Có 80+ tài liệu thuộc chủ đề "toán học trong kỹ thuật"
tailieu.vn Xem trực tuyến Tải xuống
The positive sense of the boundary is defined to be the sense for which the domain lies to the left of the boundary.. R, i.e., the interior of the disk of radius R >. 0 centered at the point a. A neighborhood of the point at infinity is understood as the set of points z such that |z| >. If...
tailieu.vn Xem trực tuyến Tải xuống
If f (z) and g(z) are analytic functions in a simply connected domain D and z = a and z = b are arbitrary points of the domain D, then the formula of integration by parts holds:. C AUCHY ’ S THEOREM FOR A SIMPLY CONNECTED DOMAIN . If a function f(z) is analytic in a simply connected domain D...
tailieu.vn Xem trực tuyến Tải xuống
(10.1.2.48). If a is an essential singularity of f (z), then to obtain res f (a), one has to find the coefficient c –1 in the Laurent expansion of f (z) in a neighborhood of a.. A function f (z) is said to be continuous on the boundary C of the domain D if for each boundary point z 0...
tailieu.vn Xem trực tuyến Tải xuống
α that do not pass through the point z = –d/c (α. Re(λd/c)) are taken to the circles |w – w 0. Re(λd/c) passing through the point z = –d/c are taken to the straight lines. r that do not pass through the point z = –d/c (r ≠ |z 0 + d/c|) are taken to the circles |w –...
tailieu.vn Xem trực tuyến Tải xuống
Cauchy-type integral.. is called a Cauchy-type integral.. The Cauchy-type integral is a function analytic at any point z that does not lie on C.. If the curve C divides the plane into several domains, then, in general, the Cauchy-type integral defines different analytic functions in these domains.. One says that the function f (ξ) satisfies the H¨older condition with exponent...
tailieu.vn Xem trực tuyến Tải xuống
Inverse Laplace transform.. Given the transform f(p), the function 2 f (x) can be found by means of the inverse Laplace transform. The integral in inversion formula is understood in the sense of the Cauchy principal value: c+i∞. 0, then the left-hand side of is equal to 1 2 [f (x 0 –0)+f(x 0 +0)]. For brevity, we write the...
tailieu.vn Xem trực tuyến Tải xuống
444 I NTEGRAL T RANSFORMS TABLE 11.4. Main properties of the Fourier transform. Function Fourier transform Operation. i n 2 f u (n) (u) Differentiation of the transform. Asymmetric form of the Fourier transform. Alternative Fourier transform.. Sometimes it is more convenient to define the Fourier transform by f ˇ (u). In this case, the Fourier inversion formula reads f...
tailieu.vn Xem trực tuyến Tải xuống
Areas of application of integral transforms (first in the last column come references to appropriate sections of the current book). (direct and inverse) Paragraphs 8.1.5-2 and 8.4.4-1 Computation of coefficients. Paragraphs 12.4.1-3 and 12.4.2-6;. Paragraphs 16.5.2-1 and 16.5.3-2;. 1 and 2, McGraw-Hill, New York, 1954.. and Roth, R., The Laplace Transform, World Scientific Publishing Co., Singapore, 1984.. Bracewell, R., The...
tailieu.vn Xem trực tuyến Tải xuống
see Paragraph The substitution z = y 1– a brings it to a linear equation, z x + (1 – a)f (x)z = (1 – a)g(x), which is discussed in Paragraph 12.1.2-5. With this in view, one can obtain the general integral:. y 1– a = Ce – F + (1 – a)e – F e F g(x) dx, where...
tailieu.vn Xem trực tuyến Tải xuống
Reduction of the Abel equation of the second kind to the canonical form.. In turn, equation can be reduced, by the introduction of the new independent variable. (12.1.6.5) Here, the function R(z) is defined parametrically (x is the parameter) by the relations. Therefore the function R(z) in the right-hand side of the Abel equation can be identified with the two-parameter...
tailieu.vn Xem trực tuyến Tải xuống
12.1.10-3. Runge–Kutta method of the fourth-order approximation.. This is one of the widely used methods. All methods described in Subsection 12.1.10 are special cases of the Runge–Kutta method (a detailed description of this method can be found in the monographs listed at the end of the current chapter).. In practice, calculations are performed on the basis of any of the...
tailieu.vn Xem trực tuyến Tải xuống
0, k[f (x)] –1/4 exp. Consider an equation of the form. Then the leading terms of the asymptotic expansions of the fundamental system of solutions of equation are given by the formulas. 0, the asymptotic solution of this equation, satisfying the boundary conditions y(0. C 2 , can be represented in the form. Now let us take a look at...
tailieu.vn Xem trực tuyến Tải xuống
where Q(u, v) is given by . Problems with boundary conditions of the third kind.. where Q(u, v) is defined by . Let us note some special properties of the Sturm–Liouville problem that is the mixed boundary value problem for equation with the boundary conditions. Let y n = y n (x) be nonzero solutions of the linear equations. Then...
tailieu.vn Xem trực tuyến Tải xuống
(12.3.3.5) On substituting x = x 0 , the initial conditions and the expression of y xx (x 0 ) of into the right-hand side of equation we calculate the value of the third derivative:. The subsequent derivatives of the unknown are determined likewise.. The thus obtained solution can only be used in a small neighborhood of the point x...
tailieu.vn Xem trực tuyến Tải xuống
Form of the solution sought. One looks for periodic solutions of the equation. secular terms are eliminated through selection of the constants ω k Method. see also the problem in the method of scaled parameters. see Paragraph 12.3.5-4, Item 2. 0 cos ϕF dϕ, F = f(a cos ϕ,–aω 0 sin ϕ). Method of two-scale. see Paragraph 12.3.5-5, Item 2....
tailieu.vn Xem trực tuyến Tải xuống
We now consider an equation of the general form. εy xx = F(x, y, y x subject to boundary conditions . For the leading term of the outer expansion y = y 0 (x. In the general case, when using the method of matched asymptotic expansions, the position of the boundary layer and the form of the inner (extended) variable...
tailieu.vn Xem trực tuyến Tải xuống
The eigenvalue problem for the equation y xx + λ(1 + x 2 ) –2 y = 0. The solution of the Cauchy problem with f(x. The first iteration for the first eigenvalue is determined by and results in the value λ with the relative error Δ λ/λ . Consider the eigenvalue problem for the equation. For the relative error...
tailieu.vn Xem trực tuyến Tải xuống
where C is an arbitrary constant. where J 0 (x) is the Bessel function of the first kind and I 0 (x) is the modified Bessel function of the first kind.. this formula is only valid for finite initial values of the function and its derivatives.. Solutions to linear differential equations with polynomial coefficients can sometimes be represented as a...
tailieu.vn Xem trực tuyến Tải xuống
By convention, for clearness and convenience of interpretation of the results, t will be used to designate the independent variable and will be treated as time. A solution x = x(t), y = y(t) of system plotted in the plane x, y (the phase plane) is called a (phase) trajectory of the system.. A solution to system will be sought...
tailieu.vn Xem trực tuyến Tải xuống
Euler system of ordinary differential equations.. A homogeneous Euler system is a homogeneous linear system of ordinary differential equations composed by linear combinations of the following terms:. Such a system is invariant under scaling in the independent variable (i.e., it preserves its form under the change of variable x → αx, where α is any nonzero number). The substitution x...