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| age | ||
| visits | member for | 1 year, 8 months |
| seen | Apr 24 at 10:58 | |
| stats | profile views | 121 |
Undergraduate theoretical physics student.
Interests:
Physics: all areas of fundamental physics.
Math: all areas related to the above.
(specifically, the interface of theoretical physics and mathematics)
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Oct 17 |
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Resource request: history of and interconnections between math and physics @WillieWong: I want to see if - and how - it is possible, since on some <underline>very rare</underline> situations it is the most effectual use of the tool. (Pun very much intended! :) ) |
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Oct 17 |
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Resource request: history of and interconnections between math and physics @Willie Wong: [Note on formatting]: I would prefer to underline the word "seriously" in the question- (both because it more precisely & accurately conveys the 'sense' of my question, and because I want to know how to do it on SE, (I perused Meta.Stackoverflow to find out how - but couldn't make out head or tails.)) -cheers |
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Oct 16 |
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Resource request: history of and interconnections between math and physics I have this book. - Looking specifically for a book / paper / online video / (and/or other) educational resource on this specific topic (math-physics link) and their historical origin, and interrelations. |
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Nov 21 |
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Show that the function $h(x) = \int^x_0 g(t) \, dt$ is $C^2$ but not $C^3$ at $x = 0$ Yes; now I see. Thank you. |
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Nov 21 |
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Show that the function $h(x) = \int^x_0 g(t) \, dt$ is $C^2$ but not $C^3$ at $x = 0$ Function $f$ is $C^1$ but not $C^0$ at $x = 0$ as $f'(x) = \frac{1}{3}x^{-2/3}$ for $x \neq 0$, and 'undefined' for $x = 0$. |
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Oct 5 |
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Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ So, -to clarify this whole debate-, in the space of real numbers I cannot write an expression like $S$, and claim that I can recast it [as $1 + x^2\,S$] unless the expression is convergent, (or defined within a specified radius of convergence)? |
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Oct 5 |
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Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ ... of convergence.] |
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Oct 5 |
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Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ @t.b.: Within the radius of convergence the series is identified with / [or possibly mathematically equivalent to(?)] $\frac{1}{1-x^2}$. My question is: [preliminarily, and from receiving various responses: can we recast it the way we have if $S$ is not convergent?, and] (if we can recast it regardless or not the series is convergent or divergent [which I now believe the answer is a 'no' to]) how is it mathematically justified? .. [Part of my question I think has already been answered since I don't believe you can recast it as above if the series isn't convergent [or outside its domain... |
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Oct 5 |
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Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ If I understand correctly, the series in the question can be recast only if it is convergent; otherwise it can't. [via @Jyrki | ... so, how then are "formal power series" related to the "usual" series above? ... |
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Oct 5 |
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Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ @Jyrki: Ok. Thanks for the response. -- May I courteously request a formal answer to the above question? (Use whatever mathematics you like, as long as your answer is completely rigorous.) - cheers |
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Oct 5 |
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Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ Hi @Mark, thanks for the response. | May I know the formal power series version of your response? (I think that's what I'm searching for.) |
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Oct 5 |
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Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ @Jyrki: This would only apply if the series converged. | However, for this series, the recasting is valid regardless of convergence or divergence of the series, (i.e. the issue revolves around the specific infinite number of terms for the series - which allows recasting). ... My question is: how (in a rigorous mathematical sense) is this mathematically meaningful / (possible)? |
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Oct 5 |
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Mathematical reason for the validity of the equation: $S = 1 + x^2 \, S$ Well, I'm basically trying to understand why we can do this recasting. Normally - (I'm assuming) - we cannot recast a finite series. But the above recasting is allowed since the terms go to infinity. ... What is the exact mathematical (i.e. rigorous) basis which allows this manipulation? |
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Oct 5 |
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General question on relation between infinite series and complex numbers Hi @anon, your answer seems well-thought-out and valuable to someone with complex analysis under their belt, but I'm afraid currently it's somewhat over my head. (I'm still giving it +1 since I will review it once I gain sufficient knowledge / background in the subject.) - cheers |
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Sep 30 |
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General question on relation between infinite series and complex numbers Can you provide a summary which is comprehensible to a 2nd-3rd year undergrad with modest knowledge of real analysis, and almost none of complex analysis (apart from complex numbers and some basic identities)? |
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Sep 29 |
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Show that the geometric series $a + ar +ar^2 + \cdots + ar^{n-1} + \cdots$ converges if and only if $|r| < 1$ [Plus, as per FredrikMeyer's comment ["You want to show that $r^n \rightarrow 0$ as $n \rightarrow \infty$ provided |r|<1. So let ϵ > 0 be given. Then $r^n \lt ϵ$ is equivalent to n ≥ ln ϵ/ ln r. Such an n exists."] I'm not sure why you would claim a "formal" $\epsilon$-$\delta$ proof of the statement would not be applicable.] |
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Sep 29 |
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Show that the geometric series $a + ar +ar^2 + \cdots + ar^{n-1} + \cdots$ converges if and only if $|r| < 1$ @Arturo: The issue of the epsilon-delta proof comes from the following: [comment # 5] "first I derive the formula for the finite series as $S_n = a\frac{(r^n−1)}{r−1}$; then, I claim that, if |r| < 1, convergence of the series is equivalent to claiming that: $\lim_{n \rightarrow \infty} |S_{n+1}−S_n| = \lim_{n \rightarrow \infty} |ar^n| = 0$. But I'm wondering how to prove this." ... I'm wondering if this is a valid approach to the initial problem [the first part of]? |
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Sep 29 |
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Show that the geometric series $a + ar +ar^2 + \cdots + ar^{n-1} + \cdots$ converges if and only if $|r| < 1$ Srivatsan, +1. Thanks. Appreciate greatly. |
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Sep 29 |
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Show that the geometric series $a + ar +ar^2 + \cdots + ar^{n-1} + \cdots$ converges if and only if $|r| < 1$ @Arturo: Can you provide your perspective on how best to prove the above [in question body]? I think my general approach is in the right direction, but maybe you can suggest a better method / methods. -cheers |
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Sep 29 |
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Show that the geometric series $a + ar +ar^2 + \cdots + ar^{n-1} + \cdots$ converges if and only if $|r| < 1$ @Fredrik: I've never done a formal $\epsilon - \delta$ proof before. Can you provide a detailed version for a complete newbie to fully understand. ... (Would appreciate greatly.) :) |
