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Strange function

  1. Aug 5, 2010 #1
    how this function [tex] f(x)= 0^{-x} [/tex]

    should be understood? , if x is NEGATIVE, we find no problems, since 0 raised to any power is 0

    but how about x being a positive real number ? , or x being a PURE COMPLEX or complex number ?

    could we consider a 'regularization' to this f(x) so [tex] f(x)_{reg}=0 [/tex] is always 0
  2. jcsd
  3. Aug 5, 2010 #2


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    I would imagine the same way you would understand [tex]\sqrt{x}[/tex]. It's assumed that the domain is restricted only to numbers that make sense.

    If you have some context where you think you really need to plug 1 into the function, you should post the full setup here
  4. Aug 5, 2010 #3
    In the real case, the function

    [tex]f(x,y)=y^x=e^{x\log y}[/tex]

    is defined only for [tex]y>0[/tex]

    In the complex case, put

    [tex]y=\rho e^{i\alpha}\qquad\textrm{and}\qquad x=a+ib[/tex]


    [tex]y^x=(\rho e^{i\alpha})^{(a+ib)}[/tex]

    and, after some calculations, you find



    [tex]R=\rho^ae^{-\alpha b}\qquad\textrm{and}\qquad\beta=b\log\rho+\alpha a[/tex]

    So in the complex case you must have [tex]\rho\neq 0[/tex], which means [tex]y\neq 0[/tex].

    I hope I didn't meke calculation errors...try it yourself! :redface:
    Last edited: Aug 5, 2010
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