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Homework Help: Help me simplify this!

  1. Mar 25, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]f(E) = \left(\frac{E_c}{E} \right)^{1/2} + \frac{E}{kT}[/tex]

    Need to take second derivative with respect to E for Taylor expansion about [tex]E_0[/tex] where [tex]E_0 = [\frac{1}{4}E_c(kT)^2]^\frac{1}{3}[/tex], which is the Gamow peak.

    2. Relevant equations

    3. The attempt at a solution

    So for the first derivative i got [tex]-\frac{E_c^\frac{1}{2}}{2E^\frac{3}{2}} + \frac{1}{kT}[/tex]

    I know this is correct since when i replace [tex]E[/tex] with [tex]E_0[/tex] is comes out to 0 which is correct since [tex]E_0[/tex] is a peak, plus it was a problem hint.

    For the second derivative i get [tex]\frac{3E_c^\frac{1}{2}}{4E^\frac{5}{2}}[/tex]

    Im pretty sure my derivative is correct, but when i replace [tex]E[/tex] with [tex]E_0 = [\frac{1}{4}E_c(kT)^2]^\frac{1}{3}[/tex] and try to simply i get this...

    [tex]\frac{3}{4} \* \frac{4^\frac{5}{6}}{[E_c(kT)^5]^\frac{1}{3}}[/tex]

    Can this be simplified any further other than the obvious numerical evaulation for 4 raised to 5/6?
  2. jcsd
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