what is the porpuse of Fourier analysis?
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An example application: in signal processing you often want to know the frequency components of a signal (e.g., for a cellphone, radio, etc).  Fourier analysis helps to identify the frequencies that are present in a signal, so that you  can manipulate it to your convenience.
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yes, it's a sine wave, and velocity is integral of that, ie a cosine\n(this is considering only fist order effects)
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Back in the stone ages, we used "Real and Complex Analysis" by Walter Rudin 1966 McGraw Hill.\n\nI think Rundin may have written another book with the title "Functional Analysis", but I'm not sure.
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The notation 's', 'p', 'd', and 'f' originate from a now-discredited system of categorizing spectral lines.\n\ns= sharp\np= principal\nd= diffuse\nf= fundamental
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Mathematical Analysis!
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Have fun:\nhttp://www.math.nyu.edu/~griffith/teaching/06numanalysis/labs/bisection/
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There is an application to convert an AC current to DC.\nAssume that f is a sigmoid function, then integration of\nf^2 over one period gives a constant number which is considered as flattened sigmoid and is analogous with converting AC to DC.
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Through the fourier map generator.
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From a software developer point of view. This will be really hard to do and come up with a algorithm for seperating human voices from the regular music. They are sampled in digital files and to find which wave is the vioice is too hard for a computer to tell apart.
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One of the beauties when using Fourier transforms, is the ability to work with a spectrum of modes and act on each of the components individually with a filter. By sampling the function over a period  with a finite number of values  , where  is the size of the sampling interval, the spectrum gets truncated at the shortest wavelength  called Nyquist critical wavelength: this corresponds to exactly 2 mesh points per wavelength, which does however not mean that shorter wavelengths  do not contribute to the Fourier coefficients (4.1#eq.1). Figure 4.3#fig.1 illustrates how they get aliased back into the lower components of the spectrum
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It is divergant. Because the limit of the sentance to infinity is 1/2 and not zero. ( It should be zero so that the summation converges)
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It is Parseval's theorem you seek. "The total energy contained in a waveform x(t) summed across all of time t is equal to the total energy of the waveform's Fourier Transform X(f) summed across all of its frequency components f."\n\nFor a periodic signal, take the energy in each frequency for one period.  Sum those and you have the total energy.  The maximum amplitude during the period would be if all of that energy occured at once (i.e. an  impulse).  Convert that to dBm or DBW and you have your answer in dB.\n\nLook at the last equation in the link in "sources".
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Basin Analysis is the systematic study of the processes that lead to the development of basins. Its principles are derived from the Theory of Plate Tectonics and its applications range from oil and mineral exploration to understanding the behavior of earthquakes. Many past geologic events are understood from Basin Analysis because that is where sediments from erosion are preserved.
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Quantitative methodologies... statistical analysis?   How many pages allowed on here... there are books and books about it... try reading literature on the above... infinite (well almost!) sources available on line, at the library and in book shops
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A Fourier Transform is a mathmatical way to convert a time domain signal to a frequency domain signal and vice versa. What does that mean?\n\nSuppose you want to detect a certain tone. An FT (Fourier Transform) could take the input from a microphone. Imagine that the input is a sine wave at one frequency (say 1kHz). Then the output will be a "spike" at 1kHz since the x axis of the FT is frequency. Feed the spike to the inverse transform and you'd get the sine wave back out.\n\nSo frequency detection is one application. If you put a square wave in, you'd find that the square wave has a spike at the fundamental frequency and spikes at the odd harmonics. If you don't belive that, try sketching some sine waves at f, 3*f, 5*f, 7*f, and add them together (easy with Excel or Matlab) and watch how quick it "squares" up.\n\nAnother application is filtering. Manipulating data in the frequency domain allows you to build very sharp filters... then passing it through the inverse transform produces the signal after filtering.\n\nThere are other applications such as correlation, tomography, etc.\n\nIn the real world, you probably use a Discrete FT (DFT) and probably an FFT (Fast FT) which is a specific computer algorithm for a DFT. DFT means that the input is sampled as it would be for a computer. Computers don't read sine waves, they read little points on sine waves every X microseconds. The frequency they read the signal must be at least twice the frequency of interest, or you won't get proper results.
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FFT's are totally cool. Sometimes this mathematical method is run on a computer, and other times on a specialized signal-processing chip.\n\nIt is a high speed way to convert changing data, for example an audio signal (louder and softer), or a photo (pixels change as you look from point to point).\n\nThe FFT converts the data from the realm of  time (say for audio) or space (like for a still photo) into frequency. And it can also run the other way, and convert frequency data back.\n\nAll FFT techniques involve complex numbers. These numbers are represented as two 16 bit numbers. A 16 bit number has 65,536 possible combinations. Another way to describe this is that a 16 bit number has a "resolution" of 65,536 parts.\n\nDepending on what kind of data you are measuring, there are standard and also one-of-a-kind ways of converting analog information into a digital number.\n\nFor example, in the telephone industry, an international group called the CCITT sets the representation of audio signals into digital. A common standard is the CCITT recommendation G.711, "Pulse Code Modulation (PCM) of Voice Frequencies,".  This standard, voted on in Geneva, 1972 is still in widespread use in phones and audio.\n\nIn some cases, "proprietary" (private, one-of-a-kind) schemes are used to represent signal data in 16 bit binary numbers.\n\nUsually the FFT device could care less about WHAT your coding scheme is, it just chews up the incoming  16 bit pairs (16 bits for real component and a second 16 bits for the imaginary component) and spits out the answer. It is up to you to interpret the results based on your coding scheme.\n\nBest of luck!
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If you break it into real and imaginary parts, you get integrals of\ncos(x^2) and sin(x^2) from -infty to infty. These integals are finite. The anti-derivatives are called Fresnel integrals and show up in studies of diffraction. The infinite integrals are sqrt(pi/2) each.