By Alan Jeffrey, Hui Hui Dai
The vast additions, and the inclusion of a brand new bankruptcy, has made this vintage paintings through Jeffrey, now joined by means of co-author Dr. H.H. Dai, an excellent extra crucial reference for researchers and scholars in utilized arithmetic, engineering, and physics. It offers easy access to special formulation, relationships among capabilities, and mathematical ideas that variety from matrix concept and integrals of usually taking place capabilities to vector calculus, usual and partial differential equations, distinctive capabilities, Fourier sequence, orthogonal polynomials, and Laplace and Fourier transforms. throughout the guidance of this version complete virtue was once taken of the lately up to date 7th version of Gradshteyn and Ryzhik's desk of Integrals, sequence, and items and different vital reference works. feedback from clients of the 3rd version of the instruction manual have led to the growth of many sections, and due to the relevance to boundary worth difficulties for the Laplace equation within the aircraft, a brand new bankruptcy on conformal mapping, has been further, whole with an atlas of worthy mappings. a special function of the fourth variation is the supply of a CD-ROM that offers prepared entry to the main usually used elements of the publication, including worthwhile notes for clients of the instruction manual.
- complete assurance in reference type of the branches of arithmetic utilized in technology and engineering<BR id="CRLF">- equipped to make effects concerning integrals and services effortless to locate<BR id="CRLF">- effects illustrated by way of labored examples <BR id="CRLF">- certain CD-ROM that offers entry to the main often used elements of the book<BR id="CRLF">- CD-ROM includes invaluable notes for clients
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Extra info for Handbook of Mathematical Formulas and Integrals (4th Edition)
We will use the contour shown in Figure 3, where the branch point is enclosed in a small circle about the origin while the complex s-plane is cut along the negative real axis to make the function single valued inside the contour. Let CR1 denote the large circular arc and CR2 denote the small circle around the origin. π Then on CR1 s = γ + Reiθ for π2 ≤ θ ≤ 3π 2 , and for subsequent use we now set θ = 2 + φ, so iφ iφ s = γ + iRe with 0 ≤ φ ≤ π. Consequently, ds = −Re dφ, with the result that |ds| = Rdφ.
4! 12. ex = 1 + x + 13. e−x Logarithmic series 14. 15. 16. x2 x3 x4 x5 + − + − ··· 2 3 4 5 x2 x3 x4 ln(1 − x) = − x + + ··· + + 2 3 4 x3 1+x x5 x7 =2 x+ ln + + + ··· 1−x 3 5 7 ln(1 + x) = x − [−1 < x ≤ 1] [−1 ≤ x < 1] = 2 arctanh x [|x| < 1] 12 17. Chapter 0 ln 1−x 1+x = −2 x + x3 x5 x7 + + + ··· 3 5 7 Quick Reference List of Frequently Used Data = −2 arctanh x [|x| < 1] Hyperbolic series 18. sinh x = x + x3 x5 x7 + + + ··· 3! 5! 7! [|x| < ∞] 19. cosh x = 1 + x4 x6 x2 + + + ··· 2! 4! 6! [|x| < ∞] 20.
We omit the details, but straightforward reasoning using the standard Bromwich contour shows that f(x) = L−1 1 s cosh s ∞ =1+ 4 cos [(2n + 1)πx/2] (−1)n+1 . π n=0 2n + 1 To understand why this periodic representation of f(x) has occurred, notice that F (s) = 1/[s cosh s] is the Laplace transform of the piecewise continuous function 0, f(x) = 2, 0, 0
Handbook of Mathematical Formulas and Integrals (4th Edition) by Alan Jeffrey, Hui Hui Dai