Handbook of mathematics for engineers and scienteists part 87 doc

... GeneralFormofIntegralTransforms.SomeFormulas 435 11.1.1. IntegralTransformsandInversionFormulas 435 11.1 .2. Residues.JordanLemma 435 11 .2. LaplaceTransform 436 11 .2. 1. LaplaceTransformandtheInverseLaplaceTransform ... LinearSystemsofOrdinaryDifferentialEquations 528 12. 6.1. Systems of Linear Constant-CoefficientEquations 528 12. 6 .2. Systems of Linear Variable-CoefficientEquations...

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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 ... 3.21b). 13. For secants, AC ⋅ AD = AB ⋅ AE = m 2 – R 2 (Fig. 3.21c). 14. For a tangent and a secant, AF ⋅ AF = AC ⋅ AD (Fig. 3.21c). 3.1.3-2. Segment and sector. Aplanefigure bo...

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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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... stated Discriminant D, formula (12.6.1.29) Roots of quadratic equation (12.6.1.27), λ 1 and λ 2 Conditions for coefficients of differential equations (12.6.1.25) Type of equilibrium points or shape of phase ... trajectories of a system of differential equations near an equilibrium point of the center type. 12.6. LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 535 By conve...

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... Theorems of stability and instability. In the cases where the theorems of stability and instability by first approximation fail to resolve the issue of stability for a specific system of nonlinear ... solution in the form u k = Ψ(u 3–k ), where k = 1, 2 and Ψ(u) is an arbitrary function of one variable. 2 ◦ . For linear equations (13.1.1.1) with the functions f , g,andh inde...

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... f, g,andh of equation (13.1.1.1) be continuously differentiable functions of x, y,andw in G and let x = h 1 (ξ), y = h 2 (ξ), and w = h 3 (ξ) be continuously differentiable functions of ξ for ... slightly). y x line of discontinuity characteristics characteristics Figure 13.7. Mutual arrangement of characteristics and lines of discontinuity in the case of a stable shock...

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... η x  .(13.1.3.13) This is a continuous function of η for fixed x and y. It can be shown that for fixed x and with the exception of a countable set of values of y, function (13.1.3.13) has a unique minimum with ... At the lines of discontinuity, y = y(x), the left and right limit values of w exist: w(x, y–0) and w(x, y + 0). 13.1. LINEAR AND QUASILINEAR EQUATIONS 565 1...

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... Jacobian of F , Φ ,and with respect to w, p,andq ,and[ Φ, Ψ]isthe Jacobi–Mayer bracket. In this case, relations (13.2.1.1) and (13.2.1.10) form a parametric representation of the complete integral of ... Method of separation of variables. Equations of special form. The method of separation of variables implies searching for a complete integral as the sum or product of fu...

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... of auxiliary problems for equation (14.4.4.1) and problems for associated functions ϕ(x )and (y) that determine particular solutions of the form (14.4.4.3). The abbreviation HBC below stands for ... detail). 14.5.1. Laplace Transform and Its Application in Mathematical Physics 14.5.1-1. Laplace and inverse Laplace transforms. Laplace transforms for derivatives. The Laplace t...

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