Hi, thanks for the detailed notes on the book. It is truly helpful!
I have a question in the derivation of Eq. 2.53. You introduced a function $T(s) = S(1-s)$ and then used Taylor expansion of T around 0. It seems to me that this implicitly assumes $S'(1)$ exists and takes some non-zero value.
However, if we check the derivative of $S(x)$ using its functional form 2.58, we can show that
a. $S'(1) = -1$ when $m = 1$.
b. $S'(1) = 0$ when $0 < m < 1$.
c. $S'(1)$ does not exist (is infinite) when $m > 1$
This seems to render the Taylor expansion invalid for most of $m$ values. Can you help me figure out if there is anything wrong in my reasoning?
Thanks!
Hi, thanks for the detailed notes on the book. It is truly helpful!
I have a question in the derivation of Eq. 2.53. You introduced a function$T(s) = S(1-s)$ and then used Taylor expansion of T around 0. It seems to me that this implicitly assumes $S'(1)$ exists and takes some non-zero value.
However, if we check the derivative of$S(x)$ using its functional form 2.58, we can show that
a.$S'(1) = -1$ when $m = 1$ .$S'(1) = 0$ when $0 < m < 1$ .$S'(1)$ does not exist (is infinite) when $m > 1$
b.
c.
This seems to render the Taylor expansion invalid for most of$m$ values. Can you help me figure out if there is anything wrong in my reasoning?
Thanks!