Handbook of mathematics for engineers and scienteists part 51 doc

Handbook of mathematics for engineers and scienteists part 2 doc

... GeneralFormofIntegralTransforms.SomeFormulas 43511.1.1. IntegralTransformsandInversionFormulas 43511.1 .2. Residues.JordanLemma 43511 .2. LaplaceTransform 43611 .2. 1. LaplaceTransformandtheInverseLaplaceTransform ... LinearSystemsofOrdinaryDifferentialEquations 528 12. 6.1. Systems of Linear Constant-CoefﬁcientEquations 528 12. 6 .2. Systems of Linear Variable-CoefﬁcientEquations 539 12. 7. NonlinearSystemsofOrdinaryDifferentialEquations 5 42 12. 7.1. ... Coefﬁcients 514 12. 4 .2. Linear Equations with Variable Coefﬁcients 518 12. 4.3. AsymptoticSolutionsofLinearEquations 522 12. 4.4. CollocationMethodandItsConvergence 523 12. 5. NonlinearEquationsofArbitraryOrder...

Handbook of mathematics for engineers and scienteists part 7 docx

... right-handsides 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 ... b2).1.6. MEAN VALUES AND INEQUALITIES OF GENERAL FORM 131.6. Mean Values and Inequalities of General Form1.6.1. Arithmetic Mean, Geometric Mean, and Other Mean Values.Inequalities for Mean Values1.6.1-1. ... 3b)+nab. A large number of formulas for the summation of various ﬁnite series can be found inSection T1.1.10 ARITHMETIC AND ELEMENTARY ALGEBRA1.4. Binomial Theorem and Related Formulas1.4.1. Factorials....

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 ... 3.21b). 13. For secants, AC ⋅ AD = AB ⋅ AE = m2– R2(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 bounded by two radii and ... legs are of equallength. The line segment connecting the midpoints of the legs is called the median of thetrapezoid. The length of the median is equal to half the sum of the lengths of the bases,m...

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 ... xn–1n=1≤j<i≤n(xi– xj).5.2.2-6. Determinant of a sum and a product of matrices.The determinant of the product of two matrices A and B of the same size is equal to theproduct of their determinants,det(AB)=detA ... Equivalence transformation.Matrices A and A of size m × n are said to be equivalent if there exist nondegeneratematrices S and T of size m × m and n × n, respectively, such that A and A are...

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 abasis 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 trace of its matrix in ... trace of the operator A.In the ﬁnite-dimensional case, λ is an eigenvalue of a linear operator A if and only if λ isa root of the characteristic equation (5.6.3.2) of the operator A. Therefore,...

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 thetheory of functions of a complex variable.7.2.8-5. Calculation of improper integrals using the Laplace transform.The following classes of ... Asymptotic Formulas for the Calculation of ImproperIntegralsBelow are some general formulas, involving arbitrary functions and parameters, that maybe useful for determining the asymptotic behavior of...

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 twoforms, 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 passesthrough the sphere center, ﬁnd the volume of the ﬁgure the cylinder cuts out of the sphere.The volume of this ... the right-hand side and the inner integralon the right-hand side exist.7.3. DOUBLE AND TRIPLE INTEGRALS 3217.3.2-2. Change of variables in the double integral.1◦.Letx = x(u, v)andy = y(u,...

Handbook of mathematics for engineers and scienteists part 65 doc

... property) and transforms any pair of points symmetric about a circleCinto a pair of points symmetric about the image of the circleC(preservation of symmetricpoints).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 equalto the order of the zero or minus the order of the pole, respectively.The residue of a ... FUNCTIONS OF COMPLEX VARIABLE4. If f(z) is the quotient of two analytic functions,f(z)=ϕ(z)ψ(z),in a neighborhood of a point a and ϕ(a) ≠ 0, ψ(a)=0,butψz(a) ≠ 0 (i.e., a is a simple poleof...

Handbook of mathematics for engineers and scienteists part 74 doc

... thechange of variable z =A2x2+A1x+A0+y (the constants A2, A1 ,and A0are selected usingthe method of undetermined coefﬁcients). In particular, the nonhomogeneous boundaryconditions of ... continuous in x for ﬁxed s, with x1≤ x ≤ x2 and x1≤ s ≤ x2.2◦. G(x, s) is a solution of the homogeneous equation (12.2.4.6), with r = 0,forallx1< x < x2exclusive of the point ... conditions of the general formα1yx+ β1y = 0 at x = x1,α2yx+ β2y = 0 at x = x2.(12.2.5.2)It is assumed that the functions p, px, s ,and q are continuous, and p and s are...

Handbook of mathematics for engineers and scienteists part 82 docx

... statedDiscriminant D,formula (12.6.1.29)Roots of quadraticequation (12.6.1.27), λ1 and λ2Conditions for coefﬁcients of differential equations (12.6.1.25)Type of equilibrium points orshape of phase ... trajectories of a system of differential equations near an equilibrium point of the centertype.12.6. LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 535By convention, for clearness and convenience of ... SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 539TABLE 12.5Classiﬁcation of equilibrium points for systems of constant-coefﬁcient equations (12.6.1.25);the symbols ◦ and ∗ indicate stable and unstable...

Handbook of mathematics for engineers and scienteists part 84 docx

... Theorems of stability and instability.In the cases where the theorems of stability and instability by ﬁrst approximation fail toresolve the issue of stability for a speciﬁc system of nonlinear ... solution in the formuk= Ψ(u3–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 independent of theunknown ... to some special types of linear and quasilinear ﬁrst-order partial differential equations; Ψ(u)is an arbitrary function. The subscripts x and y indicate the corresponding partial derivativesNo.EquationsGeneral...

Handbook of mathematics for engineers and scienteists part 85 doc

... f, g,andh of equation (13.1.1.1) be continuously differentiable functions of x, y,andw in G and let x = h1(ξ), y = h2(ξ), and w = h3(ξ) be continuously differentiablefunctions of ξ for ... slightly).yxline of discontinuitycharacteristicscharacteristicsFigure 13.7. Mutual arrangement of characteristics and lines of discontinuity in the case of a stable shockwave.The position of the point of ... dependence of f on w leads to atypical nonlinear effect: distortion of the proﬁle of the traveling wave.We further consider the domain x ≥ 0 and assume* that f > 0 for w > 0 and fw>...

Handbook of mathematics for engineers and scienteists part 86 docx

... ηx.(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 withthe exception of a countable set of values of y, function (13.1.3.13) has a unique minimumwith ... 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 56513.1.3-6. Hopf’s formula for the generalized ... interpretation: the rate of change of the totalvalue of w distributed over the interval (y1, y2) is compensated for by the “ﬂux” of thefunction F (w) through the endpoints of the interval.Let...

Handbook of mathematics for engineers and scienteists part 87 doc

... Jacobian of F , Φ ,and with respect to w, p,andq ,and[ Φ, Ψ]istheJacobi–Mayer bracket. In this case, relations (13.2.1.1) and (13.2.1.10) form a parametricrepresentation 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 orproduct of functions of various arguments. ... Equations of the form (13.2.1.14) are frequently encountered in various ﬁelds of mechanics, control theory, and differential games, where the variable x usually plays therole of time and the variable...

Handbook of mathematics for engineers and scienteists part 92 doc

... of auxiliary problems for equation (14.4.4.1) and problems for associatedfunctions 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 MathematicalPhysics14.5.1-1. Laplace and inverse Laplace transforms. Laplace transforms for derivatives.The Laplace transform of an arbitrary (complex-valued) ... widely used to solve linear problems of mathematicalphysics. The Laplace transform and the Fourier transform are in most common use (its and other integral transforms are considered in Chapter 11...

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