Approximating the Cosine Integral?

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    Cosine Integral
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Discussion Overview

The discussion revolves around the approximation of the cosine integral function, Ci(x), particularly focusing on the search for a semi-quick method to approximate it using asymptotic series or other means. Participants explore the definitions and implications of asymptotic expansions, as well as the limitations of expressing Ci(x) through elementary functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks a method to approximate Ci(x) for large values, expressing uncertainty about the definition of asymptotic expansion.
  • Another participant asserts that Ci(x) cannot be expressed through elementary functions and suggests looking for asymptotic behavior and series expansions.
  • A participant questions the existence of an asymptotic series for Ci(x), prompting a discussion about the conditions under which such series can be formed.
  • Clarifications are made regarding specific formulas referenced in external materials, with one participant indicating that a proposed series does not apply to Ci(x) but rather to another function.
  • There is a request for alternative methods to calculate Ci(x) without relying on tables, highlighting a desire for practical computation techniques.
  • A suggestion is made to check for the power series of Ci(x), with a caution about the convergence radius affecting its applicability for arbitrary arguments.

Areas of Agreement / Disagreement

Participants express differing views on the existence of an asymptotic series for Ci(x), with some asserting it does not exist due to the nature of sine and cosine functions, while others seek clarification on the potential for approximations. The discussion remains unresolved regarding the availability of a straightforward method for calculating Ci(x).

Contextual Notes

There are limitations regarding the definitions and conditions under which asymptotic series can be applied, as well as the convergence of power series for Ci(x). Specific mathematical references are mentioned, but their applicability is debated.

Manchot
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Does anyone know of a semi-quick way of approximating Ci(x)? I tried to find an asymptotic expansion for it, but had little luck. Truth be told, I'm not even sure exactly what the definition of asymptotic expansion is. I discovered it while learning about ways of approximating harmonic numbers and Stirling's approximation, and only know that it works for large values. Basically, I'd like it to be valid for high numbers, and this is why I'm trying to use this type of series. Can anyone offer any insight? Thanks a bunch!

Edit: By "semi-quick" I meant in terms of simple, nice, elementary functions. I don't care if there are a finite number of terms or not, because to me, more terms = more accuracy.
 
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As u might have guessed,u cannot express Ci(x) through elementary functions.Asymptotic behavior,series expasion and a lot more can be found here and the next pages.

Daniel.
 
Yeah, I knew that it couldn't be expressed in a finite number of elementary functions. But what prevents there from being an asymptotic series which approximates it?
 
Manchot said:
But what prevents there from being an asymptotic series which approximates it?

What do you mean??For large values of the argument,you can approximate the function with its asymptotic series and that's it...

Daniel.
 
Maybe I'm being unclear about what I mean. I'm trying to find what its asymptotic series is. Now, I see on the pages you linked to (page 232) something labelled "Asymptotic series," but it doesn't say what they refer to. Should I assume that 1/z (1-2!/x²+4!/x^4...) is Ci(x)'s asymptotic series?
 
Manchot said:
Maybe I'm being unclear about what I mean. I'm trying to find what its asymptotic series is. Now, I see on the pages you linked to (page 232) something labelled "Asymptotic series," but it doesn't say what they refer to. Should I assume that 1/z (1-2!/x²+4!/x^4...) is Ci(x)'s asymptotic series?

No.The formula u've indicated (no.5.2.34,page 233) is not for Ci(x) (else it would have been written at the left of the equality sign),but for another function,namely f(z).U needto use formulas 5.35,page 233 and 5.2.9,page 232 to get your desired result,which DOES NOT EXIST,since "sine" and "cosine" do not have well defined asymptotic behavior and hence don't have asymptotic series...

Daniel.
 
Well, then, if there is no asymptotic expansion, does anyone know a relatively easy way to calculate it? (without using tables?) Any help is greatly appreciated.
 
Check the same site for its power series.And pay attention to its convergence radius,which means you cannot compute the function using its power series for arbitrary arguments.

Daniel.
 

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