Asymptotic expansions of the sine function

Thread starter Robin04
Start date Jul 14, 2019
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Function Sine

In summary, an asymptotic expansion of the sine function is a mathematical series that represents its behavior as the input approaches infinity or zero. It is derived using the Taylor series expansion and can be truncated to a desired level of accuracy. Some common examples include the Maclaurin, Fourier, and Laplace series. These expansions are significant as they allow for quick and accurate approximations of the sine function and can be applied in various contexts. However, there are limitations as they are only accurate for inputs close to the point of expansion and may require additional terms for desired accuracy.

Jul 14, 2019

Robin04

Homework Statement: ##\sin{(a + b)}\sim f_1(a)+f_2(b)##

Relevant Equations: -

There are no restrictions for ##a,b,f_1,f_2##. One solution is the first order Taylor series expansion of course with ##f_1(a)=a,f_2(b)=b##, but are there any other solutions? I tried the Bhaskara formula but I couldn't express it in this form.

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Jul 14, 2019

pasmith

Science Advisor

Homework Helper

Asymptotic in what limit?

Jul 14, 2019

Robin04

pasmith said:

Asymptotic in what limit?

Depends on the expansion, it's not specified at this point yet.

1. What is an asymptotic expansion of the sine function?

An asymptotic expansion of the sine function is a mathematical representation of the behavior of the sine function as the input approaches infinity or zero. It involves expressing the function as a sum of terms, with each term becoming increasingly small as the input grows or shrinks.

2. How is an asymptotic expansion of the sine function useful?

An asymptotic expansion of the sine function is useful in approximating the value of the sine function for extremely large or small inputs, as it provides a simpler expression that is easier to evaluate. It is also used in various fields of science and engineering, such as in signal processing and differential equations.

3. Can the sine function be approximated accurately with an asymptotic expansion?

Yes, the sine function can be approximated accurately with an asymptotic expansion for a wide range of inputs. However, the accuracy of the approximation decreases as the input gets closer to the boundaries of the function's domain.

4. Are there different forms of asymptotic expansions for the sine function?

Yes, there are multiple forms of asymptotic expansions for the sine function, such as the Taylor series expansion, the Laurent series expansion, and the Laplace's method expansion. Each form has its own advantages and is suitable for different types of applications.

5. How is the convergence of an asymptotic expansion of the sine function determined?

The convergence of an asymptotic expansion of the sine function is determined by analyzing the behavior of the terms in the expansion as the input approaches infinity or zero. If the terms become increasingly small, the expansion is said to converge. However, if the terms do not approach zero, the expansion does not converge and cannot be used to approximate the sine function.