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Approximation formula proof

  1. May 4, 2014 #1
    Please help me prove the approximation formula below given in my book. This is not homework question.


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  2. jcsd
  3. May 4, 2014 #2


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    You just neglect ##\tau_1## or ##\tau_2##, whatever is smaller. Note that a larger value will lead to a larger exponential as well (for positive t).
  4. May 5, 2014 #3
    Thank you. However, that doesn't solve the problem. What need to be proved is different.
  5. May 5, 2014 #4

    If τ1>>τ2, then [itex]e^{-t\frac{(τ_1-τ_2)}{τ_1τ_2}}<1[/itex].

    From this, it follows that, in the numerator, [itex](τ_2/τ_1)e^{-t\frac{(τ_1-τ_2)}{τ_1τ_2}}<<1[/itex]

    Also, in the denominator, [itex](τ_2/τ_1)<<1[/itex]

    So the term in parenthesis approaches unity.

  6. May 6, 2014 #5


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    Why? It shows that the big is a correct approximation.
  7. May 6, 2014 #6
    I guess the analysis I did in #4 did not work for the OP? (It was just a more fleshed-out version of what mfb was saying).

  8. May 7, 2014 #7
    Thank you all.
    I think you misread the question a bit. The expression on the right hand side of the equation is [tex]e^{-\frac{t}{τ}}[/tex] with [tex]τ = τ_1 + τ_2[/tex].
  9. May 7, 2014 #8
    You're right. But that doesn't matter much. The same general procedure could be used for this.

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