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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 2

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 2 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Now we compute the derivative. 4- log z e 1 Log r lớ dz dr e-ie 1 r _ 1 z Solution 8.14 The complex derivative in the coordinate directions is d e-ie d - Ỉ e-ie d dz dr r dỡ We substitute f u iv into this identity to obtain the Cauchy-Riemann equation in polar coordinates. e- f - w df_ i df I-- dr r dỡ ur ivr -- ue ive r We equate the real and imaginary parts. 11 ur - ve vr ue 1 ur - ve ue -rvr Solution 8.15 Since w is analytic u and V satisfy the Cauchy-Riemann equations ux vy and uy -vx. 414 Using the chain rule we can write the derivatives with respect to x and y in terms of u and v. É uxi v4 dx du dv I y vy I dy du dv Now we examine ộx iộy. ộx 1ộy ụA vA 1 uy T vy Tv ộx lộy ux 1uy Tu vx 1vy Tv ộx 1ộy u iuy T 1 vy ivx Tv We use the Cauchy-Riemann equations to write uy and vy in terms of ux and vx. ộx 1ệy ux ivx T 1 ux ivx Tv Recall that w ux ivx vy iuy. ộx 1 y He 1 v dz Thus we see that d T d T dw 1 dộ dộ du dv d dx dy We write this in operator notation. d d c dw -1 i d d du dv d dx dy .