xxviii PREFACE
formula 5 in Subsection 3.1.2. At the end of each chapter, we present a list of main and
additional literature sources containing more detailed information about topics of interest
to the reader.
Special font highlighting in the text, cross-references, an extensive table of contents,
and an index help the reader to find the desired information.
We would like to express our deep gratitude to Alexei Zhurov for fruitful discussions
and valuable remarks. We also appreciate the help of Vladimir Nazaikinskii and Grigorii
Yosifian for translating several chapters of this book and are thankful to Kirill Kazakov and
Mikhail Mikhin for their assistance in preparing the camera-ready copy of the book.
The authors hope that this book will be helpful for a wide range of scientists, university
teachers, engineers, and students engaged in the fields of mathematics, physics, mechanics,
control, chemistry, biology, engineering sciences, and social and economical sciences.
Some sections and examples can be used in lectures and practical studies in basic and
special mathematical courses.
Andrei D. Polyanin
Alexander V. Manzhirov
Main Notation
Special symbols
= equal to
≡ identically equal to
≠ not equal to
≈ approximately equal to
∼ of same order as (used in comparisons of infinitesimals or infinites)
< less than; “a less than b” is written as a < b (or, equivalently, b > a)
≤ less than or equal to; a less than or equal to b is written as a ≤ b
 much less than; a much less than b is written as a  b
> greater than; a greater than b is written as a > b (or, equivalently, b < a)
≥ greater than or equal to; a greater than or equal to b is written as a ≥ b
 much greater than; a much greater than b is written as a  b
+ plus sign; the sum of numbers a and b is denoted by a + b and has the property

sum,
n

k=1
a
k
= a
1
+ a
2
+ ···+ a
n

product,
n

k=1
a
k
= a
1
⋅ a
2
⋅ ⋅ a
n
∂ symbol used to denote partial derivatives and differential operators; ∂
x
is the
operator of differentiation with respect to x
xxix


a if a ≥ 0
–a if a < 0
a vector, a = {a
1
, a
2
, a
3
},wherea
1
, a
2
, a
3
are the vector components
|a| modulus of a vector a, |a| =
√
a ⋅ a
a ⋅ b inner product of vectors a and b, denoted also by (a ⋅ b)
a × b cross-product of vectors a and b
[abc] triple product of vectors a, b, c
(a, b) interval (open interval) a < x < b
(a, b] half-open interval a < x ≤ b
[a, b) half-open interval a ≤ x < b
[a, b] interval (closed interval) a ≤ x ≤ b
arccos x arccosine, the inverse function of cosine: cos(arccos x)=x, |x| ≤ 1
arccot x arccotangent, the inverse function of cotangent: cot(arccot x)=x
arcsin x arcsine, the inverse function of sine: sin(arcsin x)=x, |x| ≤ 1
arctan

x
2
+ 1

arctanh x hyperbolic arctangent, the inverse function of hyperbolic tangent; also denoted
by arctanh x =tanh
–1
x;arctanhx =
1
2
ln
1 + x
1 – x
(|x| < 1)
C
k
n
binomial coefficients, also denoted by

n
k

, C
k
n
=
n!
k!(n – k)!
, k =1, 2, , n
C Euler constant, C = lim

)
MAIN NOTATION xxxi
div a divergence of a vector a
e the number “e” (base of natural logarithms), e = 2.718281 ;definition:
e = lim
n→∞

1 +
1
n

n
erf x Gauss error function, erf x =
2
√
π

x
0
exp

–ξ
2

dξ
erfc x complementary error function, erfc x =
2
√
π


ν
(x) modified Bessel function of the first kind, I
ν
(x)=
∞

n=0
(x/2)
ν+2n
n! Γ(ν + n + 1)
Im z imaginary part of a complex number; if z = x + iy,thenImz = y
inf A infimum of a (numerical) set A;ifA =(a, b)orA =[a, b), then inf A = a
J
ν
(x) Bessel function of the first kind, J
ν
(x)=
∞

n=0
(–1)
n
(x/2)
ν+2n
n! Γ(ν + n + 1)
K
ν
(x) modified Bessel function of the second kind, K
ν
(x)=

d
n
dx
n
(x
2
– 1)
n
R set of real numbers, R = {–∞ < x < ∞}
Re z real part of a complex number; if z = x + iy,thenRez = x
r, ϕ, z cylindrical coordinates, r =

x
2
+ y
2
and x = r cos ϕ, y = r sin ϕ
r, θ, ϕ spherical coordinates, r =

x
2
+y
2
+z
2
and x = r sin θ cos ϕ, y =sinθ sin ϕ,
z = r cos θ
rank A rank of a matrix A
curl a curl of a vector a, also denoted by rot a
sec x secant, even trigonometric function of period 2π:secx =

y

x
first derivative of a function y = f(x), also denoted by y

,
dy
dx
, f

(x)
y

xx
second derivative of a function y = f (x), also denoted by y

,
d
2
y
dx
2
, f

(x)
y
(n)
x
nth derivative of a function y = f(x), also denoted by
d

0
e
–t
t
α–1
dt
Φ(a, b; x) degenerate hypergeometric function, Φ(a, b; x)=1+
∞

n=1
a(a+1) (a+n–1)
b(b+1) (b+n–1)
x
n
n!
Δ Laplace operator; in the two-dimensional case, Δw =
∂
2
w
∂x
2
+
∂
2
w
∂y
2
,wherex
and y are Cartesian coordinates
Δx increment of the argument