Question about the maclaurin serie and laplace transform

[Lorens]

Question about the maclaurin serie and the laplace transform.

For maclaurin serie i wonder, the function used for teh maclaurin development must be derivativable to infinity?

What is the difference between the fouri transform and the laplace transform? As i understood it, it's just the same except that teh la place transform can converge for more functions.

 

[HallsofIvy]

Yes, to form the MacLaurin series, or more generally Taylor series, of a function the function must be infinitely differentiable. Of course, that doesn't guarantee that the series will converge, or that, it it does converge, it will converge to the value of the function!

In a certain sense (a very complicated sense!) the Fourier series is a complex version of the Laplace series.

 

[tacman]

The Maclaurin series is a representation of a function as an infinite sum of terms involving the function's derivatives evaluated at a specific point (usually 0). In order for the Maclaurin series to exist, the function must be infinitely differentiable at the point of expansion. This means that the function must have derivatives of all orders at that point.

The Laplace transform, on the other hand, is a mathematical tool used to transform a function of time into a function of complex frequency. It is often used in engineering and physics to solve differential equations. The main difference between the Laplace transform and the Fourier transform is that the Laplace transform can handle a wider range of functions, including those that are not periodic.

In summary, the Maclaurin series is a representation of a function using its derivatives, while the Laplace transform is a mathematical tool used to transform functions. Both have different applications and uses in mathematics and science.