Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 65', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 416 Functions of Complex Variable 4. If f z is the quotient of two analytic functions f z z in a neighborhood of a point a and p a 0 a 0 but z a 0 . a is a simple pole of f z then res f a y -. Vz a 5. If a is an essential singularity of f z then to obtain res f a one has to find the coefficient c_1 in the Laurent expansion of f z in a neighborhood of a. A function f z is said to be continuous on the boundary C of the domain D if for each boundary point z0 there exists a limit lim f z f z0 as z z0 z g D. z z Cauchy s residue theorem. Let f z be a function continuous on the boundary C of a domain D and analytic in the interior of D everywhere except for finitely many points a1 . an. Then y f z dz 2ni res f ak where the integral is taken in the positive sense of C. The logarithmic residue of a function f z at a point a is by definition the residue of its logarithmic derivative fl i f Theorem. The logarithmic derivative fz z f z has first-order poles at the zeros and poles of f z . Moreover the logarithmic residue of f z at a zero or a pole of f z is equal to the order of the zero or minus the order of the pole respectively. The residue of a function f z at infinity is defined as res f to f z dz 2ni Jr where r is a circle of sufficiently large radius z p and the integral is taken in the clockwise sense so that the neighborhood of the point z to remains to the left of the contour just as in the case of a finite point . The residue of f z at infinity is equal to minus the coefficient of z-1 in the Laurent expansion of f z in a neighborhood of the point z to res f to -c_1. Note that res f to lim -zf z z provided that this limit exists. Theorem. Ifafunction f z has finitelymanysingularpoints a1 . an in the extended complex plane then the sum of all its residues including the residue at infinity is zero res f to Y res f ak 0. . Basic Notions 417 Example 10. Let us calculate the integral ln z 2 --------az z2
