Cauchy-Riemann equation polar form

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  1. BookInquiry's Avatar
    1 17-05-2012  06:20
    • Junior Member
    • Posts: 47
    Cauchy-Riemann equation polar form
    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(rcjs\thetha) W=f(z)

    rU_r=V_\theta
    rV_r=U_\theta
    jrW_r=W_\theta <--Cauchy-Riemann equation


    U_{rr}?? U_{\theta\theta}??
    V_{rr}?? V_{\theta\theta}?? <--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(rcjs\theta) W=f(z)

    rR_r=RB_\theta
    rRB_r=-R_\theta
    jrW_r=W_\theta <--Cauchy-Riemann equation

    R_{rr}?? R_{\theta\theta}??
    B_{rr}?? B_{\theta\theta}?? <--harmonic condition??
    Last edited by BookInquiry; 18-05-2012 at 02:55.
  2. Jake22's Avatar
    2 17-05-2012  09:30
    • TSR Demigod
    • Posts: 5,176
    Re: Cauchy-Riemann equation polar form
    Do you have a question?
  3. BookInquiry's Avatar
    3 18-05-2012  02:54
    • Junior Member
    • Posts: 47
    Re: Cauchy-Riemann equation polar form
    Most harmonic conditions of them are not known.
  4. Jake22's Avatar
    4 18-05-2012  12:43
    • TSR Demigod
    • Posts: 5,176
    Re: Cauchy-Riemann equation polar form
    So you want to write the Laplace equation in polar coordinates?

    r^2W_{rr}+rW_r + W_{\theta\theta} = 0
  5. BookInquiry's Avatar
    5 19-05-2012  14:07
    • Junior Member
    • Posts: 47
    Re: Cauchy-Riemann equation polar form
    Are they related? You must have worked with some (books, lecture notes or internet materials) which make you master the topics. Could you suggest your choice to me? I want to follow your track to study.
  6. Jake22's Avatar
    6 19-05-2012  14:32
    • TSR Demigod
    • Posts: 5,176
    Re: Cauchy-Riemann equation polar form
    Any good calculus book will talk about change of variables. I don't know one to recommend off the top of my head I am afraid.
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