# Euler expansion of double exponential?

• Clifford
In summary, the conversation discusses using the Euler expansion to estimate a variable that grows as a single exponential and then considering a double exponential fit. The possibility of expanding the matrix exponential is mentioned as a way to handle the double exponential case.
Clifford
Simple question,

I have used the euler expansion to estimate a variable that grows as a single exponential.

In excerpted form:

for (i=1;i<npts; i++)
{
}

where dt is the step size and tau is the 'time constant.'

Now, however, I think that the data would be better fit with a double exponential.

adapt = a(1) * exp(-tau1*X) + a(3) * exp(-tau2*X);

I am unsure how to expand this analogously to the single exponential.
thanks!

Clifford

Assuming that a(1) and a(3) are some incremental values, you can define your system as an autonomous system as, $$\dot{x} = Ax$$ where $$A$$ is a $$3 \times 3$$ matrix and $$x \in \mathbb{R}^3$$, then expand the matrix exponential. And take the first state.

## 1. What is the Euler expansion of double exponential?

The Euler expansion of double exponential is a mathematical formula used to represent a function that has two exponential terms. It is an extension of the well-known Euler's formula, which represents a function with a single exponential term.

## 2. How is the Euler expansion of double exponential derived?

The Euler expansion of double exponential is derived using the Taylor series expansion, which is a way to represent a function as an infinite sum of its derivatives. By applying this method to the double exponential function, we can arrive at the Euler expansion.

## 3. What are the applications of the Euler expansion of double exponential?

The Euler expansion of double exponential has many applications in mathematics, physics, and engineering. It is used to solve differential equations, approximate functions, and analyze complex systems. It is also used in signal processing, control systems, and financial modeling.

## 4. How does the Euler expansion of double exponential compare to other expansions?

The Euler expansion of double exponential is similar to other types of expansions, such as the Binomial expansion and the Fourier series. However, it has the advantage of being able to represent functions with two exponential terms, which makes it more versatile in solving problems in various fields.

## 5. Are there any limitations to the Euler expansion of double exponential?

Like any mathematical formula, the Euler expansion of double exponential has its limitations. It may not be applicable to all functions, and the accuracy of the expansion may decrease as the number of terms increases. It is also important to be aware of the convergence and divergence of the series when using it to approximate a function.

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