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Complex Numbers

  1. Jan 2, 2008 #1
    [SOLVED] Complex Numbers

    A very happy new year to all at PF.

    1. The problem statement, all variables and given/known data
    Kreyszig, P.665 section 12.3,

    A function f(z) is said to have the limit l as z approaches a point [tex]z_0[/tex] if f is defined in the neighborhood of [tex]z_0[/tex] (except perhaps at [tex]z_0[/tex] itself) and if the values of f are "close" to l for all z "close" to [tex]z_0[/tex]; that is, in precise terms, for every positive real [tex]\epsilon[/tex] we can find a positive real [tex]\delta[/tex] such that for all z not equal to [tex]z_0[/tex] in the disk [tex]|z-z_0|<\delta[/tex], we have

    [tex]|f(z)-l|<\epsilon[/tex]...(2)

    that is, for every z not equal to [tex]z_0[/tex] in that [tex]\delta[/tex] disk, the value of f lies in the disk (2).

    I think this means that if you were to plot z and f(z), for all values of z near the point [tex]z_0[/tex] (within the [tex]\delta[/tex] disk), but not necessarily at [tex]z_0[/tex] itself (as the function may not exist at [tex]z_0[/tex]), the value of f(z) would be very close to l (but not necessarily l, as [tex]f(z_0)[/tex] may not exist) and that this value of f(z) would be inside the [tex]\epsilon[/tex] disk.

    Is that right?
     
  2. jcsd
  3. Jan 2, 2008 #2

    mjsd

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    Homework Helper

    sounds good
     
  4. Jan 2, 2008 #3
    Congrats, you just understood what a limit is.
     
  5. Jan 3, 2008 #4
    Thank you.
     
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