# Tagged Questions

Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials or sines and cosines are used.

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### Given any sequence $(a_n)_{n \in \mathbf{N}}$ is $\sum_{n \geq 0} a_n e^{2 \pi i n z}$ holomorphic on the upper half plane?

I've seen quite often that people consider some arbitrary sequence $(a_n)_{n \in \mathbf{N}}$ (say of real numbers), and form the sum $\sum_{n \geq 0} a_n e^{2 \pi i n z}$, $z \in \mathbf{H}$. Usually ...
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### Poissions Equation (Laplace)

\begin{align} u''_{xx}&+u_{yy}= x, \quad 0<x<1, \quad 0<y<1,\\ \\ u(x,0)&=u(x,1) = 0, \\ u(0,y)&=u(1,y) = 0,\\ \end{align} Having some problems with Poissons Equation. I'...
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### Fourier Series of a sum of two functions [closed]

Is the Fourier series of a sum of two functions $f,g$ the term by term sum of the Fourier Series?
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### Identity for the sum of products of Sinc functions

The Sinc function is defined as follows: $$\mathrm{sinc}(t) = \begin{cases} \frac{\sin(\pi t)}{ \pi t} & \mathrm{if} \quad t \neq 0, \\ 1 & \mathrm{otherwise.} \end{cases}$$ I want to show the ...
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### For fourier series g(x), prove that the fourier series for the integral G(x) can be found by term-by-term integration of g(x)

I want to prove that if I have a fourier series of the form $g(x) = a_0/2 + {\sum_i}^\infty a_icos(ix) + b_isin(ix)$, the fourier series of G(x) $-x*a_0/2$ can be found by simply integrating g(x) ...
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### Why can we calculate the Fourier series of $x^2$ in any interval $[-l,+l]$?

We know that a function must satisfy Dirichlet's Conditions before it can be expanded in Fourier series. And Dirichlet's Conditions strictly require a function to be periodic in the interval in which ...
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### How to find the inverse Fourier transfmation of $\exp(-sk)/k$.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of $\exp(-sk)$ which is $$\frac{\sqrt2}{\sqrt \pi}\frac{x}{x^2+ s^2}$$ ...
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### Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
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### Combination of even and odd functions

Can someone help me how to show that any function $f(x)$ defined on a symmetrically placed interval can be written as a sum of an even and a odd function? What is the special role played by "...
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### Evaluating infinite series $\sum_{n=0}^{\infty} \frac{1}{a^{2}+(2n+1)^2}$

I have no idea to approach this problem. Mathematica gave the sum to be $$\sum_{n=0}^{\infty} \frac{1}{a^{2}+(2n+1)^2} = \frac{\pi}{4a} \tanh(\frac{a \pi}{2})$$ How can I analyze this?
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### Fourier series on incomplete data [closed]

Given a periodic function that's only partly specified, e.g.: $$f(\theta)=\begin{cases}1 & \text{if } \cos(\theta)>a\\ -1 & \text{if } \cos(\theta)<-a\end{cases}$$ Obviously the ...
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### Given Fourier coefficients of a function , find the function

Given these Fourier coefficients: $$X[k]=\begin{cases} 1 & \text{, k even}\\ 2 & \text{, k odd}\\ \end{cases}$$ I want to find the analytical expression for the function. What i tried was ...
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### Is there a general rule to find period of multiplied functions?

We know that $g(x)$ and $f(x)$ are both periodic and trigonometric functions and we also know its period interval. How can we find the period of the function $f(x)g(x)$?
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### Fourier series in spherical coordinates?

I'm reading an article and he just state: let $f\left(\theta,\varphi\right)$ be of this form $$f\left(\theta,\varphi\right)={\sum}g_{m}\left(\theta\right)e^{im\varphi},$$ I'm on the unitary ...
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### inverse fourier transform of w*e^w

I have the function \begin{align} F^{-1}\{{λe^{-|λ|}}\} \end{align} How can we find the inverse Fourier transform? The correct answer is: \begin{align} \frac{-2ix}{π(1+x^2)^2} \end{align} Can ...
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### What effect does sampling time have on a Fourier Series sum?

What effect would the sampling time of this Fourier sum have on it's accuracy? Is this to do with Nyquists theorem? or am I heading in the wrong direction with this question? Cheers
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### Fourier sine and cosine series: reconstruction is shifted with respect to measured data

I am working in strain analysis. Strain in a mechanical testing machine is captured by strain gages. Signals are like the slim line in the graph below showing strain versus time. The data are of the ...
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### Prerequisites for Fourier Series/Self-Study?

What would be the prerequisites for a typical Introduction to Fourier Series taught at a university. Is an introduction to a rigorous treatment of calculus typically expected or not? So far I've ...
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### How to prove this function is entire?

Given a function which Fourier coefficient decay fast as $k^{-k}$, for example \begin{align} f(x):= \sum_{k=1}^\infty \frac{1}{k^k} \exp(2\pi \, i\,k\,x) . \end{align} How can we prove this function ...
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### Heat and Wave equation - Green's function versus Fourier series?

I am learning how to solve the heat and wave equation in bounded domains in 1 and 2D as well as in $\mathbb{R}$ and $\mathbb{R}^2$. In the latter case I have learned the representation formulas i.e. ...
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### Why does specifying an interval for a function make the function odd or even?

I am currently reading about Fourier series and Orthogonality of functions and Complete Sets of functions. Below are two extracts from the book I'm reading for which I simply do not understand: <...
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### Expanding a function in a Fourier Series

I am having an issue integrating the sin function with the variable of n, any help would be appreciated. I have deduced it to an odd sine series with the following for B_n and I am unsure how to ...
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### Solving Viscous Burgers using spectral method

I am trying to solve the Viscous Burgers equation using the spectral method. $$u_t+uu_x = Du_{xx}$$ where $D$ is a constant (chosen to be zero) and with the initial condition $$u(x,0) = exp(-x/0.2)^2$$...
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### Fourier coefficients of a triangle function

I'm trying to find the Fourier coefficients ($c_n$), of the following function : for $x$ in $[-\pi/2;\pi/2[$ $f(x)=x$ for $x$ in $[\pi/2;3\pi/2[$ $f(x)=\pi-x$ I think its not that hard, but I keep ...
Prove the classical full Fourier series of $f(x)$ converges uniformly to $f(x)$ if $f(x)$ is continuous of period $2\pi$ and its derivative $f'(x)$ is piecewise continuous. How do I go about doing ...
### If $f \in L^1[-\pi, \pi]$ is odd and $f(x + \pi) = f(x)$ for $x \in \mathbb{R}$, then $\beta_{2k - 1} = 0, \forall{k} \in \mathbb{N}$
I'm learning about Fourier analysis and need help with the following problem: Suppose $f \in L^1[-\pi, \pi]$ and $\alpha_n, \beta_n$ are the Fourier coefficients of $f$. Show that if $f$ is odd ...