Why Do Common Functions Use Asymptotic Expansion Series?

In summary, the speaker is looking for examples of Asymptotic Expansion Series for their Mathematical Technique mini project. They have found some examples on the internet, such as Gamma Function, Error Function, Riemann Zeta Function, Exponential Integral, and Multiple Integral. However, their lecturer has asked them to explain why these examples use Asymptotic Expansion Series instead of other expansion series. The speaker is hoping for a more detailed explanation.
  • #1
kari_convention
3
0
Hello and good day. I am searching for Asymptotic Series for my Mathematical Technique mini project. I'm doing Asymptotic Expansion Series and the examples of it. I search on the internet and I found few examples of it such as Gamma Function, Error Function, Riemann Zeta Function, Exponential Integral and Multiple Integral.
But my lecturer have giving me some twist, he asked me to find why all the examples that I found, choose Asymptotic Expansion Series instead of other expension series.

Thanks.
 
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  • #2
I don't see why e.g. the Gamma function is an asymptotic expansion, so a more detailed explanation would have been helpful. However, the link also discusses your question in section 2.2.2
 

FAQ: Why Do Common Functions Use Asymptotic Expansion Series?

What is an asymptotic expansion series?

An asymptotic expansion series is a mathematical technique used to approximate a complicated function by expressing it as a sum of simpler functions. This allows for easier analysis and calculation of the original function.

When is an asymptotic expansion series useful?

An asymptotic expansion series is useful when dealing with functions that are difficult to evaluate or integrate directly. It is also commonly used in physics and engineering to approximate solutions to complex problems.

How is an asymptotic expansion series calculated?

An asymptotic expansion series is typically calculated by taking the Taylor series of a function and then using techniques such as the Euler-Maclaurin formula to simplify the series and express it in terms of simpler functions.

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

A finite asymptotic expansion series has a fixed number of terms, while an infinite asymptotic expansion series has an infinite number of terms. In practical applications, a finite series is often sufficient for obtaining a good approximation of a function.

Are there any limitations to using asymptotic expansion series?

While asymptotic expansion series can be a powerful tool for approximating functions, there are some limitations. They may not always converge, and the accuracy of the approximation may decrease as the number of terms in the series increases. Additionally, they may not be suitable for functions that have singularities or discontinuities.

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