# §1.14 Integral Transforms

## §1.14(i) Fourier Transform

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

1.14.1

(Some references replace by .)

If is absolutely integrable on , then is continuous, as , and

### ¶ Inversion

Suppose that is absolutely integrable on and of bounded variation in a neighborhood of 1.4(v)). Then

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

In many applications is absolutely integrable and is continuous on . Then

### ¶ Convolution

For Fourier transforms, the convolution of two functions and defined on is given by

1.14.5

If and are absolutely integrable on , then so is , and its Fourier transform is , where is the Fourier transform of .

### ¶ Parseval’s Formula

Suppose and are absolutely integrable on , and and are their respective Fourier transforms. Then

(1.14.8) is Parseval’s formula.

### ¶ Uniqueness

If and are continuous and absolutely integrable on , and for all , then for all .

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

These are defined respectively by

1.14.9
1.14.10

### ¶ Inversion

If is absolutely integrable on and of bounded variation (§1.4(v)) in a neighborhood of , then

### ¶ Parseval’s Formula

If , is of bounded variation on and as , then

where and are respectively the cosine and sine transforms of .

## §1.14(iii) Laplace Transform

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

Alternative notations are , , or even , when it is not important to display all the variables.

### ¶ Convergence and Analyticity

Assume that on is piecewise continuous and of exponential growth, that is, constants and exist such that

Then is an analytic function of for . Moreover,

1.14.19.

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

### ¶ Inversion

If is continuous and is piecewise continuous on , then

Moreover, if in some half-plane and , then (1.14.20) holds for .

### ¶ Translation

If , then

Also, if then

where is the Heaviside function; see (1.16.13).

### ¶ Differentiation and Integration

If is piecewise continuous, then

If also exists, then

### ¶ Periodic Functions

If and for , then

Alternatively if for , then

### ¶ Derivatives

If is continuous on and is piecewise continuous on , then

1.14.27

If and are piecewise continuous on with discontinuities at () , then

Next, assume , , , are continuous and each satisfies (1.14.18). Also assume that is piecewise continuous on . Then

1.14.29

### ¶ Convolution

For Laplace transforms, the convolution of two functions and , defined on , is

1.14.30

If and are piecewise continuous, then

1.14.31

### ¶ Uniqueness

If and are continuous and , then .

## §1.14(iv) Mellin Transform

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

1.14.32

Alternative notations for are and .

If is integrable on for all in , then the integral (1.14.32) converges and is an analytic function of in the vertical strip . Moreover, for ,

1.14.33

Note: If is continuous and and are real numbers such that as and as , then is integrable on for all .

### ¶ Inversion

Suppose the integral (1.14.32) is absolutely convergent on the line and is of bounded variation in a neighborhood of . Then

If is continuous on and is integrable on , then

### ¶ Parseval-type Formulas

Suppose and are absolutely integrable on and either or is absolutely integrable on . Then for ,

When is real and ,

### ¶ Convolution

Let

1.14.39

If and are absolutely integrable on , then for ,

## §1.14(v) Hilbert Transform

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

### ¶ Inversion

Suppose is continuously differentiable on and vanishes outside a bounded interval. Then

### ¶ Inequalities

If , , is integrable on , then so is and

where when , or when . These bounds are sharp, and equality holds when .

### ¶ Fourier Transform

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

where is given by (1.14.1).

## §1.14(vi) Stieltjes Transform

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

1.14.47

Sufficient conditions for the integral to converge are that is a positive real number, and as , where .

If the integral converges, then it converges uniformly in any compact domain in the complex -plane not containing any point of the interval . In this case, represents an analytic function in the -plane cut along the negative real axis, and

### ¶ Inversion

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

1.14.49

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

### ¶ Laplace Transform

If is piecewise continuous on and the integral (1.14.47) converges, then

## §1.14(vii) Tables

Table 1.14.2: Fourier cosine transforms.
 , , , , , , , , , ,
Table 1.14.3: Fourier sine transforms.
 , , , , , , , , , ,
Table 1.14.4: Laplace transforms.
 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).