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Irid
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I've came up to a problem, where I would like to prove that a differentiable function f(x) can be approximated by
[tex]f(x) = f(x_0) \left(\frac{x}{x_0}\right)^{\alpha}[/tex]
where
[tex]\alpha = \frac{d \ln f(x)}{d \ln x} \Big |_{x=x_0}[/tex]
But I'm not sure this is true. The problem and solution can be found at:
http://crydee.sai.msu.su/~konon/Book/Book.html
see Problem 8.2, the book is in Russian. If you can't find it, the problem and solution are respectively at pages
http://crydee.sai.msu.su/~konon/Book/ch3L/node12.html
http://crydee.sai.msu.su/~konon/Book/ch4L/node9.html
The idea depicted in that book is to use a Taylor expansion for a function [tex]\ln f(x)[/tex], but use a variable [tex]\ln x[/tex] instead of [tex]x[/tex], as if the function really was [tex]\ln [f(\ln x)][/tex], but that's not shown. Anyway, say I Taylor expand this function:
[tex]\ln [f(\ln x)] = \ln [f(\ln x_0)] + \frac{d \ln [f(\ln x)]}{d \ln x} \Big |_{x=x_0} (\ln x-\ln x_0) + \cdots = \ln[f(\ln x_0)] + \ln \left(\frac{x}{x_0}\right)^{\alpha} + \cdots[/tex]
OK, now I put those two terms together and obtain
[tex]f(\ln x) = f(\ln x_0) \left(\frac{x}{x_0}\right)^{\alpha}[/tex]
Looks like it, but if we let [tex]\ln x = z[/tex], then
[tex]f(z) = f(z_0) e^{\alpha(z-z_0)} \approx f(z_0) [1 + \alpha(z-z_0)][/tex]
after approximating exponent for small [tex]\Delta z[/tex] and this is just Taylor series, i.e. it gives nothing new and no expected new awesome cool approximation. Am I missing something, or the textbook is rubbish?
Ironically, the author complains that no textbook gives this approximation, but it is used everywhere in astrophysics.
[tex]f(x) = f(x_0) \left(\frac{x}{x_0}\right)^{\alpha}[/tex]
where
[tex]\alpha = \frac{d \ln f(x)}{d \ln x} \Big |_{x=x_0}[/tex]
But I'm not sure this is true. The problem and solution can be found at:
http://crydee.sai.msu.su/~konon/Book/Book.html
see Problem 8.2, the book is in Russian. If you can't find it, the problem and solution are respectively at pages
http://crydee.sai.msu.su/~konon/Book/ch3L/node12.html
http://crydee.sai.msu.su/~konon/Book/ch4L/node9.html
The idea depicted in that book is to use a Taylor expansion for a function [tex]\ln f(x)[/tex], but use a variable [tex]\ln x[/tex] instead of [tex]x[/tex], as if the function really was [tex]\ln [f(\ln x)][/tex], but that's not shown. Anyway, say I Taylor expand this function:
[tex]\ln [f(\ln x)] = \ln [f(\ln x_0)] + \frac{d \ln [f(\ln x)]}{d \ln x} \Big |_{x=x_0} (\ln x-\ln x_0) + \cdots = \ln[f(\ln x_0)] + \ln \left(\frac{x}{x_0}\right)^{\alpha} + \cdots[/tex]
OK, now I put those two terms together and obtain
[tex]f(\ln x) = f(\ln x_0) \left(\frac{x}{x_0}\right)^{\alpha}[/tex]
Looks like it, but if we let [tex]\ln x = z[/tex], then
[tex]f(z) = f(z_0) e^{\alpha(z-z_0)} \approx f(z_0) [1 + \alpha(z-z_0)][/tex]
after approximating exponent for small [tex]\Delta z[/tex] and this is just Taylor series, i.e. it gives nothing new and no expected new awesome cool approximation. Am I missing something, or the textbook is rubbish?
Ironically, the author complains that no textbook gives this approximation, but it is used everywhere in astrophysics.
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