tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 8

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 1 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The argument of ez is a function of y alone. arg ez arg ex iy y 2nn n E Z In Figure are plots of ez and a branch of arg ez . 0 y 5 0 -5 5 y Figure Plots of ez and a branch of arg ez . Example Show that the transformation w ez maps the infinite strip TO x TO 0 y n onto the upper half-plane. Method 1. Consider the line z x ic TO x TO. Under the transformation this is mapped to w ex lc elcex TO x TO. This is a ray from the origin to infinity in the direction of elc. Thus we see that z x is mapped to the positive real w axis z x in is mapped to the negative real axis and z x ic 0 c n is mapped to a ray with angle c in the upper half-plane. Thus the strip is mapped to the upper half-plane. See Figure . Method 2. Consider the line z c iy 0 y n. Under the transformation this is mapped to w ec iy ec eiy 0 y n. 254 Figure ez maps horizontal lines to rays. This is a semi-circle in the upper half-plane of radius ec. As c TO the radius goes to zero. As c TO the radius goes to infinity. Thus the strip is mapped to the upper half-plane. See Figure . 3 2 1 1 1 Figure ez maps vertical lines to circular arcs. .