# fourier

1. ### Fourier-Series, where is my mistake?

Hello, I've got a piecewise defined, 4-periodic function . I have calculated the coefficients as seen in the pictures . The problem is, when I plot the Fourier series , it seems not to be correct, as you can see in this picture (blue: Fourier series, black: actual function): I am looking...
2. ### Half range cosine FS expansion

Hi everyone, my university maths lecture notes suggest that in finding a fourier series expansion for a function it is easiest to find the even extension. As such, I mirrored the function across the y-axis (as shown in working link) and have been trying to solve it thusly, however piecewise...
3. ### Fourier transform of rect function and its inverse Fourier transform

Hi I have calculated the FT of the rect function from -a to a and get this \frac{\sin(2\pi k a )}{\pi k} Now to perform the inverse \int^\infty _{-\infty}\frac{\sin(2\pi k a )}{\pi k} e^{2\pi i k x} dk I am not sure how to proceed from here, should I convert the sinc to and exponential?
4. ### Find Fourier coefficients of g(x) = f(2x)

Hi, I have the following problem: Assume that the function is -periodic and has the Fourier series: Find the Fourier coefficients of the function g(x) = f(2x) As far as I understood, the Fourier coefficients of this function can not be a number, because we don't have the function...
5. ### Show that function is periodic (Fourier series)

Hi, I have the following problem: Assume that the function f is 2\pi-periodic and has the Fourier series: f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx) Let k \in \mathbb{N}. Show that the function g(x) = f(kx) is 2\pi-periodic, too. I am in general quite unsure how to start...
6. ### fourier series

the ans that i gt is different with the answer provided , is my ans wrong ? i got ( cos(2n -1) / 2n-1 )instead of ( cos(n+1) / n+1 )
7. ### fourier series

i am having problem for the coefficient of bn , the asnwer given for bn is b1 = 2 / π , b3 = 2 / 3π , whereas for me , i gt b1 = -2 / π , b3 = -2 / 3π , is my ans correct or the given answer is correct ?
8. ### fourier series

here's the question : i think the solution given is wrong .... i have done the correction beside , is my working correct ?
9. ### Improper Integral convergence

While finding the Fourier Transform of the unit step function u(t) , I came across the following integral: \int_{0}^{\infty}e^{-i\omega t}dt = \left[-\frac{e^{-i\omega t}}{i \omega}\right]_{0}^{\infty} The textbook says that the integral will not converge.Can anyone explain the reason why so?
10. ### fourier series

is the author wrong ? i was told that the f(x) = 0.5(a_0) +Σ(b_n)sin (nπx / L ) ........ but , in the example(photo2) , the author ignore the L , which the author gave f(x) = 0.5(a_0) +Σ(b_n)sin(nπx ) +...... P/ s : i have tried to make some correction beside the working , is it correct ?
11. ### fourier series

for the bn , i think there's something wrong with it, integrate sin , we will get -cos , right ?
12. ### formula for fourier series

why the formula of a_n and b_n for periodic function is 2/T .... ? why not 1/ T .... ? as we can see the 'coefficient' for the a_n and b_n for periodic function 2L are 1/ L
13. ### Fourier cosine series

why the author need to differentiate the a1 using L'hopital rule ? the working already showed that the a1 = 0 , am I right ?
14. ### fourier series

what does the f(X) = x + 2 π mean here ? we already have f(x) at -π and π above ? why we have additional f(x) = x +2π below ?
15. ### Fourier Series

Hi, I need help knowing how G(t)= 4E/T*t[u(t)-u(t-T/4)]+E[U(t-T/4)-U(t-T/2)]+(2E-2E/T*t)[u(t-T/2)-U(t-T)] can be simplified into 2E/T{2tU(t)-2(t-T/4)-(t-T-2)+(t-T)U(t-T} I would be grateful if someone could show me the steps. I know the above is a bit difficult to interoperate so I've added...
16. ### Expand function to Fourier series

Hello there, I'm trying to calculate the Fourier transform of the function $x exp(-x^2/2)$. I'm using this definition of the Fourier transform: $$\hat{f}(k) = \frac{1}{2\pi} \int _\Re f(x) \exp ( + ikx) \mathrm{d}x$$ so the question is, how to calculate...
I have recently encountered this problem in my studies of Sobolev spaces and generalized functions (distributions), on which I can say I might have some intuition but cannot stumble across a final solution, but first some notations: $\mathcal{S}$ is the Schwartz class of functions and \$...