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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 9

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 2 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Since the nearest singularity is at z 1 the Laurent series will converge in the annulus 0 z 1 2. z2 - 1 z - 1 1 2 - 1 z 1 2 12 z 1 2 1 2 z 1 _ 1 2 12 z 1 1 4 1 1 z 1 2 _ 1 i 1 z 1 V 4 I 2 i n 0 1 1 . V -- z 1 n 4 J 2 n 0 This geometric series converges for 1 z 1 2 1 or z 1 2. The series expansion of f z is -1 2 1 1n f z z 1 2 12 z 1 z 1 4 2n z 1 n n 0 z4 -1 2 1 x 1n z2 1 z 1 2 12 z 1 z 1 4 52 2n z 1 for z 1 2 n 0 Laurent Series about z X. Since the nearest singularities are at z 1 the Laurent series will converge in 694 the annulus 1 z X. 4 2 z z2 z2 1 1 1 z2 ro 1 n n 0 z 0 E 1 nz2 n 1 n - 1 E 1 n 1z2n n - This geometric series converges for 1 z21 1 or z 1. The series expansion of f z is z4 1 E for 1 z X z2 1 2 f n - tt Solution 13.7 Method 1 Residue Theorem. We factor P z . Let m be the number of roots counting multiplicities that lie inside the contour r. We find a simple expression for P z P z . P z c H z zk k 1 nn P z c En z zj k i 7 1 j k .