>|a|
tsin(at) 2as(s2+a2)2, s>|a|
tcos(at) s2-a2(s2+a2)2, s>|a|
-bt--att ln(s+as+b), s>-a, s>-b
2(1-cosh(at))t ln(1-a2s2), (s+a)>0
2(1-cos(at))t ln(1+a2s2), s>0
sin(at)t arctan(as), s>0
Table 1.14.5: Mellin transforms.
f(x) 0xs-1f(x)x
{1,x<a,0,xa ass, a0, s>0
{ln(a/x),x<a,0,xa ass2, a0, s>1
11-x πcot(sπ), 0<s<1, (Cauchy p. v.)
11+x πcsc(sπ), 0<s<1
ln(1+ax) πcsc(sπ)sas, |pha|<π, -1<s<0
ln|1+x1-x| πtan(12sπ)s, -1<s<1
ln(1+x)x πcsc(sπ)1-s, 0<s<1
arctanx -πsec(12sπ)2s, -1<s<0
arccotx πsec(12sπ)2s, 0<s<1
1+xcosθ1+2xcosθ+x2 πcos(sθ)sin(sπ), -π<θ<π, 0<s<1
xsinθ1+2xcosθ+x2 πsin(sθ)sin(sπ), -π<θ<π, 0<s<1

§1.14(viii) Compendia

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