# Help with asymptotic expansions, please

• zeba255
In summary, we integrated the function by parts and obtained an asymptotic expansion for the cosine integral function, Ci(x), which is (sinx/x)-(cosx/x)-(1.2sinx/x)+(1.2.3cosx/x^4)+R(x). This expansion is valid for large x, specifically when x>1, due to the behavior of the integrand being too large for small x values.
zeba255
Help with asymptotic expansions, please!

## Homework Statement

What is the behavior of asymptotic expansion of cosine integral function? Under what conditions should it be valid approximation to Ci(x)

Ci(x)=- ∫∞x (cost/t)dt {in words} (Integral from x to infinity)(cost/t)dt

## Homework Equations

-Intefral x to infinty(e^it/t)dt

## The Attempt at a Solution

I have integrated the functions by part, i got the right answer with the right asymptotic expansion i.e. Ci(x) = (sinx/x)-(cosx/x)-(1.2sinx/x)+(1.2.3cosx/x^4)+R

(x) where R(x) is the remainder. This asymptotic expansion is valid for large x, i.e. x>1. This is because for small x, the integral would not be convergent as the rate of change of the integrand is too large.

## What is an asymptotic expansion?

An asymptotic expansion is a mathematical technique used to approximate a function or expression for large or small values of the independent variable. It involves representing the function as a series of simpler terms, such as polynomials, and truncating the series to a desired degree of accuracy.

## Why are asymptotic expansions useful?

Asymptotic expansions are useful because they allow us to simplify complex functions and make them more manageable for analysis. They also provide approximations that can be used in situations where the exact solution is difficult or impossible to obtain.

## What is the difference between a finite and infinite asymptotic expansion?

A finite asymptotic expansion involves truncating the series after a certain number of terms, while an infinite asymptotic expansion includes an infinite number of terms. Finite expansions are typically easier to calculate, but may be less accurate than infinite expansions.

## How do I determine the accuracy of an asymptotic expansion?

The accuracy of an asymptotic expansion can be determined by comparing it to the exact solution or by using error estimates. These estimates can be obtained by analyzing the convergence properties of the series and identifying the dominant terms.

## What are some common applications of asymptotic expansions?

Asymptotic expansions are commonly used in physics, engineering, and other sciences to approximate solutions to differential equations, integrals, and other mathematical problems. They are also used in statistical analysis, numerical methods, and in the study of complex systems.

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