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§1.14 Integral Transforms

Contents
  1. §1.14(i) Fourier Transform
  2. §1.14(ii) Fourier Cosine and Sine Transforms
  3. §1.14(iii) Laplace Transform
  4. §1.14(iv) Mellin Transform
  5. §1.14(v) Hilbert Transform
  6. §1.14(vi) Stieltjes Transform
  7. §1.14(vii) Tables
  8. §1.14(viii) Compendia

§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)=f(x)=12πf(t)eixtdt.

(Some references replace ixt by ixt). The same notation 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)=(f)(x).

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)|dt.

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))=12πF(x)eixudx,

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)=12πF(x)eixtdx.

Convolution

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

1.14.5 (fg)(t)=12πf(ts)g(s)ds.

If f(t) and g(t) are absolutely integrable on (,), then so is (fg)(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 and square integrable on (,), then

1.14.6 (fg)(t)=12πF(x)G(x)eitxdx,
1.14.7 F(x)G(x)dx=f(t)g(t)dt,
1.14.7_5 F(x)G(x)¯dx=f(t)g(t)¯dt,
1.14.8 |F(x)|2dx=|f(t)|2dt.

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

Poisson’s Summation Formula

See §1.8(iv) and especially (1.8.14).

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

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

1.14.9 c(f)(x) =cf(x)=2π0f(t)cos(xt)dt,
1.14.10 s(f)(x) =sf(x)=2π0f(t)sin(xt)dt.

In this subsection we let Fc(x)=cf(x), Fs(x)=sf(x), Gc(x)=cg(x), and Gs(x)=sg(x).

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)dx,
1.14.12 12(f(u+)+f(u)) =2π0Fs(x)sin(ux)dx.

Parseval’s Formula

Suppose f(t) and g(t) are absolutely and square integrable on [0,), then

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

§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)(s)=f(s)=0estf(t)dt.

Convergence and Analyticity

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

1.14.18 |f(t)|Meαt,
0t<.

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

1.14.19 f(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)=12πilimTσiTσ+iTetsf(s)ds,
σ>α.

Moreover, if f(s)=O(sK) 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(sa)=fa(s),

where fa(t)=eatf(t). Also, if a0 then

1.14.22 fa+(s)=easf(s),

where fa+(t)=H(ta)f(ta) and H is the Heaviside function; see (1.16.13).

Differentiation and Integration

If f(t) is piecewise continuous, then

1.14.23 (1)ndndsnf(s)=fn(s),
n=1,2,3,,

where fn(t)=tnf(t). If also limt0+f(t)/t exists, then

1.14.24 sf(u)du=f1(s),

where f1(t)=f(t)t.

Periodic Functions

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

1.14.25 f(s)=11eas0aestf(t)dt.

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

1.14.26 f(s)=11+eas0aestf(t)dt.

Derivatives

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

1.14.27 (f)(s)=s(f)(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)(s)=s(f)(s)f(0+)k=1nestk(f(tk+)f(tk)).

Next, assume f(t), f(t), , f(n1)(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))(s)=sn(f)(s)sn1f(0+)sn2f(0+)f(n1)(0+).

Convolution

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

1.14.30 (fg)(t)=0tf(u)g(tu)du.

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

1.14.31 (fg)=(f)(g).

Uniqueness

If f(t) and g(t) are continuous and f(s)=g(s), 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)=f(s)=0xs1f(x)dx.

If xσ1f(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+it)=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 xσ1f(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))=12πilimTσiTσ+iTusf(s)ds.

If f(x) is continuous on (0,) and f(σ+it) is integrable on (,), then

1.14.35 f(x)=12πiσiσ+ixsf(s)ds.

Parseval-type Formulas

Suppose xσf(x) and xσ1g(x) are absolutely integrable on (0,) and either g(σ+it) or f(1σit) is absolutely integrable on (,). Then for y>0,

1.14.36 0f(x)g(yx)dx=12πiσiσ+iysf(1s)g(s)ds,
1.14.37 0f(x)g(x)dx=12πiσiσ+if(1s)g(s)ds.

When f is real and σ=12,

1.14.38 0(f(x))2dx=12π|f(12+it)|2dt.

Convolution

Let

1.14.39 (fg)(x)=0f(y)g(xy)dyy.

If xσ1f(x) and xσ1g(x) are absolutely integrable on (0,), then for s=σ+it,

1.14.40 0xs1(fg)(x)dx=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(x)=1πf(t)txdt,
1.14.42 f(x) =limy0+1πtx(tx)2+y2f(t)dt,
1.14.43 f(x) =limϵ0+1πϵf(x+t)f(xt)tdt.

Inversion

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

1.14.44 f(x)=1πf(u)uxdu.

Inequalities

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

1.14.45 |f(x)|pdxAp|f(t)|pdt,

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 12πf(u)eiuxdu=i(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)=𝒮f(s)=0f(t)s+tdt.

Sufficient conditions for the integral to converge are that s is a positive real number, and f(t)=O(tδ) 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, 𝒮f(s) represents an analytic function in the s-plane cut along the negative real axis, and

1.14.48 dmdsm𝒮f(s)=(1)mm!0f(t)dt(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+𝒮f(σit)𝒮f(σ+it)2πi=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) 12πf(t)eixtdt
{1,|t|<a,0,otherwise 2πsin(ax)x
ea|t| 2πaa2+x2, a>0
tea|t| 2π2iax(a2+x2)2, a>0
|t|ea|t| 2πa2x2(a2+x2)2, a>0
ea|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
eat2 12aex2/(4a), a>0
sin(at2) 12asin(x24aπ4), a>0
cos(at2) 12acos(x24aπ4), a>0
Table 1.14.2: Fourier cosine transforms.
f(t) 2π0f(t)cos(xt)dt, x>0
{1,0<ta,0,otherwise 2πsin(ax)x
1a2+t2 π2eaxa, a>0
1(a2+t2)2 π2(1+ax)eax2a3, a>0
4a34a4+t4 πeaxsin(ax+14π), a>0
eat 2πaa2+x2, a>0
eat2 12aex2/(4a), a>0
sin(at2) 12asin(x24aπ4), a>0
cos(at2) 12acos(x24aπ4), a>0
ln(1+a2t2) 2π1eaxx, a>0
ln(a2+t2b2+t2) 2πebxeaxx, a>0, b>0
Table 1.14.3: Fourier sine transforms.
f(t) 2π0f(t)sin(xt)dt, x>0
t1 π2
t1/2 x1/2
t3/2 2x1/2
ta2+t2 π2eax, a>0
t(a2+t2)2 π8xaeax, a>0
1t(a2+t2) π21eaxa2, a>0
eatt 2πarctan(xa), a>0
eat 2πxa2+x2, a>0
teat 2π2ax(a2+x2)2, a>0
teat2 (2a)3/2xex2/(4a), |pha|<12π
sin(at)t 12πln|x+axa|, a>0
arctan(ta) π2eaxx, a>0
ln|t+ata| 2πsin(ax)x, a>0
Table 1.14.4: Laplace transforms.
f(t) 0estf(t)dt
1 1s, s>0
tnn! 1sn+1, s>0
1πt 1s, s>0
eat 1s+a, (s+a)>0
tneatn! 1(s+a)n+1, (s+a)>0
eatebtba 1(s+a)(s+b), ab, s>a, s>b
sin(at) as2+a2, s>|a|
cos(at) ss2+a2, s>|a|
sinh(at) as2a2, s>|a|
cosh(at) ss2a2, s>|a|
tsin(at) 2as(s2+a2)2, s>|a|
tcos(at) s2a2(s2+a2)2, s>|a|
ebteatt ln(s+as+b), s>a, s>b
2(1cosh(at))t ln(1a2s2), (s+a)>0
2(1cos(at))t ln(1+a2s2), s>0
sin(at)t arctan(as), s>0
Table 1.14.5: Mellin transforms.
f(x) 0xs1f(x)dx
{1,x<a,0,xa ass, a0, s>0
{ln(a/x),x<a,0,xa ass2, a0, s>1
11x 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+x1x| πtan(12sπ)s, 1<s<1
ln(1+x)x πcsc(sπ)1s, 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 (2015), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).