Linked Questions

Both a power series and a Fourier series can be used to approximate a function. How do you use these differently?

I've understood Taylor series as being the representation of a "transcendental" function, using power functions with coefficents represented by appropriate derivatives. (Or maybe it is the MacLauren ...

Taylor series expansion of function, $f$, is a vector in the vector space with basis: $\{(x-a)^0, (x-a)^1, (x-a)^3, \ldots, (x-a)^n, \ldots\}$. This vector space has a countably infinite dimension. ...

Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a ...

I'm taking my first real analysis course and I'm trying to get a better feel about analytic functions. My understanding is that an analytic function is one which can be written as a power series. My ...

Suppose I have a function $f$ defined on the unit circle. Under what conditions can I define a new function $g$ defined on a subset of the complex plane containing the unit disk such that $f = g$ on ...

Let $f : \mathbb C\rightarrow \mathbb C$ be an analytic function : $f(z)= \sum a_n z^n$ It holds that $$a_n z^n= \frac{1}{2 \pi}\int_{-\pi}^{\pi}f(ze^{it})e^{-int}dt$$ and $$\sum_{n=0}^{\infty}|...

Assume we have two fourier series $$f(x) = A_0 + \sum_{i=1}^{N}\alpha_i\cos(2ix)$$ $$g(x) = A_0 - \sum_{i=1}^{N}\alpha_i\cos(2ix)$$ Obviously there are many realtions between those functions (the same ...

I am stuck on how to calculate the value of the following sum: $\sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$ I am aware that you need to find the corresponding function whose Fourier series is represented ...

I am not a Mathematician - am just a software developer though I did some "Math" back in the day as part of my undergrad studies millions of years ago. Recently I had to revisit Fourier analysis of ...

What kind of information is available in a Fourier series expansion of a real analytic function that is not (readily) available in a power series? When would one know to work with one over the other?

Assume I have some function $t\to f(t)$ that is well behaved enough to have a Fourier Transform $w\to \mathcal F\{f(t)\}(w)$ as well as a power series expansion $$p(t) = \sum_{k\in \mathbb Z^+} c_kt^k$...

I was just trying to make something out from $$f(z)=u(z)+iv(z)$$ so what I did is the following: $$f(z)=u(z)+iv(z)=\sum_n c_nz^n=\sum_nc_nr^ne^{in\theta}=\sum_nc_nr^n(\cos(n\theta)+i\sin(n\theta))$$ ...