Handbook of mathematics for engineers and scienteists part 167 doc
Handbook of mathematics for engineers and scienteists part 167 doc
... The parameters a, b, p, m, and n below in Integrals 5–8 can assume arbitrary values,except for those at which denominators vanish in successive applications of a formula.Notation: w = axn+ ... x2)–3/2dx = a–2x(a2– x2)–1/2.T2.1.2-6. Integrals involving arbitrary powers. Reduction formulas.1.dxx(axn+ b)=1bnlnxnaxn+ b.2.dxx√xn+ a2=2anlnxn/2√xn+ ... x2+12a√2arctanax√2a2– x2.1130 INTEGRALS2.dx(a + x)(b + x)=1a – blnb + xa + x, a ≠ b.Fora = b, see Integral 2 with n =–2 inParagraph T2.1.1-1.3.xdx(a + x)(b + x)=1a – ba...
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Handbook of mathematics for engineers and scienteists part 2 doc
... VariousFormsoftheFourierTransform 44311.4.1. FourierTransformandtheInverseFourierTransform 44311.4.2. FourierCosineandSineTransforms 44511.5. OtherIntegralTransforms 44611.5.1. Integral Transforms ... . . . . . . 427ReferencesforChapter10 43311. Integral Transforms 43511.1. GeneralFormofIntegralTransforms.SomeFormulas 43511.1.1. IntegralTransformsandInversionFormulas 43511.1.2. Residues.JordanLemma ... Residues.JordanLemma 43511.2. LaplaceTransform 43611.2.1. LaplaceTransformandtheInverseLaplaceTransform 43611.2.2. Main Properties of the Laplace Transform. Inversion Formulas for SomeFunctions 43711.2.3....
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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....
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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 ... 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 ... angles α, β, γ ,and between a straight line AB and the directions to points D and C are known at points A and B (Fig. 3.18a). Suppose also that the distance a = AB (or b = DC) is known and the task...
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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...
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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,...
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Handbook of mathematics for engineers and scienteists part 49 doc
... notation for the Laplace transform:f(p)=Lf(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...
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Handbook of mathematics for engineers and scienteists part 51 doc
... rv.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,...
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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,butψz(a) ≠ 0 (i.e., a is a simple poleof...
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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 formα1yx+ β1y = 0 at x = x1,α2yx+ β2y = 0 at x = x2.(12.2.5.2)It is assumed that the functions p, px, s ,and q are continuous, and p and s are...
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