Handbook of mathematics for engineers and scienteists part 165 pptx
Handbook of mathematics for engineers and scienteists part 165 pptx
... π. 15. ∞ k=1 1 k 2n+1 sin(kx)= (–1) n–1 (2π) 2n+1 2(2n + 1)! B 2n+1 x 2π , where 0 ≤ x ≤ 2π for n = 1, 2, ; 0 < x < 2π for n = 0;andtheB n (x) are Bernoulli polynomials. 1120 FINITE SUMS AND INFINITE SERIES 16. ∞ k=1 1 k2 k =ln2. 17. ∞ k=0 (–1) k n 2k = n 2 n 2 + ... a)=2 n cosh n x 2 sinh nx 2 + a . 10. n k=0 C k n cosh(kx + a)=2 n cosh n x 2 cosh nx 2 + a . 1...
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Handbook of mathematics for engineers and scienteists part 18 pptx
... equations of a curve. Parametric equations of a curve on the plane have the form x = ϕ(t), y = ψ(t), (4.2.1.4) where x and y are treated as the coordinates of some point A for each value of the ... coordinate system OXY has the form x a + y b = 1,(4.3.1.13) where a and b are the x-andy-intercepts of the line, i.e., the signed distances from the points of intersection of th...
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Handbook of mathematics for engineers and scienteists part 21 pptx
... directions of principal axes and of their conjugate chords are called the principal directions of a second-order curve. The angle between the positive direction of the axis OX and each of the two ... roots λ 1 and λ 2 of the characteristic equation (4.4.5.4) are the eigenvalues of the real symmetric matrix [a ij ] and, as a consequence, are real. Obviously, the invariants...
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Handbook of mathematics for engineers and scienteists part 23 pptx
... the surface that is formed by points of both surfaces in (4.5.2.18) and does not contain any other points. 4.6. Line and Plane in Space 4.6.1. Plane in Space 4.6.1-1. Equation of plane passing through ... z 0 ) and perpendicularly to a vector N =(A, B, C)hastheform A(x – x 0 )+B(y – y 0 )+C(z – z 0 )=0,or(r – r 0 ) ⋅ N = 0,(4.6.1.1) where r and r 0 are the position vectors of th...
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Handbook of mathematics for engineers and scienteists part 56 pptx
... series ∞ n=0 kx k–1 can be obtained from the left-hand side of the second formula in (8.3.3.7) for m = 0 and n = 1. Substituting S(x)=1/(1 – x) into the right-hand side of that formula, we get ∞ k=0 kx k–1 = d dx 1 1 ... subtraction, multiplication, and division of power series. 1. Addition and subtraction of power series.Twoseries ∞ n=0 a n x n and ∞ n=0 b n x n wit...
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Handbook of mathematics for engineers and scienteists part 76 pptx
... = 1; z p and C are arbitrary constants; and the a j (j ≥ 3) are uniquely defined in terms of α, β, z p ,andC. 2 ◦ . Two solutions of equation (12.3.4.8) corresponding to different values of the ... z p and C are arbitrary constants; the coefficients a j (j ≥ 7) are uniquely defined in terms of z p and C. 2 ◦ . In a neighborhood of a fixed point z = z 0 , the solution of the Ca...
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Handbook of mathematics for engineers and scienteists part 91 pptx
... the above holds for α = 1, 2, and β = 0,–1,–2, 14.4. Method of Separation of Variables (Fourier Method) 14.4.1. Description of the Method of Separation of Variables. General Stage of Solution 14.4.1-1. ... Derivation of equations and boundary conditions for particular solutions. The approach is based on searching for particular solutions of equation (14.4.1.1) in th...
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Handbook of mathematics for engineers and scienteists part 102 pptx
... constant, for some k and m. Transformation (15.3.4.2) is a combination of a translation transformation in t and scaling transformations in x and w. It should be emphasized that these transformations ... differential equations. 15.5. Method of Generalized Separation of Variables 15.5.1. Structure of Generalized Separable Solutions 15.5.1-1. General form of solutions. The clas...
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Handbook of mathematics for engineers and scienteists part 110 pptx
... 15.5. Algorithm for the construction of exact solutions by a nonclassical method for second-order evolution equations. Here, ODE stands for ordinary differential equation and PDE for partial differential equation. which ... condition of compatibility of equations (15.10.1.4) and (15.10.1.5). The compatibility condition is a differential equation for ϕ(y)andisa consequence...
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Handbook of mathematics for engineers and scienteists part 111 pptx
... operators of (15.10.2.4) in which the partial derivatives w t , w xx , w xt ,andw tt must be expressed in terms of x, t, w,andw x with the help of relations (15.10.2.12) and (15.10.2.2) and those ... to t and (15.10.2.2) with respect to t and x, and then equating the resulting expressions of the third derivatives w xtt and w ttx to one another: D t F =D x [D t G]. (15.10.2...
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