Questions tagged [fourier-series]
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A Fourier series is a decomposition of a periodic function as a linear combination of sines and cosines, or complex exponentials.
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Kindly assist with with fourier series square waves
enter image description here Engineering mathematics 2022 fourier-series- 1
Using non-integer inputs for the Fourier Series
I am a beginner in the topic of the Fourier Series. I have been doing some reading and, according to this website (), the following ... fourier-series fourier-transform- 1
How do I compute this limit: $\sum_{k = 1}^{\infty} \frac{\sin k}{k}$? [duplicate]
My task is to compute the limit of $$\sum_{k = 1}^{\infty} \dfrac{\sin k}{k}$$, with Fourier-theory. The only thing I know is that $$\dfrac{\sin k}{k}$$ are the coefficients of the Fourier series $$\... calculus limits convergence-divergence fourier-series- 11
Computing integral for Fourier coefficient.
Aim: Find Fourier coefficients of $\phi(x)=3\cos(2x)$, $x\in[-\pi,\pi]$. I have written $3\cos(2x)=\frac{3}{2}(e^{2ix}+e^{-2ix})$ using the exponential form of cosine. Then used the formula for ... integration complex-numbers fourier-analysis fourier-series- 53
Expressing a constant function in terms of a Fourier sine series (for initiaal boundary problem in wave equation)
So my question is motivated by the following initial boundary problem for the wave equation: \begin{align} u_{tt} &= u_{xx} \\ u(t, 0) &= u(t, \pi) = 0 \\ u(0, x) &= 1 \\ u_t(0, x) &= ... partial-differential-equations fourier-analysis fourier-series- 15
Can I design a function according to another function that they are connected with a Fourier transform relationship?
For real function $f(x)$ we have $$\mathscr{F}[\exp(-a^2x^2)\exp(if(x))]=\exp[iF(w)\phi]\lim_{R\to \infty}\int\limits_{-R}^{R}\exp(-a^2x^2+if(x)-iF(w)\phi-ixw)dx$$ Can I design a function $f(x)$ make ... complex-analysis fourier-analysis fourier-series fourier-transform- 9
Fourier series of absolutely continous function
Consider function $f$ which $2\pi$-pereodic and absolutely contionous on $[-\pi, \pi]$. Prove, that Fourier coefficients in series expansion $f(x) = \frac{a_0}{2}+\sum\limits_{n=1}^\infty(a_n\cos nx+... calculus fourier-series- 125
How to calculate the Fourier transform of exp(icf(x)) where f(x) is a Gaussian function c is a constant? [closed]
I would like to work out the Fourier transform of f(x) = exp(iexp(-a^2*x^2)*c) where c is a constant, however I find that it is beyond my ability. I would appreciate that if anyone can solve this ... complex-analysis fourier-analysis fourier-series fourier-transform- 9
Fourier transform of a periodic function $e^{i\theta(x)}$
Let $\theta: \mathbb R \to \mathbb R$ be a function such that $x\mapsto e^{i\theta(x)}$ is continuouesly differentiable and $2\pi$-periodic. In particular, we have $\theta(2\pi) - \theta(0) \in 2\pi \... fourier-analysis fourier-series- 1,727
Fourier transform of rectangular pulse
Calculate the Fourier transform of rectangular pulse given below. (Height, A; width, 2a) . I tried to calculate that but I am not sure whether it s correct or not. Question Answer fourier-series mathematical-physics- 1
Solving an Inhomogeneous heat equation subjected to some conditions by using the solution of its homogeneous version subjected to the same conditions.
Suppose the PDE is $\frac{\partial u}{\partial t}=k\frac{\partial ^2 u}{\partial x^2}$ subjected to the conditions u(0,t)=u($\pi$,t)=0 for $t\geq0$ and u(x,0)=$4\sin(3x)$. The solution in this can be ... partial-differential-equations fourier-series heat-equation- 1
Finding Fourier series of $f(x)+c$ given that of $f(x)$
So, I have the function $f(x)$ over the interval $[-\pi,\pi]$ defined as under. $$f(x)=\begin{cases}1+2x/\pi , -\pi\le x\le 0 \\ 1-2x/\pi , 0< x\le \pi\end{cases}$$ The thing is computing the ... fourier-series even-and-odd-functions- 6,308
Fourier series in the equation of the vibrating string
Consider the following problem for a vibrating string: $$ u_{tt}=c^2u_{xx}, \quad u(0,t)=u(L,t)=0, \quad u(x,0)=f(x), \quad u_t(x,0)=g(x) \tag{1} $$ Applying the method of separation of variables and ... partial-differential-equations fourier-analysis fourier-series wave-equation sturm-liouville- 73
How to decompose the $f(x)=\sin (ax)$ function into a Fourier series (when $a$ is an integer)
I have reached this stage: $\sin(ax)=\frac{2*\sin(a\pi)}{\pi} \left(\sum_{n=1}^{\infty}(-1)^n*\frac{n*\sin(nx)}{a^2-n^2}\right)$ But after all, the whole function will be equal to $0$, because with ... trigonometry fourier-series- 1
Show identity using fourier-coefficents.
I have to show that $$\sum_{i=1}^{\infty}\frac{(-1)^n}{4n^2-1} = \frac{2-\pi}{4}$$ using the fourier-coefficents of $\cos(\frac{x}{2}), x \in ]-\pi,\pi[$. I know that $c_0 = \frac{2}{\pi}$ and that $$... fourier-series- 1
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