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-   -   Approximating the Cosine Integral? (http://www.physicsforums.com/showthread.php?t=59617)

Manchot Jan12-05 10:21 PM

Approximating the Cosine Integral?
 
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.

dextercioby Jan12-05 10:33 PM

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.

Manchot Jan12-05 10:47 PM

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?

dextercioby Jan12-05 10:51 PM

Quote:

Quote by Manchot
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.

Manchot Jan12-05 10:55 PM

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!/x+4!/x^4...) is Ci(x)'s asymptotic series?

dextercioby Jan13-05 01:47 AM

Quote:

Quote by Manchot
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!/x+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.

Manchot Jan13-05 10:18 PM

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.

dextercioby Jan13-05 10:27 PM

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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