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 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 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 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 94 docx

... ξ, t) for parabolic equations of the form (14.7.2.1) are found by formula (14.7.2.3), and that for hyperbolic equations of the form (14.8.2.1), by formula (14.8.2.3)Type of problem Form of boundary ... Λ1(x, t )and 2(x, t) that occur in the integrands of the last two termsin solution (14.8.2.2) are expressed in terms of the Green’s function of (14.8.2.3). Thecorresponding formulas for Λm(x, ... PROBLEMS FOR PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE 621TABLE 14.7Expressions of the functions Λ1(x, t )and 2(x, t) involved in the integrands of thelast two terms in solutions (14.7.2.2) and...

Handbook of mathematics for engineers and scienteists part 95 docx

... values of the functions g(x), h1(x), and h2(x)onthearcAB, cut off on the given curve (14.8.4.2) by the characteristics x = x0 and y = y0 ,and on the values of a(x, y), b(x, y), c(x, y), and ... functions Λ1(x, t )and 2(x, t) that occur in the integrands of the last two termsin solution (14.8.3.2) are expressed in terms of the Green’s function of (14.8.3.3). Thecorresponding formulas will ... valueproblems.The general and special properties of the Sturm–Liouville problem (14.8.3.4) are de-tailed in Subsection 12.2.5. Asymptotic and approximate formulas for eigenvalues and eigenfunctions...

Handbook of mathematics for engineers and scienteists part 97 docx

... values of the coefﬁcients s1, s2, k1,andk2in (14.12.1.3) and (14.12.1.4), one can obtain the ﬁrst, second, third, and mixed boundary value problems for equation (14.12.1.1).The solution of ... x1, , xn and t but isindependent of z.Let= (x, y, z – ζ, t, τ ) be a fundamental solution of the Cauchy problem for equa-tion (14.11.3.7), i.e.,∂∂t= Lx,t[ ]+∂2∂z2 for t > τ ... functions of some nonstationary boundary value problems for equation (14.11.3.7) via the fundamentalsolution of the Cauchy problem.TABLE 14.13Representation of the Green’s functions of some...

Handbook of mathematics for engineers and scienteists part 121 docx

... integrals, tables of direct and inverse Laplace transformscan be applied(see Sections T3.1 and T3.2), and, in many cases, to ﬁnd the inverse transform,methods of the theory of functions of a complex ... case of equation (16.2.3.1) for K(x)=–A sin(λx).We ﬁrst apply the table of Laplace transforms (see Subsection T3.1.6) and obtain the transform of thekernel of the integral equation in the formK(p)=–Aλp2+ ... EQUATIONSSolution of the equation for the transformFigure 16.1. Scheme of solving Volterra integral equations of the second kind with difference kernel by means of the Laplace integral transform; R(x)...

Handbook of mathematics for engineers and scienteists part 129 docx

... 4λˆf(s)2λ.(16.5.3.19)The inverse transform y(x)=M–1{ ˆy(s)} obtained by means of the Mellin inversion formula(if it exists) is a solution of equation (16.5.3.18). To different signs in the formula for theimages ... pn(t) are, for instance, continuous functions of t on the interval [a, b],system (16.5.3.6) becomes a system of nonlinear algebraic equations for A1, , Am.The number of solutions of the integral ... (16.5.3.37) and (16.5.3.39) is small, and the kernelin (16.5.3.39) remains the same in the course of the solution.The successive approximation method that consists of the application of formulas(16.5.3.36),...

Handbook of mathematics for engineers and scienteists part 136 docx

... values of y(x)form + 1 ≤ x ≤ m +2 are determined by the right-hand side of (17.4.1.8) for 0 ≤ x≤ 1, the quantities y(x+1), , y(x +m–1) are determined by the initial condition(17.4.1.6), and y(x ... ≤ x ≤ m.(17.4.1.6)The values of y(x)form ≤ x ≤ m + 1 are calculated by substituting the initial values(17.4.1.6) into the right-hand side of equation (17.4.1.5) for 0 ≤ x ≤ 1.Wehavey(x + m)=fx, ... increasing and continuous for 0 ≤ x ≤ a;(ii) f (0)=0 and f(x)<x for 0 < x < a;(iii) the derivative f(x) exists, has bounded variation on the interval 0 < x < a ,and limx→+0f(x)=1....

Handbook of mathematics for engineers and scienteists part 137 docx

... multiplication (division) of the equation and the results of its differentiation by suitablefunctions. Sometimes it is useful to take logarithm of the equation or the results of itstransformation (see ... us expand (17.5.1.1) in powers of the small parameter a in a neighborhood of a0,taking into account (17.5.1.2), and then divide the resulting equation by a – a0 and pass tothe limit for a ... in a neighborhood of a and |ϕ(a)| < 1;3) the function F is analytic in a neighborhood of the point (a, b )and| ∂F∂b(a, b)| < 1.Then the formal power series solution of equation (17.4.4.1),y(x)=b...

Handbook of mathematics for engineers and scienteists part 138 docx

... equationsoften arise when themethod of functional separation of variables is used for ﬁnding solutions of nonlinear equations of mathematical physics.1◦. Consider a functional equation of the formf(t)+g(x)+h(x)Q(z)+R(z)=0,wherez ... one of the two functions f(t )and (t) is prescribed and the other is assumed unknown;also one of the functions g(x )and (x) is prescribed and the other is unknown, and thefunctions h(x), Q(z), and ... The method of elimination of an argument is much simpler than that of differentiation in aparameter, since the former only requires to solve algebraic (transcendental) equations of the form (17.5.4.3)with...

Handbook of mathematics for engineers and scienteists part 161 docx

... interval.An estimator θ∗ of a parameter θ of the distribution of an observable random variable X(the estimator is a statistic, i.e., function of a random sample) is itself a random variablewith ... value of thepopulation parameter to be estimated; i.e.,P (θ∗L≤ θ ≤ θ∗R)=γ,(21.2.3.1)where θ∗L and θ∗Rare the lower and upper (left and right) boundaries of the conﬁdenceinterval of ... thefunction TINV(2α, n) in EXCEL software. The function qt(γ, n) in MATHCAD software can also be used.Example 6. For the F -distribution with n1 and n2degrees of freedom, the γ-quantile φγcan...

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