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...

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... 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...

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... 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...

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... 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...

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... 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...

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... 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...

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... 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...

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... 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...

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... 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...

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... 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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