Cosine integral

Other integrals of importance arc the sine and cosine integrals Ci(a ), Si (a ), which are defined by the equations... 

For a perpendicular transition, evaluation of the relevant direction-cosine integrals gives... 

Let us note for future purposes that since Cvv(o)) is an even function of to, Eq. (191) can be rewritten as a Fourier cosine integral,... 

Integrals of e qmnential and trigonometric functions appear so frequently, they have become widely tabulated (Abramowitz and Stegun 1965). These functions also arise in the inversion process for Laplace transforms. The exponential, sine, and cosine integrals are defined according to the relations ... 

This is called the Fourier sine integral. By similar arguments, we can derive the Fourier cosine integral as... 

Equations 5.59 and Equation 5.60 are the Fourier sine and cosine integral representations of/(x), respectively. 

Find the Fourier cosine integral representation of the function... 

Ci( ) 1 cosr or sometimes defined as V negative of this integral Cosine integral function... 

The inverse of the transformation Ca / is given by the Fourier cosine integral formula... 

Another view of this theme was our analysis of spectral densities. A comparison of LN spectral densities, as computed for BPTI and lysozyme from cosine Fourier transforms of the velocity autocorrelation functions, revealed excellent agreement between LN and the explicit Langevin trajectories (see Fig, 5 in [88]). Here we only compare the spectral densities for different 7 Fig. 8 shows that the Langevin patterns become closer to the Verlet densities (7 = 0) as 7 in the Langevin integrator (be it BBK or LN) is decreased. 

In case the curve y = fix) is symmetrical with respect to the origin, the a s are all zero, and the series is a sine series. In case the curve is symmetrical with respect to the y axis, the fc s are all zero, and a cosine series results. (In this case, the series will be valid not only for values of x between — c and c, but also for x = — c and x = c.) A Fourier series can always be integrated term by term but the result of differentiating term by term may not be a convergent series. 

The Fourier sine transform F, is obtainable by replacing the cosine by the sine in these integrals. 

Interchange integration with summation and use the cosine analog of the sine orthogonality result in Equation (3.102) to get... 

The frequency associated with F n) is v . This frequency should be equal to n times the basis frequency, which is equal to l/(27 ) (this is the period of a sine or cosine which exactly fits in the measurement time). Thus v = n/(2T ) = n/(2NAt). It should be noted that in literature one may find other conventions for the normalization factor used in front of the integral and summation signs. 

Performing the indicated integration then leads to the probability of incident molecules striking the wall coming from angle Q, which is P(Q) dQ, in the form of a simple cosine function,... 

Data analysis in phase fluorometry requires knowledge of the sine and cosine of the Fourier transforms of the b-pulse response. This of course is not a problem for the most common case of multi-exponential decays (see above), but in some cases the Fourier transforms may not have analytical expressions, and numerical calculations of the relevant integrals are then necessary. 

The integrations shown in Eqs. (14.12) and (14.13) are performed on a digital computer, and the problem of numerical integration again rears its ugly head. The problem is made particularly dilTicult by the oscillatory behavior of the sine and cosine terms at high values of frequency. 

The results of the integrations depend on the spectral density, which is defined as the cosine Fourier transform of the dynamical friction Eq. (8) ... 

Working independently, A.Abakonovicz in 1878 and C.V. Boys in 1882 devised the integraph, an instrument that drew the integral of an arbitrary function when the latter was plotted on a suitable scale on paper. A device for finding trigonometric functions (sines and cosines), known as harmonic analyzer was devised in 1876 by Lord Kelvin. 

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