applications of fourier series?
117874
Fourier series are used throughout mathematics, physics and engineering. \n\nAll of the analysis of power systems and alternating current involve fourier series.\n\nRadio transmissions and frequency bands use fourier series\n\nSolving the wave equation, the heat equation, Maxwell's equation etc all can use fourier series to studies waves and heat and electromagnetism.\n\nFourier series is a standard tool in mathematics, in places like number theory (involved in producing secure communications).\n\nSonar systems use fourier series. Navigating submarines and finding other vessels is done use fourier series.\n\nRadar systems use fourier series. They are used for weather forecasting and speed guns used by the police.\n\nRepresenting a solution as a sum of sines and cosines is valuable whenever one is dealing with simple harmonic oscillators, which occur throughout physics. \n\nFourier series are the gateway to more sophisticated tools like Fourier transforms and other special function transforms. \n\nObviously, one could make a very long list. But his should give you some idea.
681952
Here you go:\nhttp://en.wikipedia.org/wiki/Fibonacci_series
13310
It's the limit of a type of convergent infinite sum (that's why it's symbol is an elongated 's').\n\nIt's applications range from finding the solutions to differential equations, being used to calculate an area, volume (in the sense we know it, and in a generalised space of n-dimensions).  In Fourier Analysis it's used to approximate functions using cos and sin, etc.  But, by far, the best use of it has to be answering exam questions.\n\nNot helpful, but fairly interesting.  So I thought.  I'll go now.
601386
Through the fourier map generator.
222928
There is no function that exists that has as its derivative e^(x^2).  As someone else pointed out, you can use the Maclaurin series representation and integrate term by term.  However, this series converges very slowly and would require a lot of terms for two place decimal accuracy, especially if x is large.
885756
Let F(x) = Sum n=0toInfty [anx^n] be the generating function of the sequence (an). Written in terms of F, the recurrence relation and the initial condition a0 =1 mean that, for every x (within the radius of convergence) \n\nF(x) = 1 + 8xF(x) + Sum n=0toInfty [10^n x^(n+1)].\n\n(To see this, just use the RR to substitute for an in terms of a(n-1) in the series that defines F; you get the same series but times 8x, plus the other bits.)\nThis gives an equation for F(x) which solves to \n\nF(x) = (1-9x)/[(1-8x)(1-10x)]\n\nExpand this into partial fractions \n\nF(x) = (1/2)[1/(1-8x) + 1/(1-10x)]\n\nand rewrite these as sums of geometric series to get \n\nF(x) = Sum n=0toInfty [(1/2)(10^n + 8^n) x^n]\n\nfor every x, which means that \n\nan = (1/2)(10^n + 8^n)\n\nfor every n.
776321
the question is unclear.\nDo you want an explicit formula for Y[n or do you just want to write out the convolution]
501273
Didn't got the question do you want the solution for all the problems of goodman's Introduction to Fourier optics book? Well I do not think I have them any more if you want a specific problem I can help you. I can even try to get a chapter solution for you. Just tell me each one, I can try to get it for you, I do not garantee but I think I may have a way to got them just send me an email (I have a yahoo email). If you have the solutions and you don't understand tell me each problem I can do it in more detail probably, not garantee at 100% but at a good 90% .
451479
hav u ever heard of the Principle of Mthematical Induction ?\nthat could solve ur problem in a jiffy!!\n\nHowever if u havn't, dont worry!!\n\nNote that, the sum in ur series(i.e. 1+1/2+1/3+...+1/n),where n>=2, is always less than 2, but greater than 1. so it can't b an integer.\nwanna know why?\ntry cramming up some theorems about Sequence & series of real numbers!! its a regulation part of any under-graduate level Real analysis course.
680780
You can use infinite series...\n0 = 0+0+0+...\nsubstitute 0 = (1-1)\nso \n0 = (1 - 1) + (1 - 1) + (1 - 1) + ...\n\nuse associatve law of addition so that\n0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... \n\nthen cancel out leaving you with \n0=1\n\nerror: using associative law on on infinite sum that does not converge
120506
i'm giving the final answer not hte full steps ok\n(a sin(wt)-w cos(wt))/w*(w^2+a^2)  + e^(-at)/(a^2+w^2)\nby using convolution theorem
212677
Little reply: No.\n\nP.S. 12.08 AM GMT\nThis gives the Fresnel Integral S(x), more exactly\n\nS(x)=Integral(.5Pi*x^2)\n\n(the integral is from 0 to x), and it is not possible to reduce it in a finite way.\n\nHowever S(x) has an asymptotic expansion for x>>1\n\n.5-(cos(.5Pix^2))/(x*Pi)\n\nwhich gives limit S(infinity)=0.5
787465
Write out the exponentials and use the series for e^x. Other than that, do your own homework.
1031386
Do you mean Fourier transform? \n\nIt's based on the idea that all signals are composed of simple sinusoidal signals at different frequencies.  The fourier transform converts a signal from time-domain, to frequency-domain. So when you go to graph the signal, instead of seeing the strength of the signal vs. time (time domain), you see it in terms of frequency (frequency domain). The Fourier transform provides an easy way to convert a time-domain signal into frequency domain, and vice-versa.\n\nThis is really nice when you want to see what parts of a signal will pass through a filter, or seeing what happens to your signal after a frequency shift, or checking sampling rates, etc.,etc.
767320
For the first one: yes, you can multiply the expression term by term. If f(x) is a function given by [n=0, ∞]∑(a_n x^n), and g(x) is any function, then f(x) * g(x) = [n=0, ∞]∑(g(x) a_n x^n). g(x) is just a constant in terms of n, so distributing it across the terms of the series will not change its convergence properties.\n\nFor the second question: you seem to be describing a Laurent series. These series are described in more detail on the Wikipedia:\n\nhttp://en.wikipedia.org/wiki/Laurent_series
117843
Sure.  The derivative of x^3 is 3x^2.  That should get you started.
603036
The idea is that the y_n form an orthonormal basis so every function can be expanded as a series of the form \nsum a_n y_n.\n\nHow is this known? Typically, the inverse operator of L can be shown to be a compact operator and the general theory of compact operators can be invoked. Some care may be needed with the convergence properties of the series you obtain. Only mean-square convergence (L_2) is guaranteed. \n\nIn any case, the fact that the y_n are eigenfunctions for L and that L is linear makes the determination of the A_n from the B_n much easier. The convergence details can either be ignored or dealt with separately.