Calculating Fourier Coefficients and Series for f(x)

AI Thread Summary
The discussion revolves around calculating Fourier coefficients and series for a piecewise function defined over specific intervals. The user is confused about the function's oddness and the resulting Fourier series, which includes both sine and cosine terms despite the expectation of only sine terms for odd functions. They express difficulty with the integral calculations and seek general strategies for tackling such problems. The conversation highlights the importance of correctly identifying the function's properties and suggests that the user may have overlooked something in their calculations. Overall, the user is looking for guidance on simplifying the process of finding Fourier coefficients.
broegger
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Hi!

I have to calculate the Fourier coefficients c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx and the Fourier series for the following function:

<br /> f(x)=<br /> \begin{cases}<br /> \frac{2}{\pi}x + 2 &amp; \text{for } x\in \left[-\pi,-\pi/2\right]\\<br /> -\frac{2}{\pi}x &amp; \text{for } x\in \left[-\pi/2,\pi/2\right]\\<br /> \frac{2}{\pi}x - 2 &amp; \text{for } x\in \left[\pi/2,\pi\right]<br /> \end{cases} <br />

Since this function is odd the Fourier series should only contain \sin{x} (right?), but I keep getting a series containing both sine and cosine. Furthermore I'm having big trouble with the integrals; are there any "tricks" when doing such integrals?
 
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is the function really odd?
what is f(pi/2)?what is f(-pi/2)?
are they equal?

-- AI
 
no.. f(pi/2) = -f(pi/2) => f is odd?
 
Note to self : "should not study some dumb subject like software engineering, post something at physicsforums, listen to music and chat ... all at the same time"

whoops! apologies broegger!

anyways, back to ur question ...
could u post ur working ?
prolly u overlooked something ...
since u seem to have the problem well understood, u should have got the answer by now.

-- AI
 
nope.. I can't get the right answer.. I'd rather not post my working, since it's is very messy :/ I'm not asking someone to do the calculations; I would just like a general (the easiest) way to deal with such problems...
 
grunt it out...no easy way to get your coefficients.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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