Asymptotic expansion

In mathematics, an asymptotic expansion is a special kind of series that helps us understand and guess the value of functions. It is also called an asymptotic series or Poincaré expansion, named after Henri Poincaré. When we stop an asymptotic series after just a few terms, it can give a good guess about the value of a more complicated function, especially when the input to the function gets very large or very small.

The idea of asymptotic expansions was developed by Poincaré and Stieltjes in 1886. These series are often written as power series, using powers of numbers that can be positive or negative. There are tools to create these expansions, like the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Sometimes, doing integration by parts many times leads directly to an asymptotic expansion.

Even though an asymptotic series may not add up perfectly, stopping it at the right place can give a useful guess. The error in this guess can be made very small. There are even more advanced ways to reduce the error further, using methods like Borel resummation.

Formal definition

An asymptotic expansion is a special way to guess what a function is doing. It uses a list of simpler functions to get closer to the real answer.

When the numbers you put into the function are very big or very small, an asymptotic expansion helps. It adds up the simpler parts in a certain way. Each new part grows slower than the one before it. By doing this, you can get a very good guess of what the function’s value is. This idea is used in many parts of math and science.

Examples

• Gamma function (Stirling's approximation): This is a way to guess what a special math function called Gamma is for large numbers.
• Exponential integral: Another example of using math to guess a special function when numbers get really big.
• Logarithmic integral: This shows how we can guess a math function related to logarithms.
• Riemann zeta function: This example uses a special series to guess the zeta function, a famous math function.
• Error function: This is a way to guess another important math function.

Each example shows how math can help us make good guesses about tricky functions when numbers get really large.

Worked example

Asymptotic expansions happen when we use a special kind of math series in places where it doesn’t usually work. For example, we might start with a simple series that only works for numbers smaller than one. By changing and adding parts to this series, we can create a new expression that helps us estimate values for bigger numbers.

Using a method called integration by parts, we can find a formula that gives us a closer look at these estimates. The formula shows how small pieces of the series change, helping us understand how accurate our estimates are. This way, we can get better guesses for certain math problems by using just a few parts of the series instead of all of them.

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Properties

Sometimes, there is only one way to find the details of a guess for a math idea. This means the numbers you use to make the guess are picked in just one way.

A math idea can sometimes be guessed in more than one way, using different rules.

An idea used to guess can sometimes fit more than one real thing.