Richard Feynmann, in his lectures at Caltech (available in book form), referred to the underlying relation that produced the referenced equation as the "crown jewel of algebra", I think.
Thank you arildno,
I did that integral sometime in the past 3 years and was racking my memory to come up with it; I found another nonsense solution that agreed with the known answer but as in the one I posted (after 2 days and maybe 6 hours surfing the web searching for answers), the limits of...
Area moment of inertia--circular cross section
From the bending beam calculation, the moment of inertia of the cross section with regard to a coplanor axis of rotation is used. If we have a circular "beam", the area moment of inertia of a circular disk of radius a about a diameter is I_d =...
OK, here is the same problem except make the period 2 and let the function to be modeled by Fourier series be:
f(t) =\left\{\begin{array}{cr}0&\mbox{if\;\;}-1<t<0\\sin(\pi t)&\mbox{if\quad} 0<t<1\end{array}\right
Calculations very similar to those pictured earlier above and adjusted...
Follows an attempt to display a plot of [itex]Sin(\pi t)[/tex] of period 1 using the initial constant term and 6 iterations of the Cos term. Click on the bitmap file.
y_p(t) = \mathbf\Phi(t) \ \int\mathbf\Phi^-^1 \ \vec{b} \ \dt was derived and so now let's apply it to the present example by first displaying the required inverse of the fundamental matrix without going into its calculation
\mathbf\Phi^-^1(t)...
Thanks to TALewis for the link to Tex; have perused and researched since last post and have decided to complete this explanation first using raw text and then gradually edit with Tex. UNDER CONSTRUCTION.
So with the previous replies as background, here continues the attempt to answer the...
Continued: Here is the next sequential short installment.
Referencing Reason's query setup:
"We find Y_h = Y(t)*\vec{c} " and then he gives a matrix of solutions, Y(t), arranged in column form, also known as a fundamental matrix for the associated homgeneous system.
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Would like to take a shot at this; will do so in parts. First, a simple answer is that since the column vectors of Y(t) each represent a solution to the ODE and since the product Y(t)\cdot Y^-^1(0) produces column vectors that are linear combinations of the column vectors of Y(t) then they...
Yes, except for typo--second equation should read sin(a)cos(b) = 1/2 (sin[a+b] + sin[a-b])and with the substitutions suggested these integrals tidy up rather nicely \mbox{ (hint---} B_n = 0\mbox{ for all integers,\:} n\geq 1).
I get a good form for the actual function sin(\pi t) being...
Newbie wants to bump this query:
I was having trouble with this problem as well until I discovered a simple algebra error; here is how I set it up
a_0 = \frac{1}{\frac{1}{2}}\int_0^1 Sin(\pi t) \;dt = \frac{4}{\pi}
A_N = \frac{1}{\frac{1}{2}}\int_0^1 Sin(\pi t) Cos(2 n \pi t)\; dt...