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Cauchy-Riemann equation polar form

  1. May 17, 2012 #1
    I couldn't find any book discussing all of this.

    ===================================================
    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

    ===================================================
    U+jV=f(rcjsθ) W=f(z)

    [itex]rU_r=V_θ[/itex]
    [itex]rV_r=U_θ[/itex]
    [itex]jrW_r=W_θ[/itex] <--Cauchy-Riemann equation


    [itex]U_{rr}[/itex]?? [itex]U_{θθ}[/itex]??
    [itex]V_{rr}[/itex]?? [itex]V_{θθ}[/itex]?? <--harmonic condition??

    ===================================================
    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)

    [itex]rR_r=RB_θ[/itex]
    [itex]rRB_r=-R_θ[/itex]
    [itex]jrW_r=W_θ[/itex] <--Cauchy-Riemann equation

    [itex]R_{rr}[/itex]?? [itex]R_{θθ}[/itex]??
    [itex]B_{rr}[/itex]?? [itex]B_{θθ}[/itex]?? <--harmonic condition??
     
    Last edited: May 17, 2012
  2. jcsd
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