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I'm reading M. Audin and M. Damian's book $\textit{Morse Theory and Floer Homology}$ and i'm having an issue on a proposition that is "Morse Lemma" here page 13. This proof about Morse Lemma is a little bit different compared to the standard proof as in Milnor's and most books. So i can't find out wheather this is a typo or not by looking at another book.

As i understand, it's only about the one variable calculus part that i got confused (what a poor student). The argument is as follows

In page 13 we have an expression $$ f(x) = f(0) + \frac{1}{2}f''(0) \, x^2 + \varepsilon(x) \, x^2. $$ Without much explanation in the book i assume that, the expression above is the Taylor's expansion with the remainder terms wrote as integral representation, where the linear term $f'(0)\,x$ is vanish since we're assuming $0$ is a critical point. So i assuming that above expression is appear as follows \begin{align} f(x) &= f(0) + \frac{1}{2}f''(0) \, x^2 + \frac{1}{2} \int_0^{x} f'''(t)(x-t)^2 dt\\ &= f(0) + \frac{1}{2}f''(0) \, x^2 + x^2 \Big(\frac{1}{2x^2} \int_0^{x} f'''(t)(x-t)^2 dt \Big) \\ &= f(0) + \frac{1}{2}f''(0) \, x^2 + x^2 \varepsilon(x). \end{align} So $\varepsilon(x)$ must be $\frac{1}{2x^2} \int_0^{x} f'''(t)(x-t)^2 dt $. But then why in a sentence after that expression in the book, it says that $$ \varepsilon(x) = \frac{1}{2} \int_0^{x} f'''(t)(x-t)^2 dt \quad ? $$ I'm really sorry for wasting your time if this is a silly question and it's just my ignorance. Would anyone help me with this one ? Thank you.

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    $\begingroup$ Looks like a typo. Is that formula used later in the book? $\endgroup$ – robjohn Jan 31 at 18:47
  • $\begingroup$ @robjohn Maybe. I don't think it is used anywhere. Only in that proposition. $\endgroup$ – Si Kucing Jan 31 at 19:11

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