1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Limit and diffirentiability of a function

  1. Mar 30, 2012 #1
    1. The problem statement, all variables and given/known data
    For complex numbers [itex]f[/itex] and [itex]g[/itex], and for [itex]1<p<\infty[/itex] we have [itex]\lim_{t\rightarrow 0}\dfrac{|f+tg|^p-|f|^p}{t}=|f|^{p-2}(\bar{f}g+f\bar{g})[/itex]; i.e., [itex]|f+tg|^p[/itex] is differentiable.

    I would like to show that the above statement is true.

    2. Relevant equations

    3. The attempt at a solution

    I have try several attempts in the direction of manipulating the convex function [itex]|x|^p[/itex]. But no reasonable conclusions yet.
  2. jcsd
  3. Mar 30, 2012 #2
    So, I have made some progress by rewriting the problem and using L'hopitals rule. But I am still off by a factor of [itex]\dfrac{p}{2}[/itex].

    rewriting: [itex]|f+tg|^2=f\bar{f}+tf\bar{g}+t\bar{f}g+t^2g\bar{g}[/itex]

    When I apply L'Hopitals rule I get [itex]\dfrac{p}{2}|f|^{p-2}(\bar{f}g+f\bar{g})[/itex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook