- #1
Black_Hole_93
- 2
- 0
Hi,
I have a probably very stupid question:
Suppose that there is an expression of the form $$\frac{d}{da}ln(f(ax))$$ with domain in the positive reals and real parameter a. Now subtract a fraction ##\alpha## of f(ax) in an interval within the interval ##[ x_1, x_2 ]##, i.e.
$$f(ax) \rightarrow f(ax) \left[\theta(a(x-x_1)) \theta(a(x_2-x))+ (1-\alpha) \theta(a(x_1-x)) \theta(a(x-x_2)) \right]$$
Now in the case that ##a>0## one can use ##\theta(ax) = \theta(a)\theta(x)## and after doing a case analysis for ##x \in [x_1,x_2]## and ##x\not\in [x_1,x_2]## above expression becomes
$$\frac{d}{da} ln(f(ax)) \rightarrow \frac{d}{da}ln(f(ax)) + \frac{d}{da} ln(\theta(a)) + \frac{d}{da} ln(\theta(a)) \theta(x-x_1) \theta(x_2-x) $$
I have two questions:
i) Are the steps above legitimate, as one is dealing with functionals?
It does not seem right to me that the result does not depend on ##\alpha##...
ii) How can one evaluate ##\frac{d}{da} ln(\theta(a))## ? Is the typical chain rule applicable there (which would yield 0 for a > 0)?
Thank you and best greetings :)
I have a probably very stupid question:
Suppose that there is an expression of the form $$\frac{d}{da}ln(f(ax))$$ with domain in the positive reals and real parameter a. Now subtract a fraction ##\alpha## of f(ax) in an interval within the interval ##[ x_1, x_2 ]##, i.e.
$$f(ax) \rightarrow f(ax) \left[\theta(a(x-x_1)) \theta(a(x_2-x))+ (1-\alpha) \theta(a(x_1-x)) \theta(a(x-x_2)) \right]$$
Now in the case that ##a>0## one can use ##\theta(ax) = \theta(a)\theta(x)## and after doing a case analysis for ##x \in [x_1,x_2]## and ##x\not\in [x_1,x_2]## above expression becomes
$$\frac{d}{da} ln(f(ax)) \rightarrow \frac{d}{da}ln(f(ax)) + \frac{d}{da} ln(\theta(a)) + \frac{d}{da} ln(\theta(a)) \theta(x-x_1) \theta(x_2-x) $$
I have two questions:
i) Are the steps above legitimate, as one is dealing with functionals?
It does not seem right to me that the result does not depend on ##\alpha##...
ii) How can one evaluate ##\frac{d}{da} ln(\theta(a))## ? Is the typical chain rule applicable there (which would yield 0 for a > 0)?
Thank you and best greetings :)
Last edited: