Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

How to prove Taylor's expansion for multivariable function ?

  1. Nov 7, 2009 #1
    As mentioned in the title~

    Could anyone give me a hint or an idea ?

    Thanks~
     
  2. jcsd
  3. Nov 7, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    What do you mean by "prove Taylor's expansion"? Taylor's expansion (by which I take it you mean writing a Taylor's series for the function) is defined to be a specific thing. Do you mean "proving the function is equal to its Taylor's series on the interval of convergence"? If so, what restrictions are you putting on the function. Of course, you must have f infinitely differentiable in order that its Taylor's series exist but there exist infinitely differentiable functionsm, whose Taylor's series converge for all x but that are not equal to their Taylor series anywhere except at the base point. One such is [itex]f(x)= e^{-1/x^2}[/itex] is [itex]x\ne 0[/itex], 0 if x= 0. Its Taylor's series, about x= 0, is identically 0 so converges uniformly for all x but is not equal to f(x) anywhere except at x= 0.
     
  4. Nov 7, 2009 #3
    Thank your warning~

    Let me restate my question:

    What's the form of Taylor's series about a "well-behaved" multivariable function ?

    And~how to prove it ?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: How to prove Taylor's expansion for multivariable function ?
  1. Taylor's Expansion (Replies: 3)

  2. Taylor expansions (Replies: 7)

Loading...