Passing limits through integrals

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Discussion Overview

The discussion revolves around the conditions and theorems relevant to passing limits through integrals, particularly in the context of complex functions and their behavior as a variable approaches zero. The scope includes theoretical aspects of integration and continuity, as well as the application of specific convergence theorems.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks a theorem that justifies the limit transition in the integral as z approaches zero, specifically relating to the function f and its properties.
  • Another participant questions whether the inquiry is about establishing the integral as a primitive of f(z) or about justifying the replacement of τ - z with τ as z approaches zero, suggesting that continuity of F(z) at z = 0 suffices for the latter.
  • A participant expresses a sense of relief upon realizing the simplicity of the situation, indicating a moment of self-reflection on their understanding.
  • Another participant shares their usual practice of looking up conditions for passing limits through integrals, mentioning that continuously differentiable functions are a sufficient condition, albeit overkill.
  • One participant challenges the example provided, pointing out that the function does not satisfy f(b) = 0 and questions the claim of Hölder continuity, citing the unbounded nature of the derivative of the square root function as x approaches zero.
  • A suggestion is made that the Lebesgue Dominated Convergence Theorem could be relevant in this context, highlighting the need for rigorous justification rather than assumptions about continuity.

Areas of Agreement / Disagreement

Participants express differing views on the conditions necessary for passing limits through integrals, with some asserting the simplicity of the conditions while others emphasize the need for rigorous justification. There is no consensus on the validity of the example provided or the claims about continuity.

Contextual Notes

Limitations include potential missing assumptions regarding the properties of f and the specific conditions under which theorems apply. The discussion also reflects a reliance on definitions that may not be universally agreed upon.

rsq_a
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This seems to be an elementary question, but I rarely have to deal with rigor these days, so excuse me if there's a simple answer.

I'm looking for a theorem that will allow me to state,

[tex] F(z) = \int_0^b \frac{f(\tau)}{\tau - z} d\tau \sim \int_0^b \frac{f(\tau)}{\tau} d\tau[/tex]

as [itex]z \to 0[/itex] and [itex]f(\tau): \mathbb{C} \to \mathbb{C}[/itex], but it's real along the path of integration.

I believe [itex]f(\tau)[/tex] satisfies these properties:<br /> <ul> <li data-xf-list-type="ul">[itex]f(0) = 0[/itex] and [itex]f(b) = 0[/itex]</li> <li data-xf-list-type="ul">[itex]f(\tau)[/itex] is Holder continuous along the path of integration</li> </ul><br /> Note that F(z) is well-defined and Holder continuous along the line because of the values of f at the endpoints. <br /> <br /> An example would be, <br /> <br /> [tex] \int_0^1 \frac{\sqrt{\tau}}{\tau - z} d\tau[/tex][/itex]
 
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What is it that you are looking for. Do you want a theorem that will tell you that the integral you wrote down is a primitive of f(z), or do you want a theorem that will tell you that you can replace t - z by t if z goes to zero.
Because the latter simply follows from continuity of F(z) at z = 0.
 
Huh. I guess it's that simple. Sorry, tripped over myself there.
 
Would it make you feel less stupid to know that I usually look such things up, like when I really need to know the conditions of passing limits through integrals? Otherwise I usually write something like: "I don't know the exact theorem, but f is continuously differentiable which is - although overkill - definitely a sufficient condition." :smile:
 
Your example doesn't have f(b)=0...whuuuu?
How is f(x) Holder continuous along the real line segment [0,1]?
The derivative of the square root function is unbounded as x-> 0.

A basic convergence thm I would consider is the Lebesgue Dominated Convergence Thm.

Tough going to just deem something continuous, quote a thm and call it a day.
You have to have gone to a high-tuition college to get away with that and still you
may be wrong.
 

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