Page 7 - Handbook Of Integral Equations

SOME REMARKS AND NOTATION

1. In Chapters 1–11 and 14, in the original integral equations, the independent variable is
denoted by x, the integration variable by t, and the unknown function by y = y(x).

2. For a function of one variable f = f(x), we use the following notation for the derivatives:

4
2
3
n
df d f d f d f (n) d f

f = , f = , f = , f = , and f = for n ≥ 5.
x xx 2 xxx 3 xxxx 4 x n
dx dx dx dx dx
Occasionally, we use the similar notation for partial derivatives of a function of two variables,
∂

for example, K (x, t)= K(x, t).
x
∂x
d
n
3. In some cases, we use the operator notation f(x) g(x), which is defined recursively by
dx
n n–1

d d d
f(x) g(x)= f(x) f(x) g(x) .
dx dx dx
4. It is indicated in the beginning of Chapters 1–6 that f = f(x), g = g(x), K = K(x), etc. are
arbitrary functions, and A, B, etc. are free parameters. This means that:
(a) f = f(x), g = g(x), K = K(x), etc. are assumed to be continuous real-valued functions of real
arguments;*
(b) if the solution contains derivatives of these functions, then the functions are assumed to be
sufficiently differentiable;**
(c) if the solution contains integrals with these functions (in combination with other functions), then
the integrals are supposed to converge;
(d) the free parameters A, B, etc. may assume any real values for which the expressions occurring
A
in the equation and the solution make sense (for example, if a solution contains a factor ,
1 – A
then it is implied that A ≠ 1; as a rule, this is not specified in the text).
5. The notations Re z and Im z stand, respectively, for the real and the imaginary part of a
complex quantity z.
6. In the first part of the book (Chapters 1–6) when referencing a particular equation, we use a
notation like 2.3.15, which implies equation 15 from Section 2.3.
7. To highlight portions of the text, the following symbols are used in the book:
indicates important information pertaining to a group of equations (Chapters 1–6);
indicates the literature used in the preparation of the text in specific equations (Chapters 1–6) or
sections (Chapters 7–14).

* Less severe restrictions on these functions are presented in the second part of the book.
** Restrictions (b) and (c) imposed on f = f(x), g = g(x), K = K(x), etc. are not mentioned in the text.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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