Handbook of mathematics for engineers and scienteists part 11 ppsx
Handbook of mathematics for engineers and scienteists part 1 ppsx
... 8
1. 3. PowersandLogarithms 8
1. 3 .1. PowersandRoots 8
1. 3.2. Logarithms 9
1. 4. BinomialTheoremandRelatedFormulas 10
1. 4 .1. Factorials. Binomial Coefficients.BinomialTheorem 10
1. 4.2. RelatedFormulas ... 1 10 /16 /06 2:53: 21 PM
CONTENTS
Authors xxv
Preface xxvii
Main Notation xxix
Part I. Definitions, Formulas, Methods, and Theorems 1
1. Arithmetic and Elementary Algeb...
Handbook of mathematics for engineers and scienteists part 3 ppsx
... 930
ReferencesforChapter17 935
18. Special Functions and Their Properties 937
18.1. Some Coefficients,Symbols,andNumbers 937
18.1.1. Binomial Coefficients 937
18.1.2. PochhammerSymbol 938
18.1 .3. ... Equation and a Nonlinear Wave Equation 733
15.10. DifferentialConstraintsMethod 737
15.10.1. DescriptionoftheMethod 737
15.10.2. First-OrderDifferentialConstraints 739
15.10 .3. Seco...
Handbook of mathematics for engineers and scienteists part 6 ppsx
... integer
p and q (q ≠ 0). It is only the rational numbers that can be written in the form of finite
(terminating) or periodic (recurring) decimals (e.g., 1/8 = 0.125 and 1 /6 = 0. 166 66 ).
Any integer ... m/n,wherem and n are natural numbers:
a
p
= a
m/n
=
n
√
a
m
, a ≥ 0.
6 ARITHMETIC AND ELEMENTARY ALGEBRA
1.2.2. Addition and Multiplication of Numbers
1.2.2-1. Addition of...
Handbook of mathematics for engineers and scienteists part 11 ppsx
... 1).
References for Chapter 2
Abramowitz,M.andStegun,I.A.(Editors) ,Handbook of Mathematical Functions with Formulas, Graphs
and Mathematical Tables, National Bureau of Standards Applied Mathematics, ... 1998.
Korn,G.A.andKorn,T.M.,Mathematical Handbook for Scientists and Engineers, 2nd Edition, Dover
Publications, New York, 2000.
Prudnikov, A. P., Brychkov, Yu. A., and Mari...
Handbook of mathematics for engineers and scienteists part 25 ppsx
... R
.(4.6.3.23)
Remark. To obtain formulas (4.6.3.22) and (4.6.3.23), one should rewrite the equation of the straight line
in parametric form and replace x, y,andz in equation (4.6.3.16) of the plane by their ... length of the
perpendicular drawn from this point to the plane and taken with sign + if the point and the
origin lie on opposite sides of the plane and with sign – if...
Handbook of mathematics for engineers and scienteists part 26 ppsx
... under rotations of the coordinate axes. Here Δ
ij
is the
cofactor of the entry a
ij
in Δ.
The classification of quadrics based on the invariants S, T , δ ,and and the semi-
invariants σ and Σ is given ... the coefficients, has 1 of 17 canonical forms, each of which is associated with a
certain class of quadrics (see Table 4.3).
4.7.2-3. Invariants of quadrics.
The shape of a...
Handbook of mathematics for engineers and scienteists part 28 ppsx
... = 0
by the substitution y = x + 1/x.
5.1.5-2. Equations of general form and their properties.
An algebraic equation of degree n has the form
a
n
x
n
+ a
n–1
x
n–1
+ ···+ a
1
x + a
0
= 0 (a
n
≠ ... form
p/q
,where
p
is a divisor of
a
0
and
q
is a divisor of
a
n
.If
a
n
= 1
, then all rational roots of equation (5.1.5.1) (if they exist) are integer divisors of the
free term.
THE...
Handbook of mathematics for engineers and scienteists part 33 ppsx
... ALGEBRA
Properties of the set of all solutions of a homogeneous system:
1. All solutions of a homogeneous system (5.5.1.3) form a linear space.
2. The linear space of all solutions of a homogeneous ... unknown
quantities and a basic matrix of rank r is isomorphic to the space A
n–r
of all ordered
systems of (n – r) numbers. The dimension of the space of solutions is eq...
Handbook of mathematics for engineers and scienteists part 42 ppsx
... graph of the function y = |f (x)| is obtained from that of y = f (x) by preserving
the parts of the latter for which f (x) ≥ 0 and symmetric reflection, with respect to the axis
Ox, of the parts for ... term.
The remainders in the form of Lagrange and Cauchy can be obtained as special cases of the
Schl
¨
omilch formula with p = n + 1 and p = 1, respectively.
For a = 0,...
Handbook of mathematics for engineers and scienteists part 46 ppsx
... λ ≤ λ
2
and let it has a continuous partial derivative
∂
∂λ
f(x, λ)inthe
284 INTEGRALS
7.1.6-2. Integration of the product of a polynomial by a trigonometric function.
1
◦
. General formulas:
P
n
(x)cos(ax) ... ···
–cos(ax)
P
n
(x)
a
–
P
n
(x)
a
3
+ ···
+ C.
These formulas are obtained by repeated integration by parts; see formula 4 from Para-
graph 7.1.2-2 with f (x)=P
n
(x)...
Handbook of mathematics for engineers and scienteists part 48 ppsx
... Geometric and Physical Applications of the Definite Integral
7.2.6-1. Geometric applications of the definite integral.
1. The area of a domain D bounded by curves y = f(x)andy = g(x) and straight ... convergence of the latter integral in (7.2.7.2) implies the conver-
gence of the former integral.
3. If
0 < K ≤ ∞
, the divergence of the latter integral in (7.2.7.2) implies the...
Handbook of mathematics for engineers and scienteists part 55 ppsx
... x
X.
8.2.2. Basic Criteria of Uniform Convergence. Properties of
Uniformly Convergent Series
8.2.2-1. Criteria of uniform convergence of series.
1. Weierstrass criterion of uniform convergence. A functional ... sequence for
each x and are uniformly bounded (i.e., |u
n
(x)| ≤ K with a constant K independent of n, x).
3. Dirichlet criterion of uniform convergence of function...
Handbook of mathematics for engineers and scienteists part 66 ppsx
... harmonic and bounded in D and equal to u(ξ)
at all points of continuity of this function on C.
T
HEOREM ON THE UNIQUENESS OF A SOLUTION OF THE GENERALIZED DIRICHLET PROB-
LEM.
In a given domain for ... inward normal derivative.
Green’s formula expresses the solution of the Dirichlet problem for some domain D
in terms of the logarithm of the conformal mapping of the dom...
Handbook of mathematics for engineers and scienteists part 69 ppsx
... 450 INTEGRAL TRANSFORMS
TABLE 11.5 (continued)
Summary table of integral transforms
Integral transform Definition Inversion formula
Euler
transform of
the 2nd kind*
F (x)=
1
Γ(μ)
a
x
f(t) ... are tables of the Fourier cosine transform (see Section T3.3 and the references
listed at the end of the current chapter).
3
◦
. Sometimes the asymmetric form of the Fourier cosine transform...
Handbook of mathematics for engineers and scienteists part 96 ppsx
... integrand of the last term in solution (14.10.2.4)
can be expressed via the Green’s function G(x, y, t, τ ). The corresponding formulas for H
are given in Table 14.9 for the three basic types of ... equation of special form
–L
x
[G]=δ(x – y)(14.10.3.5)
with the homogeneous boundary condition
Γ
x
[G]=0 for x S.(14.10.3.6)
14.11. CONSTRUCTION OF THE GREEN’S FUNCTIONS.GENERAL FORMU...
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