Difference between revisions of "The Road to Reality Study Notes"

→‎6.4 The "Eulerian" notion of a function?: Slight grammatical adjustment in ultimate paragraph of section 6p4
(→‎6.4 The "Eulerian" notion of a function?: Slight grammatical adjustment in ultimate paragraph of section 6p4)
Tags: Mobile web edit Mobile edit
Line 269: Line 269:
For the second method, the power series of $$f(x)$$ is introduced, <math>f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + …</math> For this series to exist then it must be $$C^\infty$$-smooth.  We must take and evaluate derivatives $$f(x)$$ to find the coefficients, thus an infinite number of derivatives (positive integers) must exist for the power series to exist.  If we evaluate $$f(x)$$ at the origin, we call this a power series expansion about the origin.  About any other point $$p$$ would be considered a power series expansion about $$p$$. (Maclaurin Series about origin, see also [https://en.wikipedia.org/wiki/Taylor_series Taylor Series] for the general case)
For the second method, the power series of $$f(x)$$ is introduced, <math>f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + …</math> For this series to exist then it must be $$C^\infty$$-smooth.  We must take and evaluate derivatives $$f(x)$$ to find the coefficients, thus an infinite number of derivatives (positive integers) must exist for the power series to exist.  If we evaluate $$f(x)$$ at the origin, we call this a power series expansion about the origin.  About any other point $$p$$ would be considered a power series expansion about $$p$$. (Maclaurin Series about origin, see also [https://en.wikipedia.org/wiki/Taylor_series Taylor Series] for the general case)


The power series is considered analytic if it encompasses the power series about point $$p$$, and if it analytic at all points of its domain, we call it an analytic function, or equivalently a $$C^ω$$-smooth function.  Euler would be pleased with this notion of an analytic function, which is ‘smoothier’ than the set of $$C^\infty$$-smooth functions ($$h(x)$$ from 6p3 is $$C^\infty$$-smooth but not $$C^ω$$-smooth).  
The power series is considered analytic if it encompasses the power series about point $$p$$. If it is analytic at all points of its domain, we call it an analytic function or, equivalently, a $$C^ω$$-smooth function.  Euler would be pleased with this notion of an analytic function, which is ‘smoothier’ than the set of $$C^\infty$$-smooth functions ($$h(x)$$ from 6p3 is $$C^\infty$$-smooth but not $$C^ω$$-smooth).  


* Physics in trying to understand reality by approximating it.
* Physics in trying to understand reality by approximating it.
1

edit