Handbook of mathematics for engineers and scienteists part 151 doc
Handbook of mathematics for engineers and scienteists part 151 doc
... strict
inequality) for an optimal solution of the primal problem, then the ith coordinate of the
optimal solution of the dual problem is zero, and, conversely, if the ith coordinate of the
optimal solution of ... used as a system of equations for the potentials. This system has
n + m unknowns and n + m – 1 equations. Since the number of unknowns exceeds the
number of eq...
Handbook of mathematics for engineers and scienteists part 2 doc
... VariousFormsoftheFourierTransform 443
11.4.1. FourierTransformandtheInverseFourierTransform 443
11.4.2. FourierCosineandSineTransforms 445
11.5. OtherIntegralTransforms 446
11.5.1. Integral Transforms ... . . . . . . 427
ReferencesforChapter10 433
11. Integral Transforms 435
11.1. GeneralFormofIntegralTransforms.SomeFormulas 435
11.1.1. IntegralTransformsandInversionFormulas 435
11.1.2. Resid...
Handbook of mathematics for engineers and scienteists part 7 docx
... right-hand
sides of the above formulas is called the index of summation (for series) or the index of
multiplication (for products). The 1 and n (or the m and n)aretheupper and lower limits
of summation ... b
2
).
1.6. MEAN VALUES AND INEQUALITIES OF GENERAL FORM 13
1.6. Mean Values and Inequalities of General Form
1.6.1. Arithmetic Mean, Geometric Mean, and Other Mea...
Handbook of mathematics for engineers and scienteists part 13 doc
... area of the triangle with vertices at the center of the disk and at the endpoints
of the radii bounding the corresponding sector. One takes the minus sign for α < 180 and
the plus sign for ... legs are of equal
length. The line segment connecting the midpoints of the legs is called the median of the
trapezoid. The length of the median is equal to half the sum of the...
Handbook of mathematics for engineers and scienteists part 31 docx
... rank of the matrix A.
Properties of the rank of a matrix:
1. For any matrices A and B of the same size the following inequality holds:
rank (A + B) ≤ rank (A)+rank(B).
2. For a matrix A of size ... x
n–1
n
=
1≤j<i≤n
(x
i
– x
j
).
5.2.2-6. Determinant of a sum and a product of matrices.
The determinant of the product of two matrices A and B...
Handbook of mathematics for engineers and scienteists part 35 docx
... system of
eigenvectors of A and the subspace consisting of vectors orthogonal to all eigenvectors
of A.Inthefinite-dimensional case, an orthonormal system of eigenvectors of A is a
basis of V.
5. ... =detA.
The trace ofthe matrix of a linear operator, Tr(A), isalso independent of the basis. Therefore,
one can correctly define the trace Tr(A) of a linear operator as the tra...
Handbook of mathematics for engineers and scienteists part 49 doc
... notation for the Laplace transform:
f(p)=L
f(x)
.
Section 11.2 presents properties and methods for determining the Laplace transform,
and Section T3.1 gives tables of the Laplace transforms of ... method for the calculation of improper integrals using the
theory of functions of a complex variable.
7.2.8-5. Calculation of improper integrals using the Laplace transfo...
Handbook of mathematics for engineers and scienteists part 51 doc
... r
v
.
Remark. The formulas from Items 3 and 4 are equivalent—they define one and the same surface in two
forms, scalar and vector, respectively.
5. Calculation of volumes. If a domain U of the three-dimensional ... sphere of radius R and a right circular cylinder of radius a < R whose axis passes
through the sphere center, find the volume of the figure the cylinder cuts o...
Handbook of mathematics for engineers and scienteists part 65 doc
... property) and transforms any pair of points symmetric about a circle
C
into a pair of points symmetric about the image of the circle
C
(preservation of symmetric
points).
420 FUNCTIONS OF COMPLEX ... zeros and
poles of
f(z)
. Moreover, the logarithmic residue of
f(z)
at a zero or a pole of
f(z)
is equal
to the order of the zero or minus the order of the pole, respectiv...
Handbook of mathematics for engineers and scienteists part 74 doc
... the
change of variable z =A
2
x
2
+A
1
x+A
0
+y (the constants A
2
, A
1
,and A
0
are selected using
the method of undetermined coefficients). In particular, the nonhomogeneous boundary
conditions of ... continuous in x for fixed s, with x
1
≤ x ≤ x
2
and x
1
≤ s ≤ x
2
.
2
◦
. G(x, s) is a solution of the homogeneous equation (12.2.4.6), with r = 0,forall
x
1
< x < x
2
exclu...
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