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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 10

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 1 part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Since z-3 2 has a branch point at z 0 and the rest of the terms are analytic there w z has a branch point at infinity. Consider the set of branch cuts in Figure 7.52. These cuts let us walk around the branch points at z 2 and z 1 together or if we change our perspective we would be walking around the branch points at z 6 and z TO together. Consider a contour in this cut plane that encircles the branch points at z 2 and z 1. Since the argument of z z0 1 2 changes by n when we walk around z0 the argument of w z changes by 2n when we traverse the contour. Thus the value of the function does not change and it is a valid set of branch cuts. Figure 7.52 Branch cuts for z 2 z 1 z 6 1 2. Now to define the branch. We make a choice of angles. z 2 r1 elớ1 01 02 for z G 1. 6 z 1 r2 e1Ớ2 02 01 for z G 1. 6 z 6 r3 e1Ớ3 0 03 2n The function is w z n elớ1 r-2 e r3 e1 3 1 2 ựr e1 ớ1 Ớ2 Ớ3 2 . We evaluate the function at z 4. w 4 V 6 3 2 e1 2nra 2nra n 2 16 We see that our choice of angles gives us the desired branch. 334 Solution 7.24 1. cos z1 2 cos v cos ựz This is a single-valued function. There are no branch points. 2. z 1 -Z e-z log z 1 __ e-z ln z I i I I Arg z I 1 1 i2n z n E Z There is a branch point at z 1. There are an infinite number of branches. Solution 7.25 1. f z z2 1 1 2 z l 1 2 z l 1 2 We see that there are branch points at z 1. To examine the point at infinity we substitute z 1 z and examine the point z 0. 7 1 y 1 A 1 z2 1 2 z Since there is no branch point at z 0 f z has no branch point at infinity. A branch cut connecting z 1 would make the function single-valued. We could also accomplish this with two branch cuts starting z 1 and going to infinity. 2. f z z3 z 1 2 z1 2 z 1 1 2 z 1 1 2 There are branch points at z 1 0 1. Now we consider the point at infinity. 1Y 1Ì z-3 2 1 z2 1 2 d zj z z .