Cauchy-Riemann equation polar form

[Ask4material]

I couldn't find any book discussing all of this.

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U+jV=f(x+jy) W=f(z)

Ux=Vy
Uy= -Vx
jWx=Wy <--Cauchy-Riemann equation

Uxx+Uyy=0
Vxx+Vyy=0 <--harmonic condition

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U+jV=f(rcjsθ) W=f(z)

rU_r=V_θ
rV_r=U_θ
jrW_r=W_θ <--Cauchy-Riemann equationU_{rr}?? U_{θθ}??
V_{rr}?? V_{θθ}?? <--harmonic condition??

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RcjsB=f(x+jy) W=f(z)

RBy = Rx
RBx= -Ry
jWx=Wy <--Cauchy-Riemann equation

Rxx?? Ryy??
Byy?? Bxx?? <--harmonic condition??

===================================================
RcjsB=f(rcjsθ) W=f(z)

rR_r=RB_θ
rRB_r=-R_θ
jrW_r=W_θ <--Cauchy-Riemann equation

R_{rr}?? R_{θθ}??
B_{rr}?? B_{θθ}?? <--harmonic condition??

 

[fresh_42]

https://users.math.msu.edu/users/shapiro/Teaching/classes/425/crpolar.pdfhttp://math.mit.edu/classes/18.013A/HTML/chapter09/section04.html