Khi biên soạn cuốn sách này, tác giả đã tham khảo và cập nhật những kiến thức mới nhất được xuất bản trong vài năm gần đây trên thế giới, đồng thời dựa trên tiêu chí nhấn mạnh vào các công cụ toán học được sử dụng nhiều nhất trong các ngành kỹ thuật, đặc biệt là trong ngành Điện tử Viễn thông. | 37 As it presently stands it would appear that we must always construct a Laurent expansion for each singularity it we wish to use the residue theorem. This becomes increasingly difficult as the structure of the integrand becomes more complicated. In the following paragraphs we will show several techniques that avoid this problem in practice. We begin by noting that many functions that we will encounter consist of the ratio of two polynomials . rational functions 2 ff z h z . Generally we can write h z as 2 - 2i mi z - Z2 J - -Here we have assumed that we have divided out any common factors between g z and 1 2 so that 0 2 does not vanish at Z1 z2 Clearly 21 22 . are singularities of f z . Further analysis shows that the nature of the singularities are a pole of order mi at 2 21 a pole of order m2 at 2 22 and so forth. Having found the nature and location of the singularity we compute the residue as follows. Suppose we have a pole of order n. Then we know that Its Laurent expansion is z 1 V. . Ll fro M 18 6 ỉ - Zo n z - o n Multiplying both sides of by 2 2o n r z z-Za V aT1 an-i z - Zo d z - 2o n - o 1 d- Because F z is analytic at 2 Zo it lias the Taylor expansion F z F zn F zo z-zo --- -. 2nĩl - o l 1 - t1-8 8 Matching powers of 2 2 J in and 1-8-8 . the residue equals Res z zo n_p 18-9 38 Substituting in F z z Zo nf z we can compute the residue of a pole of order n by Res ĩ ĩ õ ĩjị ĩ For a simple pole simplifies to Quite often f z p z q z . From 1 Hospital s rule it follows that Res y ĩ íj 1. y ZJ J Remember that these formulas work only for finite-order poles. For an essential singularity we must compute the residue from its Laurent expansion however essential singularities are very rare in applications. EVALUATION OF REAL DEFINITE INTEGRALS One of the important applications of the theory of residues consists in the evaluation of certain types of real definite integrals. Similar techniques apply when the integrand .
