tailieunhanh - Ebook Complex variables and applications (8th edition): Part 2

(BQ) Part 2 book "Complex variables and applications" has contents: Applications of residues, mapping by elementary functions, conformal mapping, applications of conformal mapping, the schwarz–christoffel transformation, integral formulas of the poisson type. | Brown-chap07-v2 11/01/07 CHAPTER 7 APPLICATIONS OF RESIDUES We turn now to some important applications of the theory of residues, which was developed in Chap. 6. The applications include evaluation of certain types of definite and improper integrals occurring in real analysis and applied mathematics. Considerable attention is also given to a method, based on residues, for locating zeros of functions and to finding inverse Laplace transforms by summing residues. 78. EVALUATION OF IMPROPER INTEGRALS In calculus, the improper integral of a continuous function f (x) over the semiinfinite interval 0 ≤ x 1, the points ck (k = 0, 1, 2) lie in the interior of the semicircular region bounded by the segment z = x (−R ≤ x ≤ R) of the real axis and the upper half CR of the circle |z| = R from z = R to z = −R. Integrating f (z) counterclockwise around the boundary of this semicircular region, we see that R (1) −R f (x) dx + f (z) dz = 2πi(B0 + B1 + B2 ), CR where Bk is the residue of f (z) at ck (k = 0, 1, 2). y CR c1 c2 –R c0 O R x FIGURE 94 With the aid of Theorem 2 in Sec. 76, we find that the points ck are simple poles of f and that Bk = Res z=ck c2 z2 1 = k5 = 3 z6 + 1 6ck 6ck (k = 0, 1, .

• Toán học
• Complex variables and applications
• Complex variables
• Applications of residues
• Mapping by elementary functions
• Conformal mapping
• The schwarz–christoffel transformation
• Complex numbers
• Analytic functions
• Elementary functions