tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 9

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 1 part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | is defined on the positive real axis. Define a branch such that f 1 1 2. Write down an explicit formula for the value of the branch. What is f 1 1 What is the value of f z on either side of the branch cuts Hint Solution Exercise Find all branch points of f z z - 1 z - 2 z - 3 1 2 in the extended complex plane. Which of the branch cuts in Figure will make the function single-valued. Using the first set of branch cuts in this figure define a branch on which f 0 zự6. Write out an explicit formula for the value of the function on this branch. Figure Four candidate sets of branch cuts for z - 1 z - 2 z - 3 1 2. Hint Solution 294 Exercise Determine the branch points of the function w z2 - 2 z 2 1 3. Construct and define a branch so that the resulting cut is one line of finite extent and w 2 2. What is w -3 for this branch What are the limiting values of w on either side of the branch cut Hint Solution Exercise Construct the principal branch of arccos z . Arccos z has the property that if x G -1 1 then Arccos x G 0 n . In particular Arccos 0 2 . Hint Solution Exercise Find the branch points of z1 2 1 1 2 in the finite complex plane. Introduce branch cuts to make the function single-valued. Hint Solution Exercise For the linkage illustrated in Figure use complex variables to outline a scheme for expressing the angular position velocity and acceleration of arm c in terms of those of arm a. You needn t work out the equations. Hint Solution Exercise Find the image of the strip z 1 and of the strip 1 A z 2 under the transformations 1. w 2z2 2. w z 1 Hint Solution .