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Rational function that approximates e^x

  1. Aug 2, 2014 #1
    Is there a rational function,not series, that approximates e^x
    ?
    for example (x+1)/(x+3)
     
  2. jcsd
  3. Aug 2, 2014 #2

    HallsofIvy

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    What do you mean by "approximates"? A given function, rational or not, has a specific difference from [itex]e^x[/itex]. We can always find a function, for example by truncating the Taylor's series to a polynomial, that "approximates" [itex]e^x[/itex] to a desired degree of accuracy. The difference is that the series, [itex]\sum_{n=0}^\infty[/itex] does NOT 'approximate' [itex]e^x[/itex], it is exactly equal to it.
     
  4. Aug 2, 2014 #3
  5. Aug 3, 2014 #4

    HallsofIvy

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    That is not at all the question you asked before. However, the standard Taylor's series for the exponential, [tex]\sum_{n=0}^\infty \frac{x^n}{n!}[/tex]
    is a very "rapid" approximation to the exponential.
     
  6. Aug 3, 2014 #5

    pasmith

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    Truncating the series after a fixed number of terms is not a good method of evaluating [itex]e^x[/itex] numerically for large [itex]|x|[/itex], since one loses precision from calculating the large values of [itex]x^n[/itex] and [itex]n![/itex] and the remainder may be large.

    To the OP: Woolfram gives a couple of continued fraction expressions for [itex]e^x[/itex]; truncating these will give you a rational function approximation. But it is better to use a library function for exp if at all possible.
     
  7. Aug 7, 2014 #6

    mathwonk

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  8. Aug 9, 2014 #7

    lurflurf

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    It depends what type of approximation you want. One common one is an approximation such that the function and its derivatives match the approximation up to some degree

    $$
    e^{-x}\sim P(x)/Q(x)\\
    \text{where}\\
    P(x)=\sum_{k=0}^m \frac{(m+n-k)!m!}{(m+n)!k!(m-k)!}x^k
    \\
    Q(x)=\sum_{k=0}^n \frac{(m+n-k)!n!}{(m+n)!k!(n-k)!}x^k
    $$

    matches up to the n+m+1 derivative

    see
    http://mathworld.wolfram.com/PadeApproximant.html
    http://en.wikipedia.org/wiki/Padé_table#An_example_.E2.80.93_the_exponential_function
    http://wwwhome.math.utwente.nl/~vajtam/publications/temp00-pade.pdf
     
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