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one of the exercises is to calculate the taylor expansion at x=0 and degree 4 for some function. For example:

$$\int_{0}^{x} e^{-t^{2}} dt$$

I actually have no clue how to get started. I know how to approximate a function with taylor theorem but the integral sign confuses me. Can someone maybe briefly explain or point to a good resource for this? Thanks in advance!

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3 Answers 3

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Hint: define $f(x) = \int_0^x e^{-t^2} dt$. What are $f'(x)$, $f''(x)$, etc?

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  • $\begingroup$ I think that the derivative is $$ f(x) = e^{-x^2}$$? $\endgroup$
    – John
    Commented Aug 11, 2015 at 9:02
  • $\begingroup$ Nearly correct [it should be $f'(x)$], so now compute the higher derivatives and plug all of the derivatives in Taylor's formula. $\endgroup$
    – wltrup
    Commented Aug 11, 2015 at 9:04
  • $\begingroup$ OKe thanks a lot! $\endgroup$
    – John
    Commented Aug 11, 2015 at 9:04
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Look up the Taylor Series expansion for $e^{x}$ but instead of $x$ you have $-t^{2}$. It is a power series.

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Let $$f(x) = \int_{0}^{x} e^{-t^{2}} dt$$ and use Leibniz rule to obtain \begin{align} f^{'}(x) &= e^{- x^{2}} \\ f^{''}(x) &= - 2 x \, e^{- x^{2}} = (-1)^{1} \, H_{1}(x) \, e^{- x^{2}} \\ f^{'''}(x) &= (4 x^{2} - 2) \, e^{- x^{2}} = (-1)^{2} \, H_{2}(x) \, e^{- x^{2}}. \end{align} The general formula is \begin{align} f^{(n+1)}(x) = (-1)^{n} \, H_{n}(x) \, e^{- x^{2}} \hspace{5mm} n \geq 0 \end{align} where $H_{n}(x)$ are the Hermite polynomials. Now, $f(0) = 0$, $f^{(2n+2)}(0) = 0$, $f^{(2n+1)}(0) = (-1)^{n} ((2n)!/n!)$ for which, by use of Taylor's theorem, \begin{align} f(x) &= \sum_{m=0}^{\infty} a_{m} \, \frac{x^{m}}{m!} = \sum_{m=0}^{\infty} D^{m}(f(x))|_{x=0} \, \frac{x^{m}}{m!} \\ &= f^{(1)}(0) \, \frac{x^{1}}{1!} + f^{(3)}(0) \, \frac{x^{3}}{3!} + \cdots \\ &= \sum_{m=0}^{\infty} \frac{(-1)^{m} \, x^{2m+1}}{m! \, (2m+1)}. \end{align}

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